Units and Dimensions

Physics Workshop / Activities Manual

1017-311 University Physics I (All Sections)

Spring Quarter 2007-3

1 Day by Day Schedule for 1017-311

 Week 1, Day A: Introduction, WebAssign, Units and Conversions, Estimations, Measurements  Week 1, Day B: Measurements, Uncertainties, Propagation of Uncertainties, Graphing  Week 1, Day C: Introduction to Vectors

 Week 2, Day A: Position, Velocity, Acceleration, Graphs of 1-D Motion, Water Accelerometer  Week 2, Day B: LoggerPro Software, Motion Sensors, Vertical Motion of a Ball  Week 2, Day C: Solving 1-D Kinematics Problems

 Week 3, Day A: Motion in 2-D  Week 3, Day B: Motion on Curves, Uniform Circular Motion  Week 3, Day C: Projectile Motion

 Week 4, Day A: Introduction to Forces, Free-Body Diagrams, Newton's Second Law  Week 4, Day B: Newton's Third Law, Newton's Second Law Problems (1-D)  Week 4, Day C: Modified Atwood Machine

 Week 5, Day A: Uniform Circular Motion, Newton's Second Law Problems (2-D)  Week 5, Day B: Contact Friction, Newton's Second Law Problems (2-D)  Week 5, Day C: Variable Force, Drag Force

 Week 6, Day A: Newton's Second Law Problems  Week 6, Day B: Kepler's Laws and Gravitation  Week 6, Day C: Problems Involving Gravitation and Kepler's Laws

 Week 7, Day A: Work and Kinetic Energy  Week 7, Day B: Work Done by Variable Forces, Work in Lifting a Backpack, Work Problems  Week 7, Day C: Work Problems

 Week 8, Day A: Conservative vs Non-conservative Forces, Energy and the Simple Pendulum  Week 8, Day B: Energy Problems  Week 8, Day C: Conservation of Mechanical Energy Lab

 Week 9, Day A: Impulse and Momentum  Week 9, Day B: Collisions, Impulse and Momentum Problems  Week 9, Day C: Center of Mass Problems, Center of Mass IP

 Week 10, Day A: Energy and Momentum, Ballistic Pendulum  Week 10, Day B: Ballistic Pendulum, Energy and Momentum Problems  Week 10, Day C: Energy and Momentum Problems, Review

2 Week #1

3 Units and Dimensions

In physics we measure quantities such the length of a room or the mass of an electron. The measurement results in a “physical quantity” consisting of a pure number and a unit. Physicists also discuss dimensions of physical quantities. The System International (SI) is based on 4 fundamental dimensions, length, L, time, T, mass, M, and charge Q. All quantities in this course are combinations of the first three "dimensions." Each of these dimensions must be measured in some type of unit. The base units that we will use are the SI units, sometimes called mks. Other systems of units are the cgs and US Engineering. They are defined in the text.

Quantity Dimension SI Unit cgs US Engineering US Engineering (1) (2) Length L meter, m centimeter, cm foot. ft meter Mass M kilogram, kg gram, g slug kg-mass, kgm Time T second, s second, s second, s second, s

The base units and dimensions are used to find units of derived quantities. Some of the unit combinations have their own names.

Quantity SI Unit cgs US Engineering US Engineering (2) (1) Area L2 m2 cm2 ft2 m2 Speed L/T m/s cm/s ft/s m/s Acceleration L/T2 m/s2 cm/s2 ft/s2 m/s2 Force M L/T2 Newton, N dyne pound, lb kilogram-force, kgf 1 N = 1dyne = 1 lb = 1 kgf = 1 kg m/s2 1 g cm/s2 1 slug ft/s2 9.80665 kgm m/s2

Named Prefixes Your text lists the named prefixes for powers of 10. Most of the prefixes represent powers of 10 spaced by 103—eg. milli = 10-3, micro = 10-6, etc. Calculators often call this “Engineering notation.” The following will be the common prefixes in this course. micro µ 10-6 milli m 10-3 centi c 10-2 kilo k 103 mega M 106

4 Determining Units in a Formula: Eg. The speed, v, of a water wave in shallow water of depth h is v = A1/2 h1/2 where A is a constant. Find the dimensions and units of A. Rewrite the equation as A = v2/ h, then the dimension of A, [A] = [v2] [h] = [L/T]2 (1/[L]) = [L/T2], and so the SI units are m/s2.

In the following examples, determine the dimensions of the quantity and its SI units. 1. For a car moving through the air, the acceleration, a, is given by a = B v + C v2 where v is speed and B and C are constants. Determine the units of B and C. 2. The rate of flow of a liquid out of a tube of radius r and length x is given by ∆V/∆t = P r4 /(D x) where V is the volume and P is pressure in SI units of kg/(s2 m). Find the units of the viscosity, D. 3. When a mass is attached to a spring, the acceleration is a = k x/m where a is acceleration, x is a length, m is mass, and k is a spring constant. Find the units of k.

Converting Units

Use the units as a guide to doing conversions. For example, Convert 25.0 pounds/square inch into newtons/square meter. 1. Find some conversion factors, either in the text (usually in an Appendix or flyleaf), or at Rowlett's compendium of units at http://www.unc.edu/~rowlett/units/index.html. 1. 1 pound = 4.45 newtons 2. 1 inch = 2.54 cm 3. 100 cm = 1 m 2. Set up the units, remembering to square where needed

2 2 pounds newton жinch ц жcm ц newton 2 ҙ ҙ з ч ҙ з ч = 2 inch pound и cm ш и m ш m 3. Put in the numbers and calculate

2 2 pounds 4.45 newton ж 1inch ц ж100 cmц newton 25.0 =172438 2 ҙ ҙ з ч ҙ з ч 2 inch 1 pound и2 .54 cmш и 1 m ш m 4. Round the answer to make the result have a reasonable number of significant figures. Here I had 3 sig. fig. to start (25.0), and I assume that the conversion factors are exact, so I use 3 sig fig in the answer, 2 25.0 pounds/square inch = 172000 newtons/square meter = 172 kN/m , where I use the k = kilo to make the number easier to read. 5 Your Conversion: Rainfall is sometimes measured in "inch-acres", that is the volume of water that corresponds to a one inch depth on an acre of land. The density of water is 1 gram/cubic centimeter. Convert this density to (metric tons)/(inch-acre). Some conversion factors are: 1 metric ton (tonne) = 1000 kg, 1 inch = 2.54 cm, and 1 acre = 4840 square yards. You may work in groups at your table, but each person should turn in a sheet with the solution neatly presented.

6 Estimation: Walking the Appalachian Trail

Do the following exercise by yourself, and for now do not share your answers or reasoning with anyone at your table. Clearly label any estimates that you make, e.g. “Estimate length of stride = xxx feet.” Indicate clearly what you are doing. You have 10 minutes.

Spring has returned and you decide that with your powerball winnings you will take the summer off and walk the Appalachian Trail. The trail stretches from Springer Mountain, Georgia to Katahdin , Maine. How many strides will it take to make the walk? A stride is the distance between two successive left steps.

Write your estimate in large letters on the back of this page.

If you want to follow up with more information about the trail, check out http://www.fred.net/kathy/at.html.

More Estimation

Size and Shape of a Standard Kilogram.

The standard kilogram is a platinum-iridium alloy kept in a laboratory in Paris, France. A picture is available at http://physics.nist.gov/cuu/Units/kilogram2.html. 1. What is your estimate for the dimensions of the standard? 2. Why is it in the shape of a right circular cylinder? 3. Determine accurately the diameter and height of the cylinder.

Thickness of a piece of paper

Determine the thickness of a piece of paper from your text. Be able to describe how you did it. Ask if you need any tools. Give as precise an answer as possible.

Size of a period in the text.

Determine the diameter of a period in the text. Be able to describe how you did it. Ask if you need any tools. Give as precise an answer as possible.

Weighing the air.

Estimate the mass of air in this room.

7 Propagation of errors

These two exercises will provide some practice in unit conversion and combining errors. You may work with others in your group, but you must do the calculations yourself and hand in your own work. Write down all the data you use and show the key steps in your calculation.

1. Assume that the length of the Appalachian trail is 2100±100 miles. Look up the value of the average stride length you measured during Workshop 1, and use ½ the range in your measured values as the uncertainty (e.g., if you measured stride lengths ranging from 1.2 to 1.5 m, with an average 1.37 m, you can state this result as 1.37±0.15 m). Now calculate the number of strides it takes to walk the trail from start to finish and estimate the uncertainty on this number. Be careful with units and significant figures.

2. The mass of a particular sample of rock retrieved from Mars is m = 772.2 ± 0.2 g. The sample is in the shape of a cylinder of height h = 10 ± 0.1 cm and base radius r = 2.5 ± 0.1 cm. Calculate the density of the rock sample in units of kg m-3. You must state your result in the correct units with its associated uncertainty. The volume of a cylinder is V = π r2h.

Mass of a Yardstick

The goal of this exercise is for your group to predict the mass of a yardstick. If the actual mass lies within your predicted range, you win! (What? Well my respect, for one thing, bragging rights for another.)

It may sound easy, but here's the problem: you are not allowed to examine or touch the yardstick in question before making the prediction. Instead, you have access to the following:  five pieces of a broken yardstick (which do not add up to an entire yard)  a triple-beam balance scale  a ruler  some graph paper  the knowledge that an intact yardstick is one yard long.

To start, please fetch a triple-beam balance scale for your group. Choose a single piece of stick and practice using the scale to measure its mass. When you are comfortable using the balance, start measuring the other pieces and record the mass of each one. Some things to consider:  what other measurement will you need to make?  how will you estimate the error on each measurement?  how will you use your data to estimate the mass of the stick?

Hint1: if you can figure out the linear mass density of the stick (that is, the mass per millimeter or centimeter), then you can compute the mass of any given length. Hint2: if you know the uncertainty in the linear mass density, then you can compute the uncertainty in the mass of any given length.

What you must hand in…

Each group must submit the following materials (just one set per group, not one per student)  a cover page including a title, the date and the names of the group members  a neat table of all measurements  a page which shows one example of each type of calculation (but not all the calculations)  a good, clear graph  the predicted mass (with uncertainty) and actual mass (with uncertainty)  a sentence or two explaining the results – did the predicted mass agree with the actual mass to within the predicted uncertainty?  your (brief) assessment of the largest single source of error in the experiment, and how it could be reduced next time.

9 Vectors: We will for the moment deal with 1D and 2D cases.

A scalar is a quantity that has a value, but no direction. A scalar can be positive or negative. Scalar arithmetic is the usual stuff you learned through grade school: addition, subtraction, multiplication, division, and raising to a power. We can also take the absolute magnitude of a scalar. Normal algebraic symbols (x, y, t, m) are used for scalars. Scalar quantities in physics include mass, time, energy, charge, and temperature. A vector is a quantity in which magnitude and direction are both important. Vector quantities in physics include velocity, force, momentum, and electric field. We need (a) ways to draw vectors on a diagram, (b) ways to quantify vectors with numbers, and (c) methods to do arithmetic operations on vectors—these include “addition”, “subtraction”, and “multiplication”, we use the same words as for scalars but define the operation differently. Geometrical Representation of Vectors It is easy to describe vectors geometrically with arrows. The magnitude of the vector is indicated by its length, and the direction is the direction on the paper. The algebraic symbol used to indicate a vector is a letter with an arrow above it. In advanced texts a vector is often indicated by making the symbol boldfaced. In our text r is used as a symbol, other texts would use r. Location of a vector is immaterial—only the length and direction count. We will adopt the concepts of Quadrants from geometry, but what is important is in which quadrant a vector points, not where it is located. Several vectors are shown to the right, with a coordinate system. D A E What is the quadrant for each vector?  r r   A , B , a n d C are all quadrant I, D is quadrant III, and E is quadrant IV. B C Are any of the vectors equal—if so which?  r r A  B  C Adding vectors geometrically—Head-to-tail We define addition geometrically as placing vectors one after the other with the head of E one attached to the tail of the next. The sum A vector extends from the first tail to the last head. The order does not matter. Two D representations of the addition of the same D vectors are shown.

10 Geometrical Multiplication of a vector by a scalar. Positive scalar: The direction is unchanged, the length is multiplied by the scalar. Negative scalar: The direction is reversed, the length is multiplied by the magnitude of the scalar.

Geometrical Subtraction of vectors There are two general methods: Head-to-tail: Multiply one of the vectors by (-1) and add the results. This is shown in the left two diagrams. Tail-to-tail: Place the vectors tail to tail, then draw the difference starting from the tip of the one that is subtracted, and going to the tip of the other. The right hand diagrams show addition and subtraction.

S u b t r a c t i o n b y a d d i n g D H e a d - t o - T a i l T a i l - t o - t a i l V e c t o r A A d d i t i o n S u b t r a c t i o n a n d i t s n e g a t i v e t o t h e n e g a t i v e o f . A A A A D D - A D

Vector Components: Now we want to put numbers to vectors, again we will stick with 2D vectors.

Polar representation:  V r e p r e s e n t e d b y a m a g n i t u d e ( p o s i t i v e b y d e f i n i t i o n ) , V a n d a d i r e c t i o n , m e a s u r e d f r o m t h e p o s i t i v e x  a x i s ,   o r V ,  .  

+y An example would be (15 m, 30°). The usual convention is used where positive angles are measured counter-clockwise.

11 Cartesian Representation A vector can be broken up into the sum of two vectors, one parallel to the x-axis, one parallel to the y- axis. The scalar lengths of the two vectors above are given from trigonometry, in equation 3-5 of text. These are called the components of the vector, and can be positive, negative, or zero.   a x  a c o s  a n d a y  a s i n  (Polar to Cartesian conversion)

The Cartesian representation of a two dimensional vector is the two components, (ax, ay) It is more common in physics and engineering to write the Cartesian form with unit vectors,  ˆ ˆ a  a x i  a y j The unit vectors have magnitude 1.

Cartesian to Polar Conversion. If we know the components we get the polar form from

 2 2 a y a   a x  a y and t a n   . There is one remaining wrinkle: inverse tangent is a function a x that returns two distinct answers, the (angle) and the (angle + pi). Calculators return only one of these, it is up to you to determine the proper quadrant and pick the proper angle.  a  3 . 0 c m iˆ  6 . 0 c m jˆ  e.g. a  6 . 7 c m , t a n   2 . 0 ,   6 3  B u t t h e v e c t o r m u s t b e q u a d r a n t 3 ( b o t h n e g a t i v e ) , s o   2 4 3 .

Multiplication of two vectors Later in the course we will define the scalar product also called the dot product. This takes two vectors and produces a scalar. Even later we will define the vector product or cross product, that produces a vector from two other vectors.

Division of two vectors is not defined.

12 Vectors y Tell the quadrant for each of the vectors.  V1 .  V2 .  V3 .  V4 .  V 5 .

Are any of the vectors equal? If so, which ones?

Arrange the vectors in order of their magnitude in a statement like V1 = V2 > V3 > V4 =V5.  r r Using the vectors in the above diagram draw the vector sum S 1  V 1  V 4 , tell its quadrant and label it.  r r r Draw the vector sum S 2  V 1  V 2  3 V 3 and label it.  r Draw and label the vector, C  V 3 .  r r Draw and label the vector D  V 3  V 2 .

13 For each of the vectors in the first diagram, give the sign of the x and y components.

Sign of x-component Sign of y-component  V 1  V 2  V 3  V 4  V 5

  Suppose A  5 c m a t 3 0  and B  8 c m a t 1 5 0  . (a) Draw the vectors on the grid. y  r r (b) Sketch the sum S  A  B of the two vectors. (c) Find the components of each vector. (d) Find components of the vector sum. (e) Find the magnitude and direction of the sum.

14 Measuring 3-D Vectors

In this activity we’ll construct real position vectors to various points in the room and then determine the displacement vectors between each pair of points. Each table group will split as evenly as possible into two teams, “A” and “B” and each team will need to measure the components of two position vectors and determine a corresponding displacement vector. Set up Scattered around the room are 14 Post-it notes labeled 1A, 1B, 2A… etc. These represent the points for which we will find position vectors. Point 1A “belongs” to team A from table 1 etc. We will use a co- ordinate system whose origin is indicated by a note on the floor, somewhere near the center of the room. The x and y directions will also indicated. The z direction is perpendicular to the floor, with “up” positive. To determine these directions nearer your points, you can use the floor tiles as references for x and y, and for z, hold a meter rule lightly at one end by your fingertips, it should hang pretty nearly vertical. Procedure and measurements

Using a meter rule, your team will measure the x-, y- and z-components of the position vectors for your own point and that of the preceding team. For example, team 1A will determine components for the position vectors of point 7B and of point 1A; team 1B will do the same for points 1A and 1B; team 2A for points 1B and 2A etc (see diagrams). Measure as accurately as you can and estimate the uncertainties. Compare the position vectors you measure with the measurements of your neighboring teams to see if they agree within the uncertainties. Then use your position vector measurements to determine the components of the displacement vector that points from the preceding team’s point to your point. For example, team 1A will find the displacement vector from point 7B to point 1A. Calculate the measurement uncertainty in each component, using the error propagation rules.

z z 3A x, y, z components of position vector Displacement vector r r 3 A 2B r 2B r 2 B

Note: Calculations called for on the following results page should be shown on separate pieces of paper and not squeezed onto the results page.

15 Results: Record the components of your position vectors and the uncertainty in each of their components below.

Position vector for point ______x-component: ______± ______meters y-component: ______± ______meters z-component: ______± ______meters

Calculate the components of your displacement vector and the uncertainty in each component below.

Displacement between point and point x-component: ______± ______meters y-component: ______± ______meters z-component: ______± ______meters

Now calculate the magnitude of the displacement vector and its uncertainty, using the information recorded above.

Magnitude of displacement: ± meters

When you have computed the magnitude of your displacement vector from the components, use a piece of string to measure the actual distance between your set of points.

Distance between points: ± meters

Is your result the same as the length of string to within the uncertainty?

Suppose we now add all the displacement vectors from all of the tables. What should be the result of this vector sum, if we have done it right?

Week #2

17 Graphs of 1D Motion

1. Here is a motion diagram showing the location of a ball at various times. The direction of positive-x is to the left.

Is the velocity Negative zero positive

Is the acceleration Negative zero positive

Explain how you get your answer for acceleration.

2. Here is a graph of position versus time for a car. Sketch graphs of velocity and acceleration versus time.

18 Handwaving

Each group will be assigned one of the following graphs, position versus time or velocity versus time. You have two tasks to do:

1. Wave your hand in such a way as to match the graph, and

2. Write a detailed description of the motion. One person from each group will present the results.

When your group is done your assigned graph, try some of the others.

A B C 2 0 2 0 2 0

1 5 1 5 1 5

1 0 1 0 1 0

0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 t i m e ( s ) t i m e ( s ) t i m e ( s )

D E F 2 0 2 0 2 0

1 5 1 5 1 5

1 0 1 0 1 0

0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 t i m e ( s ) t i m e ( s ) t i m e ( s )

G H I 2 0 4 4

1 5 2 2

1 0 0 0

5 - 2 - 2

0 - 4 - 4 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 t i m e ( s ) t i m e ( s ) t i m e ( s )

19 J K L 4 4 4

- 2 - 2 - 2

- 4 - 4 - 4 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 t i m e ( s ) t i m e ( s ) t i m e ( s )

M N O 4 4 4

- 2 - 2 - 2

- 4 - 4 - 4 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 t i m e ( s ) t i m e ( s ) t i m e ( s )

20 Water Accelerometer--Observing Acceleration

By looking at a moving object you can easily see its position and have a sense of its velocity. Acceleration is harder. You will use a very simple accelerometer to see the effect of acceleration. A thin rectangular box is filled part way with water that may have added food coloring.

Part I - Calibrating the Accelerometer

1. Place the accelerometer on the table. Sketch the water in the accelerometer when it is at rest.. What is the acceleration? Use a rubber band (or washable marker) to mark the level of the water for this case.

2. Describe the water when the object moves at constant velocity. Does the direction of the velocity have any effect? Sketch the accelerometer and water level. What is the acceleration?

3. Describe the water when the object is accelerating to the right. (Push with your hand) Sketch the accelerometer. What is the direction of acceleration?

4. Describe the water when the object is accelerating to the left. Sketch the accelerometer. What is the direction of acceleration?

You can think of the water as an arrowhead pointing in the direction of acceleration.

Accelerate right Accelerate left

21 Part II Using the Accelerometer

1. A cart is rolled along a frictionless inclined track, moving uphill, momentarily stopping, and then moving downhill.

(i) When the cart is moving up the ramp, predict the direction of the acceleration (circle one)

Up the ramp Down the ramp No acceleration Other

(ii) When the cart is at its maximum height, predict the direction of the acceleration

(iii) When the cart is moving down the ramp, predict the direction of the acceleration

2. Get a track and a cart (attach a flag to the cart). Incline the one-meter long track so that one end is about 15-20 cm above the other. Clamp the accelerometer to the flag so that the water level in the accelerometer is horizontal when the cart is on the track. Give the cart a small push up the track and draw the water level for each case in part 1. Do not look at the accelerometer just after it is pushed since the water will still be sloshing around. Look close to the end of its motion uphill.

Cart moving uphill Cart at maximum height Cart moving downhill Water

Direction of a (uphill, downhill, zero)

Does your experiment confirm your prediction, or not? Make a clear summary statement of the experiment.

22 Motion of a cart rolling down a track

The goal of this activity is to analyze the motion of a cart rolling down an inclined track. The cart was released from rest at the top of the track and the distance it rolled during several time intervals was measured. The experiment was repeated three times for each interval. The results are listed in the table below. (You could set this up and do the measurements yourself, but you need very long ramp to get sufficiently accurate results.)

Time uncert. (s) Distance rolled (cm) Trial 1 Trial 2 Trial 3 Average St. dev. 0.5 0.005 6.9 7.1 7 7 0.1 1 0.005 25 25.5 24.9 25.1 0.3 1.5 0.005 58.3 58 58.4 58.2 0.2 2 0.005 100.9 101.3 101.1 101.1 0.2 2.5 0.005 157.9 157.4 157.4 157.6 0.3 3 0.005 230.1 230 230.1 230.2 0.3 3.5 0.005 315.7 315.6 315.3 315.5 0.2

Procedure For this activity, work in teams of 2 or 3 (i.e. 2 teams per table).

1. Using consecutive measurements, figure out the average velocity between each pair of positions. You should end up with 6 values for average velocity (and the time for each — what value should you use?) Make a neat table with the average velocities. 2. Using your average velocities, figure out the average acceleration between each pair of consecutive velocities. You should end up with 5 values for average acceleration (and the time for each). Make a second neat table with the average accelerations. 3. Each person in your team should now make a graph. a. One person should graph position versus time b. One person should graph average velocity versus time c. One person should graph average acceleration versus time Use the same scale for time on each graph; the vertical scale will have to change, of course. 4. Examine the graphs. Describe the shape of each graph in words, and write down an equation relating the quantities plotted. Include all numerical coefficients explicitly.

Everyone should get this far. If you have time, try answering the questions over the page… Going further… 5. Estimate the acceleration of the cart in two ways: a. use the graph of "velocity versus time" to determine the acceleration of the cart, with an uncertainty. b. Use your graph of "acceleration versus time" and/or your table of average accelerations to determine the acceleration of the cart, with an uncertainty. Do the two values agree within the uncertainties?

23 6. If you drop a cart, it falls through the air with an acceleration which is about g = 9.8 m s-2. How does that compare to the acceleration of my cart as it rolled down a ramp? Give a numerical answer. d v 7. The definition of acceleration (in a straight line) is a  . So, turning this around, we can find d t velocity from v   a d t . Assume that the acceleration of the cart does not change with time (is this consistent with your data?) Integrate (symbolically) to find the velocity as a function of time (don't forget the constant of integration). Does your result agree with what you see in the graph?  d x 8. The definition of velocity (in a straight line) is v  . So you can also integrate velocity to find d t position as a function of time. Again, does your equation agree with what you see in the graph? 9. What was the angle of the ramp? (No, I haven't told you how to do this yet ... so don't worry if you don't know how.) 10. What is the percentage uncertainty in the largest value of position? What is the percentage uncertainty in your largest computed value of average velocity? What is the percentage uncertainty in your largest computed value of average acceleration? (Hint: check the “error propagation rules” handout, or Prof. Richmond’s guide to uncertainties at http://spiff.rit.edu/classes/phys273/uncert/uncert.html".)

Report One report is required from each group/team. It should include the following: Each team must submit the following materials:  a cover page including a title, the date and the names of the team members;  neat tables of average velocity and average acceleration;  an example of the calculation you used to determine each of these quantities;  your graphs of position, average velocity and average acceleration vs time;  your comments on the shape of each graph, and the equation that applies to each one ;  if you got that far...the answers to parts 5–10.

24 Characterization of the Vernier Motion Sensor

Group Activity

The motion sensor used in this course emits ultrasonic pulses which are reflected by a target and then detected by the sensor. The sound waves leaving the sensor fill a cone-shaped beam. The time that it takes for the reflected pulses to return to the sensor is used to calculate the position of the target. The software then performs numerical differentiation on the position data to yield velocity information. Further numerical differentiation of the velocity data yields acceleration.

After familiarizing yourself with the LoggerPro software and the motion sensor, devise an experiment to determine the following characteristics of the motion sensor:

1. The closest position that can be measured 2. The absolute uncertainty in the position for a few different positions 3. The half-angle of the beam width

Write a brief report indicating your procedure and your results. Include any diagrams or graphical information deemed appropriate. Include a cover page with title, date, and your names.

25 Analyzing the Motion of a Ball Thrown Vertically Predict 1. Consider holding a ball, tossing it vertically, and then catching it. Predict the graphs of position, velocity, and acceleration for the ball and sketch these on the left on the figures below. Include the time before you toss the object and after you catch it.

Predicted Graphs Measured Graphs

26 Experiment 2. After you have made predictions, assemble your experiment. Clamp a rod to the table, a second rod at the top and the motion sensor pointing down. Set-up the LabPro for data collection and connect a motion sensor to Dig/Sonic 1. Copy the file K200BallThrownVert.xmbl to MyDocuments and open it. Zero the sensor (Cntrl-0). The experiment file is set up so that motion away from the detector (down) is negative. Finding the file: Connect to the Team Physics 311 folder in the Students section of the server SVPHY01 on the Science Building Zone. Also in the students section is an installer for the software. RIT has a site license with unlimited copies for students at RIT. You will not be able to access this from outside RIT. If you want to borrow an installation CD, talk with your instructor. If you wish to save the data, the best method is to save the file to My Documents and send it to yourself, either by ftp (you computer jocks can handle this) or by attaching a file to an e-mail using Internet explorer and pointing to webmail.rit.edu. 3. Zeroing and Calibration. Zero the sensor when it sees the floor (Control-0 is the keyboard shortcut.) Check the calibration by placing a stationary object like a book at a known distance above the floor and collecting data. 4. Toss the ball a short (20 cm) distance and collect data. You may need to adjust the position and aim of the sensor in order to get a good result. Be sure that the curves are correct before you proceed.

Analyze 5. Make a sketch of what you actually measure in the “Measured Graphs” on the previous page, or print it out. Identify (and label) the portions of the sketches when the ball is in your hand, and when it out of your hand. 6. Select (drag across) a region of the graph when the ball is out of your hand. Do an Automatic Curve Fit (under the Analyze menu, or the “f(x)” icon) and do a quadratic fit. Record the equation and coefficients and attach the proper units to them.

What is the meaning for each coefficient, (e.g. initial position)

7. Click on the velocity-time graph. Drag across a similar region as above and do a Linear Fit (Analysis menu or the “R =” icon). Record the equation with coefficients including units.

What is the meaning of each coefficient?

27 8. Click in the acceleration-time graph. Drag across a region when the ball is out of your hand and determine the average by using Statistics (Analysis menu or the “Stat” button). Record the average value, tell the name of this quantity and its units. Also record the standard deviation.

9. Do your results of the three previous parts agree? For example, do they all give the same value of initial velocity, etc?

28 Some 1D kinematics problems:

1. Indianapolis race cars make a rolling start. At the beginning of the race they are moving at 120 mph and they accelerate to 250 mph in 4.0 seconds. (a) What is the acceleration of the cars? (b) How far do they travel in this acceleration phase? Objects falling near earth with no forces acting except gravity have a downwards acceleration of 9.8 m/s2.

2. A ball is thrown upwards at 9.0 m/s toward a platform located 4.1 m above the point where you release the ball. Air resistance is negligible. (a) Find the time taken to reach the platform. (b) Find the velocity when the ball reaches the platform.

3. You notice a falcon diving past the Kodak office building. You see it pass 5 stories in 1.5 seconds, and at the end has a downwards velocity of 15 m/s. (a) What is the acceleration of the bird? (b) What was the initial speed of the bird?

4. Your coop job is with a research group studying ozone depletion. Your group will launch a rocket that has a mass spectrometer in its nose cone. You are assigned the task of determining the acceleration of the rocket so that the equipment can be designed to survive. The rocket you will have has a burn time of 30 seconds, and can be assumed to produce constant acceleration. After the fuel is exhausted the rocket will enter free-fall. You are told that the total time of flight of the rocket must be 5.0 minutes. What is the acceleration of the rocket when the engine is on, and what is the maximum altitude of the rocket? BONUS Is this problem realistic? Where is the ozone layer anyway?

29 Week #3

30 Motion in Two Dimensions :

Terms: Trajectory=path traveled position = location relative to origin displacement = difference between positions I. Velocity A car is moving clockwise along an oval track. Make a large diagram of the oval on a piece of paper. A. Pick a point to serve as the origin of your coordinate system. This point should be outside the oval. Points A and B are shown, B where B occurs slightly later in time than A. B’ occurs closer in time to A than does B.

   1. Draw and label the position vectors, ( r A , rB ), for each point. Then draw and label the displacement vector  r Discuss your result with your partners and sketch a small version to the right.

2. How is the direction of the displacement vector related to the direction of the average velocity vector?

3. If we reduce the time interval the car only moves to B’ rather than B. Draw the displacement vector for the car moving from A to B’. What mathematical word describes the relation between the displacement vector and the trajectory as the points approach each other?

4. What is the direction of the instantaneous velocity relative to the trajectory at any point? Draw and label vectors showing the instantaneous velocities at points A and B.

Does your answer depend on whether the object is speeding up, slowing down, or moving at constant speed?

B. Suppose you choose a different origin. Which of these vectors would change, which would stay the same.

Position of A, position of B, displacement from A to B, instantaneous velocity at B.

31 II Acceleration for car moving at constant speed around the oval. A car is driving around the oval at a constant speed. Two locations, C and D are shown, D at a short time later than C. Make a large drawing of this on another piece of paper. Draw and label velocity vectors, v C , vD for the car at points C and D. A. Drawr ar vectorr diagram showing the vector change in velocity, v  v D  v C .  r 1. Draw the initial velocity v and the change in velocity, v C D with their tails together and show the angle between the two vectors. Show the same angle on the previous diagram. Is the angle between velocity and the change in velocity less than, equal to, or greater than 90°?

2. If the time interval is smaller, point D is closer to point C. Will the angle in part 1 change? If so will it be closer to or farther from 90°?

3. What is the limiting value for this angle as the time interval approaches zero? (Remember that the speed is constant.)

Can you prove this result?

4. For the case of constant speed, how are the directions of average acceleration and change in velocity related?

5. For the car moving at constant speed, what is the angle between velocity and acceleration? Draw and label the accelerations, a at points C and D.

B. Parts of two ovals are shown below and a car travels around the two paths with the same constant speed. The distance and time interval between the points is the same for both. Make a sketch of the change in velocity in the two cases. Is it the same, or is it larger in one case, and if so which? What does this say about the magnitude of the accelerations? Make a clear statement.

32 C. A car moves at constant speed around the track below. Draw the acceleration vectors at the points where there are dots. Use long vectors when the acceleration is large, short vectors when it is small, and a = 0 if there is no acceleration.

33 III Acceleration for a car increasing speed. Once more around the oval, and this time the car moves from E to F around a curve, F occurring slightly later in time than E. The car is moving four times faster at F than at Er . Draw and label velocity vectors at E and F, v E , vF . A. Draw a vector diagramr showingr rhow to get the F change in velocity, Dv = vF - vE . 1. Is the angle between the velocity at E and the change in velocity less than, equal to, or greater thanr 90°? (Draw vectors vE and Dvtail-to-tail to decide.)

2. When the car is speeding up, is the angle between acceleration and velocity less than, equal to, or greater than 90°?

B. At the points marked draw the velocity vector (- - - >) and the acceleration vector (===>) for the car. On the left the car is moving at constant speed, on the right it is speeding up starting from rest (A). Make lengths of vectors consistent with magnitudes.

34 Motion on a Curved Racetrack

Theory: The text proves that the radial (centripetal)acceleration component points to the center of the v 2 circle and has a magnitude of a = when an object moves at speed v in a circle of radius r. r r This is true whether or not the object is changing speed. If the object changes speed it will also have a tangential component of acceleration.

Example. You drive a go-cart around a go cart track, consisting of two arcs (radii shown on diagram) connected by two straight segments as shown. Indicate an acceleration of zero by writing a = 0.

(a) You drive at a constant 5.0 m/s. What is your acceleration (at points W, X, Y, and Z? Magnitude should be a number. To show direction, draw the acceleration vector (==>) at each point. Be sure that your vectors are to scale relative to each other.

(b) Your friend is more adventuresome. She drives the small arc at 5.0 m/s, then accelerates on the lower straight segment from 5 m/s to 15 m/s, which takes 5.0 s, drives the large arc at 15 m/s, and then slows back to 5.0 m/s in 5.0 s on the top straight segment. Find your friend’s acceleration for the points W, X, Y, and Z for your friend. Magnitude should be a number. To show direction, draw the acceleration vector (==>) at each point. Be sure that your vectors are to scale relative to each other.

35 Water Accelerometers and Circular Motion

Consider the special path of a circle. The natural coordinates to use with a circle are plane polar coordinates, r and . These are shown on the circle. Unit vectors are also defined, rˆ and qˆ , each having magnitude of 1. The rˆ points radially outwards from the center, and the qˆ points tangentially, in the direction of increasing angle.

If the object stays in a circle, the value of r is constant. Suppose that the object has a speed v and is going around the circle in a counterclockwise direction, write an expression for the velocity using a unit vector.

Experiment: You will on your chair and rotate and measure the acceleration with a water accelerometer. Use a rubber band to mark the water level at rest.

Predicting radial acceleration. You will hold the accelerometer in a radial direction, but at two positions, one close to your body, and one at arm’s length. The center of the rotation is to the left of the page.

Near body (prediction) Arm’s Length (prediction)

Center of Circle

Predicting tangential acceleration. You will hold the accelerometer parallel to your body, and consider the case where you rotate at constant speed (moving to the right) and at increasing speed (moving to the right.) Constant speed (prediction) Increasing speed (prediction)

Now make measurements. Radial Near body (measured) Arm’s Length (measured)

Center of Circle

Tangential Constant speed (measured) Increasing speed (measured)

36 Center of Circle

37 Problems: 2D Constant Acceleration Kinematics Assume that the acceleration is completely along the y-axis.

ax  0 vx  v0x x x0  v0x t ay  0 vy  v0y  ay t 1 2 1 2 2 y  y 0  v 0 y t  2 a y t y  y 0  2 v 0 y  v y t v y  v 0 y  2 a y y  y 0  1 2 A l s o y  y 0  v y t  2 a y t w h e r e v y i s f i n a l v e l o c i t y   1. HRW 6, Q 4.7 In the figure, the path of a “cream tangerine” (i.e.

ball) is shown as it moves up past windows A, B, and C that are

identical in size and regularly spaced vertically. Rank the times needed

to pass the windows starting with the window having the largest time. F

(Answers might be A=C>B for example.) Does the regular vertical

spacing matter?

The tangerine now passes by windows D, E, F that are identical to each other and regularly spaced horizontally. Rank the times needed to pass the windows starting with the window having the largest time. Does the regular horizontal spacing matter?

2. I shoot a ball from a cannon at 15.0 m/s, 60° above the horizontal. It lands on a platform that is 8.0 m vertically above the end of the cannon. Find the horizontal distance from the end of the cannon to the point where the ball hits.

3. I throw a snowball from a roof 16.00 m high with a velocity of 15.00 m/s at some angle above the horizontal. The angle it is thrown at is (37°± 2°). Assume free fall. (a) Where does the ball land horizontally? (b) What is the velocity of the ball as it hits? Give the answer using unit vectors and also as a magnitude and direction. We want an idea of the uncertainty in the answers. Different groups should use different angles. Each table should do two of the three possible angles, 35°, 37°, and 39°

4. HRW 33P An airplane has a speed of 290.0 km/hr and is diving at an angle of 30.0° below the horizontal when the pilot releases a package of relief food. The food travels a horizontal distance of 700.0 m from its point of release to the point where it hits the ground, and we will assume “free-fall”. (a) How long is the food in the air? (b) How high was the plane when it released the food?

5. You get the opportunity to play pro-basketball with Michael Jordan. The team is down by 2 points with time running out. Michael feeds you the ball and you take a shot with 0.82 seconds remaining. It goes in, with nothing but net, as the buzzer sounds! As with all your shots, the ball leaves your hand at an angle of 30° from the horizontal. Does the game go into overtime or have you won?

38 Projectile Motion as an Example of Curvilinear Motion

A ball is thrown under the influence of gravity alone so that it is projectile motion. The trajectory and several positions of the ball are shown. Use a ruler and draw accurate vectors and vector components .

(a) Draw a vector showing the horizontal component of the velocity at each location.

(b) Draw a vector showing the velocity of the ball at each location.

(c) Draw a vector showing the acceleration of the ball at each location—use a double shafted arrow (=====>).

(d) As the ball is moving up, what is the angle between the velocity and the acceleration, < 90°, = 90°, or > 90°?

(g) When the ball is at maximum height, what is the angle between the velocity and the acceleration, < 90°, = 90°, or > 90°?

(f) As the ball is moving down, what is the angle between the velocity and the acceleration, < 90°, = 90°, or > 90°?

39 Predicting the motion of a projectile

These problems deal with the theory behind the experiments we will try in the next workshop. For both problems, draw and label the co-ordinate axes you use and clearly show all steps in the derivation, starting from the general equations of 2-d kinematics. Keep this worksheet and any other sheets used, you will need them later.

Case 1

Let’s begin with the special case in which a ball is fired horizontallyr from a gun. The gun is at a height h above the ground and the ball leaves the barrel with a velocity v 0 , as shown in the diagram. Where will the ball land?

Your job is to derive an expression for the horizontal distance (L) from the gun to the landing point in r terms of the height h and the magnitude of the initial velocity v 0 (which you can just call v0). Remember that, during its motion, the ball is subject to a downwards acceleration, g.

40 Case 2.

Now let’s consider the more general case in which the gun is tilted at an angle  to the horizontal. This time we need two equations:

 an equation that relates the muzzle velocity v0 to things we know or can measure, h, L,  and g (see diagram).

 An equation that relates L to v0, h,  and g.

The first step is to resolve the initial velocity vector into horizontal and vertical components. Then apply the equations of kinematics to both axes, remembering that only motion along the vertical (y) axis experiences an acceleration (due to gravity). Check your solutions by setting  = 0. You should recover the solution to Case 1.

41 Projectile Experiment

In the last workshop you derived an equation relating the initial velocity of a projectile fired horizontally from a height (h) to its range (L). Now we are going to do an experiment to determine the muzzle velocity of a spring gun and use this datum to make predictions for the range and time of flight of the projectile when it is fired at an angle. We will then test the predictions by taking several shots with a tilted “cannon”. Equipment Ballistic pendulum device, open-ended box, carbon paper and blank sheet, meter sticks, stopwatch. Set up one device on your table. Set-up and operation Place the ballistic pendulum device on your table and clamp in it place. Carefully remove the pendulum arm and make sure that the cannon is fixed securely in the horizontal position. Gently push the ball to the medium range setting using the rod provided. Fire a shot or two to get a rough idea of the range and then set up your recording device (carbon paper, blank sheet and box) to catch the ball. Horizontal shot First measure the height of the cannon above the floor (h). Then, taking care that the ballistic pendulum device does not move, fire 3 shots into the cardboard box and note the landing point. Measure, as accurately as possible, the horizontal distance traveled by the projectile (L1).

Use your measurements of h1 and L1 and their uncertainties to determine 1. the muzzle velocity (v0) of the projectile, 2. the uncertainty in v0. v 0

h 1  L 1

42 Predictions for a tilted cannon

Now tilt the cannon to an angle =65° and measure the new height, h2. Use the values of v0 and h2 just determined along with the angle  to predict 1. the time of flight t2 2. the range L2 Also, calculate the uncertainties on both of these quantities. The complete uncertainty analysis requires many steps. Refer to the guidelines given in the following pages of this manual. SHOW your result for v0 and your predictions for t2 and L2 to an instructor before you proceed. Tilted shot

Now take 3 more shots with the tilted cannon. Measure the new range L2, as before. But this time, two members of each team should also use a stopwatch to measure the time of flight, t2. Compare the time measurements taken by each person. How closely do they agree? Is reaction time likely to be a significant source of uncertainty? Analysis

Do your predictions agree with the measured values of t2 and L2, within the uncertainties? Make a quantitative statement. For example,

“Our prediction for the range at angle =65° was L2=1.51±0.21 m. The measured range was 1.45±0.05 m, which agrees with the prediction within the uncertainties” What is the largest source of uncertainty in the predictions for t2 and L2? Report One report is required from each team. It should include the following:

 Cover page with the names of all members

 A brief abstract (200 words max.) summarizing the experiment and stating your final result

 A section on the results for the horizontal shot including the measured values for h1 and L1, their uncertainties, your calculation of v0 and its uncertainty (show the mains steps of the calculations).

 A section showing your predictions for t2 and L2, together with your calculated uncertainties on these quantities.

 A section showing the results of the tilted shot, including the measurements for t2 (you can quote the mean and standard deviation for this) and L2.

 A concluding statement consisting of your quantitative comparison between the predicted and actual measured values of t2 and L2, and brief comments. In addition: each person in the team must attach their own copy of the previous “Projectile Theory” worksheet showing the derivation of the range and velocity equations.

43 Error analysis for the horizontal and the tilted projectile experiments. Refer to the propagation of uncertainties handout for the rules used below.

From the horizontal shot, we got a value for the initial velocity of the ball, v0, using measurements of h and L. The equation for v0 in this case is

v 0  L 2h g

We need to calculate the uncertainty in v0. Since we can assume that g is known to high precision, we only have to consider the uncertainties in L and h.  First consider the 2 h g term. Since this is essentially constant h1/2, the relative uncertainty is  2 h g  1 h  2 h g 2 h The relative uncertainty in L is just L/L. Now we can use the error propagation rule for multiplication and division to get the relative uncertainty on v0: v L  2 h g  L 1 h  0     . v 0 L 2 h g L 2 h

The absolute error on v0 is then L 1 h  v  v    0 0  L 2 h   Note: base your estimates of L, h, on the spread in the measurements, not the uncertainty on individual measurements. 

Now we can use the value of v0 we determined to predict the time of flight t and the new range L for a tilted shot, given the new value of h and the angle, . (Note: the numerical values of L and h in the following are NOT the same as for the horizontal shot). The components of velocity are

v,0,x  v0 cos

v 0,y  v 0 sin

The range for the tilted shot is given by  L  v 0 ,x t  The time of flight is found from v  v 2  2 g h  t  0 , y 0 , y (since we take the positive root). g

Now, to calculate the uncertainty on L and t, we need to know the uncertainties on v0,x and v0,y. Let’s take v0,x as an example. Using the multiplication rule, v v c o s   0 , x  0  v 0 , x v 0 c o s 

44  We know what v0 is, and we have an uncertainty in , but there is no “rule” to turn this into an uncertainty in cos. The simplest approach is just to take half the range in the cosine that corresponds to

the range in . For example, if max     and min     , then 1 cos  cos   cos  , 2 m ax m in which we can use with v0 to get v0,x.  

Now the tricky part: the error on t. The formula for t involves two quantities with uncertainties: v0,y and  h. But we can make life easier by recognizing that the fractional uncertainty on h is much smaller than that on v0,y. If you work through the numbers, you should find that the percentage error on v0,y is ≈ 8%, whereas on h it is only ≈ 0.2%. The former is clearly going to dominate the final error on t, so we can safely ignore the uncertainty on h. The absolute uncertainty on t, is the sum of the uncertainties on the two terms, multiplied by the constant 1/g:

1 t  v   v 2  2 g h g  0 ,y  0 ,y 

2 We can get v0,y using the same method as for v0,x. But the v 0 , y  2 g h term is more complicated to deal 2  with. We get the fractional uncertainty on v 0 , y from the “power” rule: 2 v 0 , y  v 0 , y  2 2  v 0 , y v 0 , y 2  2 The uncertainty on v 0 , y  2 g h is just the sum of the uncertainties on v 0 , y and 2gh, but the uncertainty on the latter is negligible, so we have 2 2  v0,y  2gh v0,y 2v0,y  v0,y . 2 Next, using the power rule again, we see that the fractional uncertainty on v 0 , y  2 g h is  v 2  2 g h 2  0 , y  1 v 0 , y  2 g h  v 0 , y v 0 , y    2 2 2 v  2 g h 2 v 0 , y  2 g h v 0 , y  2 g h 0 , y  v v  v 2  2 g h  0 , y 0 , y So, finally,  0 , y  2 v 0 , y  2 g h  and the uncertainty on t is   1 v v t  v  0 , y 0 , y .  g  0 , y 2   v 0 , y  2 g h 

Now we have t, it’s easy to get the uncertainty on L. 

45 Relative Motion Problems

1. It is a rainy day with no wind blowing. You are driving your car at a speed of 10 m/s. You notice that the rain appears to be coming at you at an angle of 50 relative to the vertical. How fast is the rain falling toward earth?

2. A small-plane pilot wishes to travel a direct route to a town located 320 miles away in a direction 10 east of south from her present position. There is a steady 30 mph wind blowing west to east during the entire trip. Her plane’s cruising speed through calm air is 95 mph. (a) How long will it take her to make the trip? (b) What must be the plane’s heading?

46 Week #4

47 Newton's Second Law and The Force Plate

A force plate is essentially a digital bathroom scale that connects to the Labpro. When you stand still on the plate, it will read a value equal to your weight, mg. Suppose now that you start with bent knees, straighten the knees and jump off the force plate. (a) Draw free body diagrams for the different parts of the motion: First, standing motionless with knees bent, Second, in the early parts of straightening your legs in order to jump, Third, your legs are straight, but you have not yet left the plate Finally, after you have left the plate. Remember to draw the diagrams with (i) the object free of its surroundings, (ii) forces drawn with their tails at the center of the free object (draw them to scale, larger forces means longer arrows!), and (iii) near the body put information about acceleration and coordinate system.

(b) Predict the shape of a graph of force exerted by the force plate versus time.

(c) Measure the actual data and sketch below, copying F105ForcePlate to MyDocuments.

(d) Comment on the two graphs. If you did not correctly predict, explain why the real curve has the shape it does.

48 Newton’s Third Law A common (but confusing) statement of Newton’s Third Law is “For every action there is an equal and opposite reaction.” In this activity you will measure forces with force sensors and try to understand the real meaning of Newton’s Third Law.

Predictions 1. Two objects, a truck on the left, and a car on the right, are on a horizontal rough surface (friction present) and are in contact. The vehicles can have their motors on or their brakes applied, or be in neutral. When the truck and car are in contact, there is a horizontal contact force between them, meaning that the truck pushes or pulls on the car, and the car pushes or pulls on the truck. Compare the magnitudes of these two forces in different situations. All motion is along a straight line. In Parts (a) to (d) the vehicles are of equal mass, in Parts e and f the truck is much more

massive than the car.

P a r t s ( a ) t o ( d ) , E q u a l m a s s P a r t ( e ) M a s s i v e t r u c k , l i g h t c a r P a r t ( f ) M a s s i v e t r u c k , l i g h t c a r

Discuss your answers with the others in your group and record your consensus answers. Summary of predictions: In parts a-d the car and truck are of equal Force magnitude of car on truck Measurements mass. In parts e and f the truck is much is (later) more massive than the car. (>, =, <) the Force magnitude of truck on car. (a) Equal masses, Truck and car push but there is no motion.

(b) Equal masses, Truck pushes car to the right at constant speed.

(c) Equal masses, Truck pushes car to the right, both are accelerating to the right

(d) Equal masses Truck pushes car and both move to the right, car sets brakes and they are slowing down, accelerating to the left.

49 In parts a-d the car and truck are of equal Force magnitude of car on truck Measurements mass. In parts e and f the truck is much is (later) more massive than the car. (>, =, <) the Force magnitude of truck on car. (e) Very Massive truck moves to the right and rear ends a light car that is also moving right. They separate after the collision. (f) Light car moves to the right and rear ends a massive truck that is also moving right. They separate after the collision.

Measurements 2. Get a computer, LabPro, two force sensors with adaptors, and two massive bars. Connect the LabPro to the computer. Use the adaptors at the ends of the force probe cables, and connect the force probes to CH1 and CH2 of the LabPro. The truck is the sensor connected to CH 1, and the car is the sensor connected to CH2. [Note: you do NOT need to use carts and a track for this experiment.] Be sure the slide switch on the force sensors is at 50N. Drag the file F200NewtonsThirdLaw.xmbl to MyDocuments from the server. Calibrate the sensors (see below). Place the sensors on the table and zero both sensors (CNTRL-0). Check the zero by collecting data with the sensors horizontal and not touching. Beware the Tare: There is a tare button on the side of the force sensor. Be very careful not to accidentally press it. If you do, you will need to re-zero the sensors., but will not need to recalibrate. Replace the hooks on the force sensor with rubber bumpers. Push on each bumper with your hand while collecting data to see how data are collected. If the truck is on the left, pushing to the left on its bumper should give a negative force. A push to the right on the car bumper should give a positive force. 3. Set up experiments to check the results that you predicted in parts (a) to (d). Enter your observations into the appropriate column in the table on the previous page.

50 To check parts e and f of Newton’s Third Law during collisions, you must make a couple of changes to the program.

Click on the clock icon (or on menu, Experiment: Data Collection) and a window will open. In the window make the length of the experiment 1 s, and collect 1000 samples per second. Then click the Triggering tab and check Enable Triggering. Check parts e and f of your predictions. Enter your observations into the appropriate column in the table on the previous page.

4. This experiment is designed to help you learn something about Newton’s Third Law. Write a clear statement of Newton’s Third Law based on what you measured.

Calibrating the force sensor: To calibrate, under the Experiment menu choose Calibrate and choose a force sensor. Be sure the slide switch is in 50 N position. Hold sensor vertically and enter zero for the force. Now hang about 500 grams (be sure to include the mass of the hanger) and enter the weight that is hung (W = m g.) Calibrate the other sensor similarly.

51 1-D Force Problems

1. A 50 kg passenger rides in an elevator that starts from rest on the ground floor of a building at t = 0 and rises to the top floor during a 10 sec interval. The acceleration of the elevator as a function of the time is shown in the figure, where positive values of the acceleration mean that it is directed upward. Give the magnitude and direction of the following forces: (a) the maximum force on the passenger from the floor, (b) the minimum force on the passenger from the floor, and (c) the maximium force on the floor from the passenger.

2. A 29.0 kg child, with a 4.50 kg backpack on his back, first stands on a sidewalk and then jumps up into the air. Find the magnitude and direction of the force on the sidewalk from the child when the child is (a) standing still and (b) in the air. Now find the magnitude and direction of the net force on Earth due to the child when the child is (c) standing still and (d) in the air.

3. Boxes A and B, with masses mA and mB can move on a horizontal frictionless surface. You push horizontally with a push P on block A. (a) Find the acceleration of the boxes. (b) Find the force between the boxes.

Answers may contain symbols already given plus any constants like g.(c) Suppose mA = 10 kg and mB = 40 kg, and P = 100 N. What are the numerical answers? (d) Suppose mA = 40 kg and mB = 10 kg, and P = 100 N. What are the numerical answers?

4. You have been hired to design the interior of a special executive express elevator for a new office building. This elevator has all the latest safety features and will stop with an acceleration of g/3 in case of any emergency. The management would like a decorative lamp hanging from the unusually high ceiling of the elevator. You design a lamp which has three sections which hang one directly below the other. Each section is attached to the previous one by a single thin wire which also carries the electric current. The lamp is also attached to the ceiling by a single wire. Each section of the lamp weighs 7.0 N. Because the idea is to make each section appear that it is floating on air without support, you want to use the thinnest wire possible. Unfortunately the thinner the wire, the weaker it is. To determine the thinnest wire that can be used for each stage of the lamp, calculate the force on each wire in case of an emergency stop.

5. Two blocks, masses m1 and m2, are connected by an ideal string passing over an ideal pulley. One block is on a horizontal frictionless surface and the other can move vertically and only touches the string. Find the acceleration of each block and the tension in the string in terms of the masses and constants.

52 Rotating Hockey Puck

A hockey puck of mass m = 0.5 kg slides without friction in a circle of radius R = 80 cm on a table. The puck is attached to a string which hangs down through a hole in the middle of the table; from the other end of the string hangs a cylinder of mass M = 4 kg. (a) How fast must the puck be moving in order to keep the cylinder suspended at rest? (b) Suppose the puck is moving at a different speed: it makes two revolutions every second. What is the puck's actual speed? (c) If the puck moves at this speed, is the cylinder moving up or down? (d) What is the acceleration of the cylinder? (Adapted from HRW 7e, Chapter 6, Problem 49.)

53 Newton's Second Law Problems: We will be working on these problems during the next several classes. Please bring them to each class.

Start all problems with a complete free body diagram.

1. Boxes A and B, with masses mA and mB can move on a horizontal frictionless surface. You push horizontally with a push P on block A. (a) Find the acceleration of the boxes. (b) Find the force between the boxes. Answers A B may contain symbols already given plus any constants like g.(c) Suppose mA = 10 kg and mB = 40 kg, and P = 100 N. What are the numerical answers? (d) Suppose mA = 40 kg and mB = 10 kg, and P = 100 N. What are the numerical answers?

2. A book of mass m is pushed against a smooth frictionless wall by a force P that makes an angle  with the vertical. Find (a) the normal force on the book and (b) its acceleration in terms of P, m, , and constants. (c) Look at limiting cases (large and small angles) to see if it makes sense. P 

3. You are skiing on a hill inclined at  to the horizontal where there is a constant frictional force f. Find the normal force and your acceleration in terms of your mass, m, f, , and constants. Look at limiting cases to see if the answer makes sense. m 4. Two blocks, masses m1 and m2, are connected by an ideal string passing 1 over an ideal pulley. One block is on a horizontal frictionless surface and the other can move vertically and only touches the string. Find the acceleration of each block and the tension in the string in terms of the masses and constants just after the hand is removed from the block. m 2

5. A Ferris Wheel has a radius of 11.0 m and rotates once every 11.0 s. What is T o p the force of the chair seat on you at each of the three positions, top, side, S i d e and bottom? Answer as a multiple of your weight (e.g. 1.4 mg). B o t t o m

6. You whirl a ball of mass 0.40 kg on a string of length 0.90 m. At the point shown the angle of the string from the vertical is 30° and the ball has a speed of 3.50 m/s. (a) Find the tension in the string. (b) What is the acceleration of the ball, tangential and radial components? (c) Now consider the ball at the top of the circle. What is the smallest speed of the ball so that the ball continues in a circle?

54 7. A ball of mass 1.34 kg is attached to a vertical rod by two strings. The top string is 1.60 m long and makes a 30° angle with the rod, while the bottom string makes a 45° angle with the rod. The rod is rotated at 1 revolution every 1.20 s. 3 0 ° Find the tension in each string. 1 . 6 m

4 5 °

8. I have a horizontal table that is very low friction. On it I place a pair of blocks, with m1 on top of m2. Between the blocks there is friction, with static coefficient µs and kinetic coefficient µk I attach a rope to the lower block and pull m 1 with a pull P acting at an angle  above the horizontal. If the pull is small the two blocks move together. If the pull is large they m 2 move relative to each other. (a) For a large pull, find the acceleration of each block. (b) For a small pull, find the common acceleration, and find the actual force of friction.

Try using m1 = 0.600 kg, m2 = 2.400 kg,  = 30.0°, µs = 0.450, µk = 0.350 and for a small pull, P = 5.000 N, large pull P = 20.000 N. What answers do you get? Do they make sense?

9. You are pushing a box uphill the hill has a pitch of 15° above the horizontal, and the static and kinetic friction coefficients between the box and the hill are 0.45 and 0.35. The box has a mass of 40 kg. You push horizontally with a push of 180 N, and the box is initially moving uphill. Find the acceleration of the box.

10. While visiting a friend in San Francisco you decide to drive around the city. You turn a corner and are driving up a steep hill. Suddenly, a small boy runs out on the street chasing a ball. You slam on the brakes and skid to a stop leaving a 50 foot long skid mark on the street. The boy calmly walks away but a policeman watching from the sidewalk walks over and gives you a ticket for speeding. You are still shaking from the experience when he points out that the speed limit on this street is 25 mph. After you recover your wits, you examine the situation more closely. You determine that the street makes an angle of 20o with the horizontal and that the coefficient of static friction between your tires and the street is 0.80. You also find that the coefficient of kinetic friction between your tires and the street is 0.60. Your car's information book tells you that the mass of your car is 1570 kg. You weigh 130 lbs. Witnesses say that the boy had a weight of about 60 lbs. and took 3.0 seconds to cross the 15 foot wide street. Will you fight the ticket in court?

55 m The Modified Atwood Machine: Using Newton’s Second Law 1

A cart of mass m1 is connected to a mass m2 by an ideal string passing over an ideal pulley,  as shown. The friction on the horizontal surface is negligible. First make predictions for the cases below, then draw free body diagrams and predict the result based on the diagram! You will make observations as described on the reverse side. Case 1: You put your hand on the cart of mass m1 and hold it at rest. m (i) What is the direction of the acceleration of m1, left, right or a=0? 2 Prediction—left, right, zero Observation--Direction and value (with uncertainty)

(ii) Compare the string tension to the force of gravity on mass m2— is T >m2g , T =m2g , or T < m2g? Initial Prediction Based on Free Body Diagram Observation--Direction and value (with uncertainty)

m2g = T = Case 2. Starting from the situation in case 1, you quickly remove your hand. (i) After you remove your hand, what is the direction of the acceleration of m1? Prediction: left, right, zero Observation--Direction and value (with uncertainty)

(ii) After you remove your hand, compare the tension in the string to the force of gravity on mass m2-- T >m2g , T =m2g , or T < m2g? Initial Prediction Based on Free Body Diagram Observation--Direction and value (with uncertainty)

Case 3. Starting from the situation in case 1, you now push m1 so that it begins to move to the left. (i) What is the direction of the acceleration of m1, in the early part of your push, while your hand is pushing, Prediction left, right, zero Observation--Direction and value (with uncertainty)

(ii) Compare the tension in the string to the force of gravity on mass m2 while your hand is pushing, Initial Prediction Based on Free Body Diagram Observation--Direction and value (with uncertainty)

Case 4 Repeat Case 3, but now consider the motion after the cart leaves your hand. Compare the results for (a) motion of m2 upwards to (b) motion of m2 downwards Prediction for acceleration up, down, zero Observation--Direction and value (with uncertainty) (a)

Prediction for tension Based on Free Body Diagram Observation--Direction and value (with uncertainty) (a)

56 Observation Set-up a track with a cart and level the track as best you can. The cart should have a force sensor mounted to it along with a single massive bar with an electronic accelerometer taped to it. The arrow of the accelerometer should point in the direction that the cart will move. Connect a string to the hook on the force sensor, pass it over a pulley and attach it to a mass hanger with attached mass totaling 200 g. Connect the sensors to the Lab-Pro (accelerometer in Ch 1, force sensor in Ch 2), and the Labpro to the computer. Drag the program F300QualitativeAtwood.xmbl from the server to My Documents and open it. Be sure the force sensor switch is on the 10 N setting. Calibrate the force sensor and the accelerometer as described below. Zero the sensors when there is no applied force and no acceleration. Make measurements to check your predictions for the four cases. First make qualitative measurements (direction of acceleration, size of tension compared to m2 g) then record actual averages as indicated below by using the STAT icon (Statistics under the Examine menu). First drag across a region of interest, then click on the icon. Record the mean and standard deviation (std dev) of the results, using the appropriate number of significant figures. Analysis Compare your predictions to the measurements. Resolve any differences. In Case 1, did the value of the acceleration match the prediction? In Case 1, did the tension have its expected value? Compare the values of the acceleration in Cases 2, 4a, and 4b. Are they equal? Compare the tensions in Cases 2, 4a, and 4b. Are they equal? Compare the accelerations in Case 3 and Case 2. Is the result what you would expect? Compare the tensions in Case 3 and Case 2. Is the result what you would expect?

Calibration: Under the Experiment menu, Calibrate, and choose the sensor. Force sensor: Hold the sensor vertically with no mass attached so that the force is zero, and enter that value. Then hang a mass of about 500 grams (remember to include the mass of the hanger!) and enter the weight of the hanging mass. Recall that W = m g.

Accelerometer: Hold the accelerometer so arrow points vertically up, and enter the gravitational acceleration of +9.8 m/s2. Reverse the accelerometer so that the arrow points down. Enter -9.8 m/s2.

57 Week #5

58 Measuring Friction

This is a brief experiment that will let you see the properties of frictional force. Set up the lab-pro and copy the file friction (located in the Labpro folder within the Team Physics 311 folder) to MyDocuments, then open it . Connect a force sensor to Channel 1. Be sure it is on the 10 N setting. Calibrate the force sensor. Obtain a track, a sheet of sand paper, a friction block, and two black bars. Measure the masses of the bars and of the block. Note: The friction blocks have wood and felt sides Only use the wood side! Connect the block and the force sensor with a string. Place one bar on top of the block. Measure the force as you pull the block at constant speed (do your best!) both on sandpaper and on metal. Find the average force in each case and then find the coefficient of friction for the case of wood- on-sandpaper and wood-on-metal. Record the average force, the standard deviation, and the number of points included in the average. Turn the block on its edge and repeat the measurements, still with one bar on top. Does the coefficient of friction depend on the surface area? Now return to the large area and place two bars on top. Repeat the measurements. Does the coefficient of friction depend on the normal force?

59 Force of a Spring

Hang a spring vertically and measure the stretch of the spring as a function of mass added to the spring. Do not exceed 250 g mass. Take 5 points.

Use Newton’s Laws to find an expression for the force of the spring in terms of the mass, m, added. U n s t r e t c h e d Make a plot (by hand!) of the force of the spring versus the stretch on the supplied paper. Each person should make a graph. One goal of this exercise is to ensure that you can quickly make a graph by hand when needed.

First decide what “force versus stretch” means. Which variable goes on the horizontal axis, and which on the vertical axis?

Second, decide on the ranges of the variables, i.e. the maximum minus minimum value for each variable.

Third, examine the graph paper supplied (a coarse graph paper) and choose a scale for each axis.

Fourth, make tick marks on the axes and label them with values. Also label the axes with the variable and the units. Give a meaningful title to your graph.

Fifth, plot the data on your graph. Use a small dot surrounded by a circle or square or diamond, () for each data point.

Sixth, if appropriate, make a fit to your data. Your data should appear to be a straight line. Draw the line and determine its slope and intercept. For this graph, what does the slope tell you?  What is the mathematical relation that lets you predict the spring force for your spring?

60 The Dependence of Air Resistance on Velocity

The force of air resistance clearly depends on the velocity of an object moving through the air: the larger the speed, the larger the drag force. But what is the exact form of this relationship? Your textbook suggests that under some circumstances, air resistance depends on the square of the velocity:

However, some other sources suggest that at low speeds, the air resistance grows linearly with velocity:

Your job today is to figure out which of these formulae more accurately fits the data from a simple experiment.

The experiment

 Create a set of objects with the same size and shape, but different mass, by stacking coffee filters: Try using stacks of 2, 4, 6, 8, 10 filters. Write your name and the number of filters on the inner bottom surface of each stack.

 Give the objects what we HOPE will be terminal velocity by having one team member stand on the third floor of the atrium and drop the stacks, one at a time. After a short acceleration, each one will (we hope) reach a constant speed for the majority of its fall.

 Have a second team member stand at the bottom of the atrium and measure the time it takes for each stack to fall from the level of the carpet on the first floor to the bottom of the atrium.

 Make two trials for each stack.

 Calculate the speed of each stack during this final portion of its flight; the distance from atrium floor to first-floor carpet is 4.0 meters. Compare your results to those of other groups. Did you find roughly the same speed for a stack with the same number of filters?

61 The model When an object has reached terminal velocity, the downward pull of gravity exactly balances the upward push of air resistance:

That means that you can calculate the force of air resistance easily, if you know the mass of the falling filters. Now, your job is to determine which of these relationships between the force of air resistance and velocity is a better fit to your measurements.

A good way to see if measurements match a theory is to make a graph on which the theory predicts there should be a straight line:

If the data lie in a line on the graph, then they agree with the theory. The slope of that line must correspond to the symbol m in the equation, and the y-intercept of the graph must correspond to the symbol b. a. If you wanted to test the first theory, in which air resistance goes like velocity, what variable should you put on the x-axis of a graph? What variable would go on the y-axis of the graph? b. Have one person make such a graph. Do your measurements lie in a straight line on this graph? c. If you wanted to test the second theory, in which air resistance goes like velocity-squared, what variable should you put on the x-axis of a graph? What variable would go on the y-axis of the graph? d. Have one person make such a graph. Do your measurements lie in a straight line on this graph?

62 Log-log plots and power laws We have only considered two very simple possibilities, in which the the force of air resistance depends on velocity to the first power, or the second power. It's quite possible that real life may be more complicated: maybe air resistance depends on some fractional power of the velocity, like

When physicists think that one quantity depends on some other quantity raised to a power, they often turn to log-log graphs. Starting with a formula in which velocity goes like some power n of the mass, they take the logarithm of both sides, and then make a graph based on that new equation.

1. Make a graph based on the final form of the equation above; you will need to make a table showing the logarithms of your measurements. 2. Fit a straight line to the data in this log-log graph. What is the equation of the line? What is the value of the power p based on this method? 3. Which of our simple models -- air resistance going like velocity or velocity-squared -- does this method support?

63 Newton's Second Law Problems: We will be working on these problems during the next several classes. Please bring them to each class.

Start all problems with a complete free body diagram.

1. Boxes A and B, with masses mA and mB can move on a horizontal frictionless surface. You push horizontally with a push P on block A. (a) Find the acceleration of the boxes. (b) Find the force between the boxes. Answers A B may contain symbols already given plus any constants like g.(c) Suppose mA = 10 kg and mB = 40 kg, and P = 100 N. What are the numerical answers? (d) Suppose mA = 40 kg and mB = 10 kg, and P = 100 N. What are the numerical answers?

2. A book of mass m is pushed against a smooth frictionless wall by a force P that makes an angle  with the vertical. Find (a) the normal force on the book and (b) its acceleration in terms of P, m, , and constants. (c) Look at limiting cases (large and small angles) to see if it makes sense. P 

3. You are skiing on a hill inclined at  to the horizontal where there is a constant frictional force f. Find the normal force and your acceleration in terms of your mass, m, f, , and constants. Look at limiting cases to see if the answer makes sense. m 4. Two blocks, masses m1 and m2, are connected by an ideal string passing 1 over an ideal pulley. One block is on a horizontal frictionless surface and the other can move vertically and only touches the string. Find the acceleration of each block and the tension in the string in terms of the masses and constants just after the hand is removed from the block. m 2

5. A Ferris Wheel has a radius of 11.0 m and rotates once every 11.0 s. What is T o p the force of the chair seat on you at each of the three positions, top, side, S i d e and bottom? Answer as a multiple of your weight (e.g. 1.4 mg). B o t t o m

6. You whirl a ball of mass 0.40 kg on a string of length 0.90 m. At the point shown the angle of the string from the vertical is 30° and the ball has a speed of 3.50 m/s. (a) Find the tension in the string. (b) What is the acceleration of the ball, tangential and radial components? (c) Now consider the ball at the top of the circle. What is the smallest speed of the ball so that the ball continues in a circle?

64 7. A ball of mass 1.34 kg is attached to a vertical rod by two strings. The top string is 1.60 m long and makes a 30° angle with the rod, while the bottom string makes a 45° angle with the rod. The rod is rotated at 1 revolution every 1.20 s. 3 0 ° Find the tension in each string. 1 . 6 m

4 5 °

8. I have a horizontal table that is very low friction. On it I place a pair of blocks, with m1 on top of m2. Between the blocks there is friction, with static coefficient µs and kinetic coefficient µk I attach a rope to the lower block and pull m 1 with a pull P acting at an angle  above the horizontal. If the pull is small the two blocks move together. If the pull is large they m 2 move relative to each other. (a) For a large pull, find the acceleration of each block. (b) For a small pull, find the common acceleration, and find the actual force of friction.

Try using m1 = 0.600 kg, m2 = 2.400 kg,  = 30.0°, µs = 0.450, µk = 0.350 and for a small pull, P = 5.000 N, large pull P = 20.000 N. What answers do you get? Do they make sense?

9. You are pushing a box uphill the hill has a pitch of 15° above the horizontal, and the static and kinetic friction coefficients between the box and the hill are 0.45 and 0.35. The box has a mass of 40 kg. You push horizontally with a push of 180 N, and the box is initially moving uphill. Find the acceleration of the box.

10. While visiting a friend in San Francisco you decide to drive around the city. You turn a corner and are driving up a steep hill. Suddenly, a small boy runs out on the street chasing a ball. You slam on the brakes and skid to a stop leaving a 50 foot long skid mark on the street. The boy calmly walks away but a policeman watching from the sidewalk walks over and gives you a ticket for speeding. You are still shaking from the experience when he points out that the speed limit on this street is 25 mph. After you recover your wits, you examine the situation more closely. You determine that the street makes an angle of 20o with the horizontal and that the coefficient of static friction between your tires and the street is 0.80. You also find that the coefficient of kinetic friction between your tires and the street is 0.60. Your car's information book tells you that the mass of your car is 1570 kg. You weigh 130 lbs. Witnesses say that the boy had a weight of about 60 lbs. and took 3.0 seconds to cross the 15 foot wide street. Will you fight the ticket in court?

65 Pirate Treasure

Our two drunken pirates have another treasure chest to haul back to their ship. By this time, however, Bill has sobered up and realizes that they need to head due East. Unfortunately, Jack is still swigging rum and pulls 40° North of East with a force 300 N. Bill isn’t quite as strong as Jack and can only manage a force of 250 N. The chest has a mass of 200 kg. Ignoring friction,

(a) find the direction in which Bill must pull in order to keep the chest moving due East (b) What is the acceleration of the chest?

The pirates finally reach the beach and since Jack can now see the ship, both of them begin hauling due East, each with the same force as before. However, the coefficient of kinetic friction between sand and the chest is 0.5. Will the pirates need help from their shipmates to keep the chest moving across the beach at a constant velocity?

Friction Challenge Problems

1. Your shoes have rubber soles. As you stand on an asphalt walkway, the coefficient of static friction between your shoes and the ground is µS = 0.95. A Martian tripod buried directly below you, begins to bore up towards the surface. The ground tilts slowly, so you are able to keep your balance and stand on your two feet…but the angle keeps increasing. At what angle will your feet start to slide along the asphalt?

2. Riddick is 85 kg of pure muscle. As he grabs onto a rope to climb up out of the underground prison, his hands squeeze the rope with a grip of 1000 N. What is the minimum coefficient of static friction between Riddick's hands and the rope, which allows him to hang (by one hand) on the rope?

3 Big block A, of mass mA = 89 kg, sits on a frictionless floor. Little block B, of mass mB = 15 kg, is held against the right side of block A. The coefficient of static friction between the two blocks is µS=0.72. At time t=0, a force F starts to push the big block A. At the same time, the little block B is released. How large must the force F be in order to keep the little block B from sliding down to the floor? If the force is exactly this minimum required magnitude, how long will it take the blocks to slide 50 m across the floor?

67 Testing Newton’s Second Law—Can We Neglect Friction? m 1 Earlier we looked at the modified Atwood’s machine, shown to the right, to get some qualitative feel for Newton’s Second Law. We analyzed this for the case of no friction, an ideal M o d i f i e d A t w o o d ’ s string, and an ideal pulley. Real systems have friction, and M a c h i n e . A s s u m e a n m 2 we must decide experimentally whether friction plays a i d e a l s t r i n g a n d p u l l e y , significant role. b u t i n c l u d e f r i c t i o n b e t w e e n Theory: t h e t a b l e a n d m 1 . Find the acceleration of the system including a constant frictional force, f = µ m1g. Be prepared to turn in your analysis at the start of the period. [Start from free body diagrams and Newton’s Second Law. Do not try to patch up an equation obtained for no friction!]

m 2 g  m 1 g The result you should obtain is a   . m 1  m 2  m 1  m 2  Experiment: Your task is to check the validity of the analysis, and to determine whether friction must be included or not. Based on your measurements you will either be able to say that friction is unimportant within the uncertainties of your experiment, or that it is important and find the coefficient of friction. Along the way you should learn something about interpreting graphs.

Look at the expression for acceleration, it contains some easily varied quantities, m1 and m2, and other quantities that are hard to vary, µ and g. You want the simplest experimental design that will test this equation. First consider the case when m1 >> m2, and rewrite the equation using this approximation. Your graph will have acceleration on the vertical axis, but you need to decide what to plot on the horizontal axis. Possible approaches are

(a) Keep m1 fixed and vary m2, [e.g. m1 = 500 g, m2 = 10, 20, 30, …, 80, 90 g]

(b) Keep m2 fixed and vary m1, [e.g. m2 = 100 g, m1 = 500, 700, 100, 1500, 2000 g]

(c) Keep (m1 + m2) fixed and vary m2 [e.g. m1 + m2 = 500 g, m2 = 10, 20, …, 90 g] Discuss in your groups the advantages of doing each approach and decide on one. Modeling: You can use the theory to predict what you should see. Use a spreadsheet like Excel to calculate the acceleration and make plots. Use the suggested values indicated above with a small coefficient of friction like µ = 0.01. What graphs give a true straight line? What are the meanings of the slope and intercept? Suggested range of values:

Keep m1 greater than 500 g and m2 less than 50 g. As you have seen with earlier labs, you need to consider how many different masses to use and how to get standard deviations on the data collected.

68 Graphing: Plot your data in such a way as to get a straight line. Draw the best-fit line and lines showing the possible range of slopes. What does the slope of this line tell you (compare with theory)? What does the intercept tell you? You will be graded on the quality of your experimental plan, your data, your graphs, and the conclusions you draw from the data.

How do I show the range of possible values? Here is a graph having nothing to do with the data that you will collect, showing data fit with a trendline and including hand drawn lines that indicate the possible range of values. The dashed lines have different slopes and pivot around the center of the data. Data points are within 2 standard deviations (twice the size of an error bar) from the lines. For a brief reference to graphing download the Excel sheet from http://www.rit.edu/~vwlsps/311L_s03/QuickGraphInstructions.xls And for more detail see http://www.rit.edu/~vwlsps/graphing/graphingpart1.html

Figure 1 Motion of a Toy Car y = 4.4237x + 27.37

50 ) m

n 40 o i t i

s 35 o P 30

20 0 2 4 6 Time, t(s)

69 Week #6

70 Planetary Orbits

You will need to use a laptop in this activity, so form two groups of 2 or 3 at each table.

The planets follow slightly elliptical orbits around the Sun. The orbits can be described in terms of the semi-major axis (a) and eccentricity (e), as shown in the diagram (note that the orbit shown is much more elliptical than that of a real planet).

The Table below lists the semi-major axis (in units 106 km) and the orbital period (in years) for each of the solar system planets.

Semi-major axis Orbital Period (T) Planet (a) (106 km) (yr) Mercury 57.9 0.241 Venus 108.2 0.615 Earth 149.6 1.000 Mars 227.9 1.88 Jupiter 778.3 11.86 Saturn 1429 29.46 Uranus 2871 84.1 Neptune 4498 164.86 Pluto* 5915 248.6 * Pluto is no longer officially a “planet”, but we will include it anyway.

Your job is to use the information in the table to analyze the relationship between the semi-major axis (a) and the period (T) of the orbits of the planets.

1. Open Excel and transfer the data above to two columns in the spreadsheet. 2. Use the spreadsheet to create two new columns containing the values of a and T in meters and seconds, respectively (1 year = 3.156107 s). 3. Now, use the LOG10 function to compute the log of both variables in two more columns. 4. Create a graph of log(T) versus log(a), with log(T) on the y-axis. The chart type should be “XY scatter”; do not choose the options that draw lines between the points. (Select the cells containing the data, click on the “Chart Wizard” button and follow the instructions.) 5. Perform a linear fit to the data, using the option to display the fit equation on the graph. (With the chart selected, choose “Add trendline…” from the Chart menu, select “Linear” and click on the “Options” button to display the equation.)

71 Carefully write down the fit equation and consider the following questions.

1. Is a linear function a good fit to the data? 2. Compare the fit equation to the equation for a straight line: y = mx + c. What can you conclude about the relationship between T and a? 3. Assume that the planets are actually undergoing uniform circular motion (not a bad approximation in most cases) in which the radius of the planet’s orbit is equal to the semi-major axis a. Decide what provides the centripetal force and then apply Newton’s 2nd law to derive a general equation relating the orbital period T to the semi-major axis, a. This should involve the mass of the Sun, M and Newton’s gravitational constant, G. 4. Compare your equation to your linear fit to the log T – log a graph. What is the significance of the numerical coefficients of the fit (i.e. m and c)? 5. Use the coefficients of your linear fit to determine the mass of the Sun. Compare your result to the actual mass (you can find this in Appendix C of the textbook). Write out your answers neatly and attach a print-out of your graph.

72 Problems involving gravitation and/or Kepler’s Laws

1. Mars has a mass 0.1075 times the mass of the Earth and a radius 0.53 times that of Earth. If you weigh 150 pounds on Earth, what would you weigh on the surface of Mars? Solve this problem without using numerical values for the mass and radius of the Earth.

2. The distance from the sun to Jupiter is 5.2 A.U. How long does it take Jupiter to orbit the sun? Express your answer in years.

3. Approximate the moon’s motion about the Earth as uniform circular motion. Calculate the magnitude of the centripetal acceleration. What is its direction? The moon’s distance is 384.4 x 103 km and the orbital period is 27.322 days.

Now calculate the magnitude of gravitational acceleration due to the earth at the moon’s distance. What is its direction? The mass of the Earth is 5.98 x 1024 kg and the gravitational constant is G = 6.67 x 10-11 Nm2/kg2.

Comment on your results.

4. Mark lives on an island lying on the Earth’s equator. His gravitational weight is 160 pounds. When he steps on the bathroom scale he is reading his apparent weight. If the Earth spun faster he would appear to weigh less. How long would the “day” be if his apparent weight was half his gravitational weight? The Earth’s radius is 6378 km.

5. A Hohmann transfer ellipse is an orbit used to transfer between two circular orbits. It is a least-energy orbit because once the orbit is established, the gravitational force of the sun will govern the motion of the spacecraft and rocket fuel would not be necessary during the bulk of the trip. Consider the Earth to be on a circular orbit of radius 1 A.U. around the sun. Consider Mars to be on a circular orbit of radius 1.5 A.U. around the sun. The perihelion point on the Hohmann transfer ellipse would lie on the Earth’s orbit and its aphelion point would lie on the orbit of Mars. (a) How long would it take to travel from Earth to Mars on such an orbit? Express your answer in months or years. (b) What is the eccentricity of this ellipse?

6. If the Earth were suddenly stopped in its orbit and released from rest, it would fall into the sun. How long would it take for the Earth to fall into the sun? (Treat the Earth and sun as point masses.)

In this one-dimensional problem the force (and hence the acceleration) is variable. It is not a trivial problem to solve analytically. Can you think of a way to replace this problem with a Kepler’s 3rd Law problem that will give you an answer very close to the solution of the analytical problem?

7. The differential gravitational force is responsible for the tides. To find an expression for the differential force, take the derivative with respect to r of Newton’s Universal Law of Gravitation and solve for dF. Then treat dF and dr as finite differences Δ F and Δ r. What is the ratio of the magnitudes of the differential forces on the Earth due to Jupiter and the moon? Assume that Jupiter is at its closest position to the Earth. When all of the planets line up on the same side of the sun should we expect catastrophic tides?

73 Pulsars

During the Fall of 1967 Jocelyn Bell Burnell, a graduate student at the University of Cambridge, and her mentor Prof. Anthony Hewish, were analyzing data from a newly constructed radio telescope when they noticed a sequence of pulses with a short (1.33 s) but extremely precise period. After ruling out man-made sources, they seriously considered the possibility that they had picked up a signal from an extraterrestrial civilization (legend has it that they nicknamed the first such source LGM – “Little Green Men”). However, it was soon realized that the pulsing sources (by this time they had discovered several more “pulsars”) were in fact the collapsed remnants of massive stars that had been destroyed in supernova explosions. These remnants – “neutron stars” – had been predicted by theories but never before detected. They were expected to be extremely dense, consisting essentially of neutrons crushed together to form something akin to a gigantic atomic nucleus, and also to be rotating extremely rapidly. At the time, the only evidence that astrophysicists had to go on, in making the connection between pulsars (the observed sources) and neutron stars (a theoretical prediction), were the measured pulse periods. But this was all that was needed. Your challenge is to combine ideas of dynamics, circular motion and gravitation to show that if a pulsar’s period is also the rotation period of the source (which we know now, it is), then it must have the density of a neutron star, if it is held together by gravity.

Assume that the pulsar is a spherical star of mass M, radius R and uniform density . It has a solid surface and it is rotating with a period, T. Consider a small chunk of matter (of mass m) sitting on the surface, at the equator of the star.

1. Find the maximum rotation speed for which the lump can remain on the star’s surface, in terms of M, R and G, the constant of gravitation.

74 2. Now find an equation relating the density,  to T, the period corresponding to the maximum rotation speed.

3. The fastest known pulsars have periods of about 1 mili-second. Use your equation to calculate the density of this maximally rotating star. For comparison, the average density of the Earth is 5500 kg m-3, while the density of an atomic nucleus is ~21017 kg m-3.

4. Calculate the weight of 1 teaspoon of pulsar matter on the Earth’s surface. The volume held by a teaspoon is 510-6 m3.

5. A typical neutron star has a mass ~ 1.4 M, but a radius of only 10 km. Calculate the gravitational acceleration (i.e., “little g”) at the surface of a neutron star.

If you are interested, you can find the story of the discovery as told by Prof. Bell Burnell herself here: http://www.bigear.org/vol1no1/burnell.htm

Mass of Earth: 5.971024 kg Radius of Earth: 6.378106 m 30 Mass of Sun (M): 1.9910 kg Radius of Sun: 6.95108 m Gravitational constant, G 6.6710-11 N m2 kg-2

75 Week #7

76 Work and kinetic energy

Introduction

1 2 The kinetic energy (K) of an object of mass m, moving with velocity v, is defined as K  2 m v . net The work done by a net force F (x) in moving the object from an initial position x0 to a final position x is x W   F n et ( x )d x x 0  If the force acts is one dimension (i.e. along the x-axis in this Force W case), the work is just the area under the curve Fnet(x) vs x, Fnet between the limits x0 and x. For a constant force, it is the area  of the shaded rectangle. x x position 0 The work–energy theorum tells us that the work done by the force is equal to the change in the kinetic

energy of the object:W  K  K0 . Another way of saying this is that the final kinetic is equal to the initial kinetic plus the work done: K  K0  W . Speed of a car Now try to use these ideas to answer the following questions.  A car moves along a straight line in the positive x direction. The net force on the car varies as the car moves, and is plotted as a function of position for two different situations in the graphs.

1. On each graph, shade in the area that represents the work between positions A and B.

2. Is the work positive or negative?

3. At which point is the car moving faster (circle one)? Top graph: A B

Bottom graph: A B

Explain your reasoning:

77 A ball of mass of 0.50 kg moves along a straight line and is acted upon by a net force of F = (–6.00 N) + (1.5 N/m) x.

When the ball is located at x = 2.00 m you notice that it has a speed of v = 4.00 m/s.

(a) Sketch the force as a function of the position, paying attention to values at the axes.

(b) What is the ball's kinetic energy at x = 2.00 m?

(c) What is the net work done as the ball moves from 2.00 m to 5.00 m? How can this be indicated on your sketch?

(d) What is the ball's kinetic energy when it reaches x = 5.00 m?

(e) What is the speed of the ball when it reaches x = 5.00 m? F

, e c r o F

Position, x

78 Work from Graphs

Graphs of a spring force as a function of its stretch (with 0 being the relaxed or unstretched length) are shown below for two different springs. For each graph, determine the work done by the spring as it changes length from 1 to 4 m. Shade in the part of the graph that represents the work you calculate.

79 Work Problems

1. Constant force in direction of motion. An iceboat of mass M = 250 kg is moving in the direction of the wind (see diagram below.) The wind exerts a constant force F = 150 N as the boat moves a distance D = 100 m.

(a) What is the work done on the boat during its motion.

(b) If the boat starts at rest and the friction is negligible, what is the final speed of the boat?

2. Constant force not in direction of motion. The same iceboat now moves at an angle  = 45° to the wind as shown in the diagram below and the wind exerts a constant force F = 150 N. The distance traveled by the boat is L=100 m.

(a) What is the work done on the boat during its motion.

(b) If the boat starts at rest and friction is negligible, what is the final speed of the boat?

 = 45°

3. Constant force, varying direction of motion. The iceboat is tethered by a rope to a post so that it can move around an arc of a circle of radius R = 100 m (see diagram over page). The wind exerts a constant force F =150 N on the boat.

(a) What is the work done by the wind on the boat as it rotates through 90°?

(b) What is the work done by the tension in the rope during this motion?

(c) If the boat starts from rest, what is its speed at the end of this motion (again, friction is negligible)?

Hints: it will help to draw a diagram of the situation when the boat has traveled an angle  around the arc.  Now use your diagram to help you figure out the component of the wind in the direction of a small step, ds, in the boat’s path around the arc.  Find an equation for the small amount of work dW done by the wind over the small step ds.  Remember that for a small angle, ds = Rd.

81 4. Multiple forces, varying direction of motion. The situation is the same as in Problem 3, except that now we are going to account for friction. The coefficient of kinetic friction between the ice and the boat’s runners is µk=0.03. (a) What is the work done by friction as the boat swings through 90°? (b) If the boat starts from rest, what is its speed at the end of this motion? Hint: the work done by the wind is the same as in Problem 3.

5. Using the dot product. A particle of dust of mass m = 2 10-3 kg has an initial speed v = 2 m s-1. An electric field inside the lab exerts a force  ˆ ˆ ˆ F 12 i  6 j  8 k (mN) on a small dust particle. The particle moves from initial position  ˆ ˆ ˆ r1 1.2 i  0.3 j  0.6 k (m)  to final position  ˆ ˆ ˆ r2  3.5 i 1.2 j  0.1 k (m)  (a) What is the work done on the dust particle by the electric force? (b) What is the work done on the dust particle by the gravitational force (“down” is direction  ˆj )?  (c) What is the change in the dust particle's kinetic energy? (d) What is the dust particle's final speed?



82 Ski Trails

Three sisters – Alice, Betty and Colleen – go skiing. Each has a mass of m = 50 kg. When they get to the top of the hill, they see three different routes to the bottom.  Route A: a constant slope of 3.81 degrees for 12,027 meters  Route B: a gentle slope of 1.83 degrees for 25,013 meters  Route C: a steep slope of 18.43 degrees for 2527 meters

Alice takes route A, Betty takes route B, Colleen takes route C. Each one starts from rest at the top of the hill and lets gravity do the work.

(a) How much work does gravity do on each sister during the descent? (b) How fast is each sister going when she reaches the bottom? (c) Joe the ski-lift operator is at the top of the hill. He walks to a sheer cliff and drops a ball, which falls straight down to the bottom of the hill. (d) How far does the ball fall? (e) How fast is the ball moving when it reaches the bottom?

The sisters meet at the bottom. “That was fun!” they cry, “let’s do it again!” They go back up to the top of the hill and ski down the same routes as before. Now, however, the snow has melted a bit and become sticky. The coefficient of kinetic friction is µk = 0.02.

(a) How much work does friction do on each sister during the descent? (b) Now how fast is each sister going when she reaches the bottom? (c) How large would the coefficient of kinetic friction have to be to prevent each sister from sliding at all? (d) If the mass of the sisters was NOT 50 kg, which of the above answers would you have to change? Why?

83 Work done in lifting a block

Equipment: Motion sensor, force sensor, LabPro, laptop, mass set, cage. Part 1: Predictions Imagine you are holding a block of mass 2 kg initially at rest. You then pull upwards, lifting the block by 50 or 60 cm, and end with the block again at rest. 1. Sketch your predictions for the position (y), velocity (v) and lifting force (F) versus time. 2. Make a prediction for the change in kinetic energy of the block.

84 Part 2: Experiment Set-up and calibration

 First locate and load the experiment file lift_block. To locate the file go to My Computer > Student Shares on svphy01 > University Physics Students > Team Physics 311 > Lab Pro.  Connect the Labpro to the computer, a motion sensor to Dig/Sonic 1 and a force sensor to CH1. Be sure that the force sensor is switched to the 50 N setting.  Place the motion sensor on the floor with the protective cage over it.  Double-click the file to start LoggerPro.  Calibrate the force sensor. Hold the force sensor vertically by itself and enter 0 N. Then hang a 2 kg mass from the sensor, enter the appropriate force and keep.  Remove the mass and zero the force sensor.  Zero the motion sensor by holding your hand motionless above it, roughly at the height of the tabletop. The motion sensor has a limited range. It will yield spurious results if the object it is sensing comes too close to it, or moves too far away. Click Collect a few times and wave your hand up and down over the sensor to find the critical points. Note the minimum and maximum distance over which the sensor gives accurate results and ensure that you keep within these limits when you take your real data. Beware the Tare: be careful not to push the “Tare” button on the side of the force sensor. If you do, zero the sensor. Measurements Hold the force sensor with the block attached vertically above the motion detector, at about the height of the tabletop. Click Collect, stay at rest for 1 sec, quickly but smoothly raise the block about 40-60 cm, and then hold the block steady at its new height. Don't pull too fast or the force sensor will give noisy results. You may need several trials to obtain good position and force graphs. Analysis When you have obtained some reasonably smooth curves, print out the graphs and compare them with your predictions for position vs time and force vs time.  Were your predicted shapes for the graphs correct? If not, sketch in what you actually observed and explain the difference. Use Statistics on the position-time graph to determine the vertical distance through which you lifted the block. You should obtain a distance and a standard deviation in the distance.  Initial height = ±  Final height = ±  Change in height = ±

85 Now make the force versus position graph active (click in its window). Under Analyze, select Integral to find the work done by your pull as you raised the block.  Work of pull = ±

What are the initial and final kinetic energies of the block in the experiment? The Work-Kinetic Energy Theorum states that: Net work done = Final KE – Initial KE

Is this verified by your experiment, within the uncertainties? Explain:

Going further (if time)

1 Modify the position vs time graph so that it shows both position and velocity as functions of time. a. Explain how the shape of the velocity curve is related to the shape of the position curve. b. Integrate velocity over time. What is the result? How is it related to your experimental measurements? 2 Modify the force vs time graph so that it shows both force and acceleration as functions of time. a. Explain how the shape of the acceleration curve is related to the shape of the force curve. b. How can you make these two curves match? i. Mathematically, what must you do to the force curve? ii. In terms of physical quantities (e.g., mass, ‘g’ etc…), what must you do to the force curve? Note: extra credit will be awarded if you complete this section. You may email the Logger Pro experiment file to yourself to work on it out of class.

Report One main report is required from each team. Hand in this worksheet with your graphs and any additional sheets attached. Each person should hand in any extra credit work individually.

86 The Work done by a Spring

Equipment. Rods, clamps, cart, track, spring, rule, set of standard weights, graph paper. The set-up

A cart of mass M is placed on a ramp tilted at angle θ. It is attached to a spring of force constant k, which is initially at its rest length (the “start” position). We hold the cart in the “start” position for a moment, then release it. The cart rolls down the ramp a distance L before coming to a momentary halt. It then starts back up the ramp, pulled by the spring.

Theory Use concepts of work and the work-kinetic energy theorem to derive an equation for the displacement L, in terms of M, , k and g. Ignore friction for the moment…  How much work is done by gravity as the cart rolls?  How much work is done by the spring as the cart rolls?  What is the cart's kinetic energy just as it reaches its lowest point on the ramp?  Write an equation, which gives the distance L in terms of other quantities.

Prediction Set up a track so that it is tilted at an angle of about 10°. Measure the tilt angle  (use trigonometry). Measure the mass M of the cart. Determine the uncertainties in both these quantities. Now choose a spring.

Your job:  Determine the spring constant, k, and the uncertainty (just as you did in week 5).

87  Use the values of M, k and θ in your equation to predict the displacement down the ramp, L, for your spring. Calculate the uncertainty in L. Measurements Once you have made your prediction and recorded it on page 3, measure the actual value of L for your spring. Release the cart at the rest length of the spring and measure the displacement when the cart comes momentarily to rest at its lowest point. Repeat this measurement several times to get an average and a standard deviation. Test your prediction Write down your data and results below

Equation for L:

Mass, M: ±

Angle, : ±

Spring constant, k: ±

Prediction for L: ±

Measured L: ±

If your prediction does not match the measurement, can you think of a reason why? What could you do to make your prediction more accurate?

Report Each team should hand in this worksheet and attach a page showing the derivation of the equation for L (neatly set out, please). Make a note of your predicted and measured values for L and keep it.

88 Week #8

89 Conservative and Non-Conservative Forces

1. Force of gravity: Motion in a Vertical Plane

Point A is at your feet, point B is 2 m vertically up and 1.5 m horizontally from the first point. Three paths between point A and point B are shown.

You move a ball from A to B along each of the paths. The ball has a mass of 4.0 kg and is motionless at the start and at the end. (a) What is the net work done on the ball in each for each of the three paths? (b) Determine the work done by the force of gravity for Path 1. (c) Determine the work done by the force of gravity for Path 2. (d) Determine the work done by the force of gravity for the diagonal line, Path 3. (e) Are your answers the same or different for the three paths? (f) Does the work of gravity equal the net work? Why or why not?

90 2. Friction Force: Motion on a Horizontal Plane Now consider the same diagram for a book on a horizontal table where there is a frictional force of 5 N on the book. B is located 2 m north and 1.5 m east of A. The book is at rest at the start and end of its motion.

(a) What is the net work done on the book for each of the three paths? (b) Determine the work done by the force of friction for Path 1. (c) Determine the work done by the force of friction for Path 2. (d) Determine the work done by the force of friction for the diagonal line, Path 3 . (e) Are your answers the same or different for the three paths? (f) Does the work of friction equal the net work? Why or why not? Which of these forces is "Conservative" and which is "Non-Conservative"? Why?

91 Using Conservation of Energy: Pendulum

Consider the following pendulum set-up. A ball of mass m is attached to a string of length L, and is pulled so that the ball is an initial height h0 from the lowest point in its swing. A rod is clamped at a height h above the lowest point so that when the pendulum swings down, the string wraps around the rod, and the ball swings around in a circle of smaller radius.

Your challenge:

Find the smallest height h0 from which the ball can be released in order that it complete the small circle without the string going slack. [Hint: you will need to consider more than just energy!]

Checking your answer: You will be provided a simple pendulum and can measure its length L. Set up the experiment so that the rod is at a height h = L/4. Does your predicted value succeed? Is your predicted value the "minimum"? Try decreasing the height by 5% and seeing if the ball goes around or if the string goes slack. In what ways does your actual experimental set up fall short of ideal?

92 Roller Coaster

A roller coaster car of total mass m =500 kg (including passengers) moves along the track shown the diagram. The track is frictionless and the car has a velocity of magnitude v1=2 m/s at the top of the first hill and h = 50 m. We will take the gravitational potential energy of the system to be zero at point C.

Answer the following questions, using the idea of conservation of mechanical energy. Do not just write down the numerical answer — first derive and write down an equation in terms of m, g, h and v1, then insert the numerical values.

(a) What is the total mechanical energy of the car at point A?

(b) What is the gravitational potential energy (GPE) of the car at point B?

(c) What is the speed of the car at point C?

(d) The car will not reach the top of the last hill. How high will it go?

Adapted from Cummings et al., Understanding Physics, Wiley 2006. Chapter 10 Problem 4.

Energy Problems

1. A cart of mass 1.8 kg rolls without friction on the track shown. At point A the cart is moving at 6.0 m/s. Find the speed of the car at points B and C.

2. The same track is used, but now there is friction acting. When the cart reaches point B it has a speed of 1.40 m/s. What work was done by friction? 3. A mass m = 0.60 kg is attached to a spring of constant k = 75 N/m and the system is hung vertically. Initially the mass is pulled down, stretching the spring by x 0 = 12 cm, and you throw the mass with an initial velocity v 0 = 1.8 m/s. (a) Find the speed of the spring when it returns to its relaxed state. (b) Find the lowest and highest points that the mass reaches. 4. A block of mass m = 3.0 kg is attached to a spring of constant k = 240 N/m. It can slide along a board tilted at an angle theta = 37° to the horizontal. Initially the block is released from rest when the spring is compressed by x 0 = 30 cm. (a) Find the speed of the block when the spring is relaxed again. (b) Find the highest point that the block reaches. 5. You decide to take the plunge and go bungee jumping. Being a college student, you do this on the cheap, and buy a bungee cord with a spring constant of k = 500 N/m and a length of L = 8.0 m. You choose a bridge where you tie the end of the cord so you can jump off. Assume that you step off the bridge with zero speed. How high must the bridge be so that you can do this fun jump more than once? (Being nervous, you want to be extra safe. What should you assume for air resistance?) 6. A spring is attached to a ceiling, and has a relaxed length of 25 cm. When a mass m = 0.80 kg is attached to the spring it stretches to an equilibrium length of L0 = 34 cm. (a) Find the spring constant of the spring. (b) I lift the mass until the spring returns to its relaxed length, and then release it. When the mass returns to the equilibrium length, what is its speed? (c) After I release the mass and it falls, what is the length of the spring when the mass reaches its lowest point?

94 7. Two blocks are connected by an ideal string passing over an ideal pulley. One block rests on a track tilted at an angle of theta to the horizontal, but the friction there is negligible. The system is released from rest. When the second block falls a distance d , find the speeds of the blocks. Use the ideas of energy!

8. A flexible chain has a length L . Most of it is on a horizontal table, but a length x is hanging off the edge. Assume no friction, and that the chain starts at rest. What is its speed when the chain just leaves the table? 9. In a new exciting ride, a roller-coaster car rolls to the bottom of a track, up a ramp, and then leaves the track. Riders are issued parachutes and wished "good luck." Suppose that the car has a speed of 9.0 m/s at the crest of a hill 8.0 m above the ground. The ramp exit is at a height of 1.0 m above the ground, and makes an angle of 30° with the horizontal. Assume that there is no air resistance or friction acting. Use ideas of energy to find (a) The speed of the car as it leaves the ramp. (b) The speed of the car when it reaches maximum height after leaving the ramp. (c) The maximum height of the cart above the ground after leaving the ramp. 10. A ball of mass m is attached to a string of length L . When the ball is at its lowest point it is given a velocity v 0, and it just barely makes a complete revolution. What must be the value of v 0? Answer in terms of m , g , and L . [Hint: you will need to use both the ideas of energy and the ideas of Newton here.]

11. A ball of mass m is attached to a string of length L . At the lowest point the ball has a speed v 0. For the moment assume that the ball is moving in a vertical circle. (a) Find the speed of the ball as a function of the angle from the vertical. (b) Find the tension as a function of the angle.

min (c) Use the result of part (b) to determine the minimum speed v 0 that allows the ball to complete the circle.

min min (d) Suppose that v 0 < v 0 , specifically v 0 = 0.8 v 0 . At what angle will the ball leave the circle? (e) After the ball leaves the circle, sketch the path of the ball until the string again becomes taut.

95 12. A foam ball of mass m is attached to a very light rod of length L (to center of ball). The rod is held horizontally, and then the combination is released. When the ball reached the lowest point, the tension in the rod is T = 2.5 m g . How much work was done by air resistance? The answer is in symbols, and only m , L , g and constants are allowed in the answer. The radius of the ball is small compared to L (not as pictured.)

13. Consider a hemisphere of ice of some unknown radius R. Your friend, of unknown mass sits at the top of the hemisphere, and a small gust of wind starts him (her) moving. At what angle, measured from the vertical, does your friend fly off the ice? BONUS question: Where do they land? 14. Two identical springs have constants of 80 N/m and are 30 cm long when relaxed (unstretched). As shown in the diagram, one end of each spring is attached to a small air hockey puck of mass 0.60 kg that can move on a frictionless table. The springs are stretched and connected to posts that are 80 cm apart. The puck is pulled 30 cm to the right and released. Find the speed of the puck when it returns to the center.

96 Neutron Star

Stars about 10 more massive than the Sun live fast and die young (after only a few million years) in massive explosions known as supernovae. During such events the outer layers of the star are blasted off into space (forming supernova remnants like the crab nebula). However, the core of the star collapses under its own weight until it is crushed into a ball of neutrons – a neutron star. Neutron stars are the densest form of matter known. A typical neutron star has a mass, MN, about 2M, the mass of the Sun 30 (M = 1.9910 kg), but a radius of only RN=10 km.

Suppose that you could drop a baseball onto the surface of a neutron star. The baseball is initially at rest and you release it at a distance >> RN. There are no non-conservative forces involved. The mass of the baseball is m=0.15 kg.

Answer the following questions, using the principles of conservation of mechanical energy. Do not just write down the numerical answer — first derive and write down an equation in terms of MN, RN, m, G and the final velocity, v, then insert the numerical values.

(a) What is the total mechanical energy of the baseball just before it is released?

(b) What is kinetic energy of the baseball, just before it hits the surface of the neutron star? (For reference the Hiroshima atomic bomb released 8.371013 J of energy.)

(c) What is the speed of the baseball just before it hits the surface (in km/s)?

(d) What is its speed as a percentage of the speed of light (c = 2.998108 m s-1)?

97 Escape velocity

The minimum initial velocity needed for an object (such as a rocket) to escape to infinity from the surface of a massive body (such as a planet) is given by

2 G M v  e s c R where G= 6.6710-11 m3 kg s-2 is the constant of gravitation, M is the mass of the body and R is its radius. Note that v only depends on the properties (mass and radius) of the massive body (not the  esc escaping object). Also notice that for a given mass, M, vesc is higher the more compact the body (smaller R).

Lets examine this by imagining we can compress the Sun into a smaller and smaller volume. The mass 30 of the Sun is M = 1.9910 kg. Calculate the escape speed in each of the following situations, and compare it with the speed of light (c=3.0108 m s-1).

5 (a) The normal-size Sun, R =6.9510 km.

R≈7x105 km

(b) The Sun when it is about the size of the Earth, R=6.95103 km (the Sun will actually reach this size when it ends its life as a white R≈7000 km dwarf).

(c) The Sun if it was about the size of a neutron star, R=10 km. (The Sun will R≈10 km never actually get this small.)

(d) The Sun if it had a radius R=3 km.

R≈3 km

(e) Now suppose you could squash it just a little more. How would vesc compare with the speed of light? What consequence(s) would follow from this?

98 Conservation of Energy as a Mass on a Spring Oscillates

The goals  Find the effective mass of the spring  Find the spring constant of the spring  Find the "phase difference" during oscillation between a) force and displacement b) force and acceleration c) force and velocity d) kinetic energy and spring potential energy  Check to see if total energy is constant over several full cycles  Find the "time constant" of energy loss over a period of three minutes Optional:  Investigate the effect of air resistance on energy loss.

Setting up the experiment a) Set the slide switch on the force probe to 10 N. Place the motion sensor on the floor. Attach the force sensor to CH 1, and the motion sensor to Dig/Sonic 1.

b) Go to "Student Shares" -> "University Physics" -> "Team Physics 311" -> "Lab Pro", Drag the file Conservation_of_Energy (note the underscores) onto your local desktop. Double-click on Conservation_of_Energy to start LabPro.

c) Calibrate the force sensor.

d) Choose one of the tiny springs. Attach the aluminum mass hanger and add a 50 g mass for a total hanging mass of 100 g. The mass should be about 15 or 20 cm below the bench top at equilibrium. Zero both sensors with the block at rest.

e) Check your set-up by lifting the mass straight up about 5 cm and releasing it. Avoid side-to-side motion. What do you expect to see for position as a function of time? Collect data; do you see

99 what you expected? If not, adjust the sensors until you get what you expect.

f) You should see the position oscillate around 0 meters (to within about +/- 0.001 m), and the force oscillate around 0 Newtons (to within about +/- 0.01 N). The amplitude of the motion should be around 5–10 cm. Look carefully; if the centers of oscillation aren't zero, you may have to adjust the readings manually to force them to zero. Adjust to zero manually: Return the system to equilibrium and zero all sensors. Collect data with the system at rest. Use Analyze:Statistics to determine the average position of the mass. Click on the oval Labpro icon to bring up the Sensors Window, and click on the motion sensor. On the pop-up menu, click Set Offset. A box will pop up with an initial value of the offset. Subtract the average position from the offset and enter this difference as the new offset, then click OK. (E.g. if the average position is 0.0123 and the initial offset is -0.612345, the new offset is -0.624645.) Collect data again and the position should have an average very close to zero. Close the Sensors window. Each time you zero the motion detector you will need to repeat this.

Use Analyze Data -> Examine to look at the highest and lowest distance the mass reaches. If these two extremes are not the same size, make appropriate adjustments and take new data. If sometimes helps to wait one minute after starting the block in motion before you collect data.

Be absolutely certain that you have the equipment properly zeroed or your results will be bogus!

Collecting a first dataset

Start the mass moving and let it oscillate for about twenty or thirty seconds; higher order vibrations will damp out during this time, leaving very smooth, simple, harmonic motion. Press Collect to gather data. You should see nice smooth sine wave shapes for position and velocity. Before you go any further, ask an instructor to verify that your data is good enough. Look at the sample posted in class.

Analysis over a few cycles

You can (and should) print out several graphs to illustrate your report. If you want to plot more than one measurement on the vertical axis, click the name, click More on the pop up menu, and check the boxes for the additional quantities you wish to plot.

1. Compare the Force and Position graphs, and give a verbal description of their relation. Are force and displacement "exactly in phase", "exactly out of phase", or something else? Explain.

2. Compare the Force and Acceleration graphs, and give a verbal description of their relation. Are force and acceleration "exactly in phase", "exactly out of phase", or something else? Explain.

3. Compare the Force and Velocity graphs, and give a verbal description of their relation. Are force and velocity "exactly in phase", "exactly out of phase", or something else? Explain.

4. Find the spring constant k by making a graph of one particular quantity versus another. Print the graph. On your printed graph, make a linear fit to the graph and determine the slope and its uncertainty. (Hint 1: the left-hand side should be "Force", in Newtons). (Hint 2: the force constant is a

100 positive value)

5. Find the "total mass" m of the moving system by making a graph of one particular quantity versus another. Again print the graph and make a linear fit. (Hint the left-hand side should again be "Force", in Newtons). How does this "total mass" compare to the mass of the block plus the hanger? If they are not the same, explain how big the difference is, and explain where the difference ("extra" mass or "missing" mass) originates.

6. Create formulas within LabPro for kinetic energy (KE) and potential energy (U).

KE = (1/2)mv2 U = (1/2)ky2

To create a new column for kinetic energy, go to Data: New Calculated Columns and enter values for the name (e.g. Kinetic Energy), short name (e.g. KE), and units (J). In the Equation box, enter the formula for kinetic energy. You should know the effective mass from part 5 (suppose you found 0.999 kg), and can get the velocity from the Variablespull down menu. The equation should look something like

0.5 * 0.999 * "velocity"^2

7. In a similar fashion, define the potential energy, U, and the total mechanical energy, E = KE + U. If you need to change a definition, select the column from Data:Column Options and an editing widow will appear.

8. Make graphs that allow you to discuss conservation of energy. Examples of graphs might be any one of the energies versus time (or position, or velocity, or acceleration), one energy versus another energy, etc. You need to try different graphs and decide what they tell you about energy conservation.

9. Is energy conserved during your measurements? There are two ways to answer this question: a. what is the "peak-to-peak" change in the total energy, expressed as a fraction of the average total energy? You should be able to state something like "total energy bounces around its mean by around +/- 2 percent." b. what is the secular change in the average total energy during the period you measured? You might compare the average energy during the first second of data to the average energy during the final second of measurements.

Analysis over a very long period

101 You ought to find that total mechanical energy is almost exactly the same over a few cycles of oscillation. But what if we monitor the system over many, many cycles? Change the Experiment ->Data Collection parameters so that you measure 10 samples per second over a period of 180 seconds (yes, that's about 3 minutes).

10. Reset the system to equilibrium and re-zero the sensors

11. Pull the weight down about 5 cm and release it carefully. Wait about 30 seconds to let the wiggles dissipate.

12. Collect measurements over the long period. While you are waiting for this to finish, you might perform calculations for the previous portions of the experiment, or start writing your report.

13. Look at (and print) a graph of Total Energy versus Time. Is energy conserved over this long period?

14. What is the "half-life" of energy? That is, how long does it take for the total energy to decay to half its original value?

15. What is the "half-life" of displacement? That is, how long does it take for the displacement to decay to half its original value? Is this the same as the "half-life" of energy? Should it be?

16. Fit the Total Energy versus Time graph with an exponential curve. Record the parameters displayed by LabPro. Re-write the equation so that it looks like this:

-t/τ E = Eo e

where t is the time in seconds and E0 the initial energy of the system. What is the time constant  of the system?

17. How much energy would be left if the spring continued to oscillate for twenty minutes?

102 Bonus section (do this if you finish everything else before the end of class)

Some of the energy lost by the spring and hanger is due to work performed by the force of air resistance. Recall that this force depends on several parameters, one of which is the cross-section area A of the moving object.

18. What is the cross-section area of your moving object? Express the area in square centimeters.

19. Create a "sail" by cutting a piece of paper so that it has a larger area -- twice the original A of your object. Tape this sail onto the bottom of the hanger so that the moving object now has double its original area. Repeat the Analysis over a long period procedure (steps 10 to 17 above). Determine the time constant for this modified object.

20. Now make the "sail" three times the original area A. Determine the time constant again.

21. (If you have time, do it again for a sail four times the original area)

22. In theory, how should the amount of energy lost to air resistance depend on the area of the object?

23. In practice, how does the amount of energy lost to air resistance depend on the area of the object?

24. Discuss.

103 Week #9

104 Center of Mass of a Group of Particles

At first sight, calculating the motion of an extended object (such as a javelin) or a group of particles (such as debris from an explosion) seems complicated. But we can make life easier by following only the motion of the center of mass of the system. Thus, we can determine the motion of a javelin by applying Newton’s Laws to its center of mass, rather than calculating the motion of each individual point.

In 2-dimensions, the center of mass of a group of N point masses (particles) is  ˆ ˆ R CoM  XCoM i YCoM j

N N N N  X C o M   x i m i  m i and Y C o M   y i m i  m i i 1 i 1 i 1 i 1

are the x- and y- components of the center of mass, respectively. Even if you are not actually dealing with particles, it is often possible to treat objects as if they are point  masses, if they are uniform and symmetric (e.g., its obvious that the center of mass of a uniform disk must be at its center), or if you are trying to find the CoM of objects which are small compared to their distance apart (e.g., Earth and Moon).

OK, so here’s the Earth and Moon. Where is the CoM? You will find the relevant data in the Appendix in your textbook. Take the origin of the x-axis to be the center of the Earth. Is the CoM inside or outside the Earth?

Earth Moon

105 The objects have masses mA = 5 kg, mB = 2 kg, mC = 4 kg, what is the position vector of their center of mass?

106 A circle of sheet metal of uniform density and radius R has a smaller circle of radius, R/2, cut out so that it just touches the edge of the big circle. Where is the center of mass of the remaining metal? (Hint: you can treat the two circles as point particles; if the cut-out piece was replaced, the CoM of the two “particles” would be the center of the larger circle, which you can take as the origin.)

107 Center of Mass of 2-D and 3-D Objects

Sometimes we cannot compute the center of mass of a system by treating its components as point masses. In these cases, we cannot avoid calculus. We can still write the center of mass vector (in 2-d) as  ˆ ˆ R CoM  XCoM i YCoM j

but now the x- and y- components of the center of mass are given by integrals over volume: 1 1  X  x d V and Y  y d V C o M M  C o M M  where M = total mass of the object,  = density. For regular shapes it is usually possible to identify a plane of symmetry that allows you to convert the integral over volume to one over just a single co-ordinate.   Uniform triangle

The triangle has a uniform density .

(a) Draw the axis of symmetry for the triangle.

(b) Where should you place your origin of co-ordinates? Draw co-ordinate axes with x horizontal and y vertical.

(c) Use symmetry to find the y co-ordinate of the center of mass. What is it?

108 To find the x-co-ordinate:

(d) You only need to find the XCoM for one half of the triangle. Explain why.

(e) What is the total mass of one of these right triangles, in terms of , L and H?

(f) Write down the equation for y(x) for the top right triangle.

(g) For this 2-d object, the XCoM co-ordinate is given by an integral over area. Divide the top right triangle into small vertical strips of width dx. Write down an equation for the area, dA, of these strips.

1 (h) Now use your answers to (d) and (f) to solve the integral X  x d A . C o M M 



109 Uniform cone

Now let’s try a 3-d object. The cone has a uniform density . Decide where to place your origin of coordinates. Let z be the vertical axis.

It is easy to determine XCoM and YCoM from symmetry, but to get ZCoM you need to evaluate the integral 1 Z  z d V . C o M M 

Luckily this can be reduced to an integral over z using the same kind of procedure you used for the triangle. It will help to remember that the area of a circle of radius x is is πx2. Also, the volume of the  1 2 cone is V  3 R H .



110 Impulse in Real Collisions

We have discussed Conservation of Momentum when a system is isolated so that the Net External Force is zero. This is rarely true in the real world. Here you can determine how to deal with such real situations.

One ball collides with another ball in the air above your head. The system will be the two balls, and the collision is one dimensional. Two forces that act on the top ball are shown in the graph below, • an external force of gravity, constant downward -2.0 N and • an internal collision force approximated as a triangle with a peak value of 1000 N that acts from 5.00 to 5.01 sec. (The sketch is NOT to scale!)

0 2 4 6 8 10 t (s)

What is the impulse of each force for the time intervals shown in the table?

Time Interval (s) Impulse of Collision force (N.s) Impulse of gravitational force (N.s) 5.00 – 5.01 4.00 – 6.00 0.00 – 10.00

Under what conditions can we neglect the impulse of the non-zero external force?

111 Impulse and Forces on Falling People

George Benedik and Felix Villars, in Physics With Illustrative Examples From Medicine And Biology, Volume 1, Addison Wesley, 1973, report that the threshold for human survival is when the pressure on impact is less than 50 pounds/square inch. If the fall is onto a person's back of area 3.8 square feet, the threshold force for death is 27,000 pounds = 1.2 x 105 N. In a fall from high altitude, the change in momentum is fixed by the terminal velocity before impact, about 120 mph, and the velocity of 0 after what we hope is not "Terminal" impact.

For the situations described below, find the average force and decide if the situations satisfy this criterion for survival?

"During one of the battalion drops, from 1200 feet on a clear, relatively warm day, an observer noted what appeared to be an unsupported bundle falling from one of the C-119 airplanes; no chute deployed from the object. The impact looked like a mortar round exploding in the snow. When the aidmen reached the spot they found a young ... paratrooper flat on his back at the bottom of a 3 1/2 foot crater in the snow, which consisted of alternating layers of soft snow and frozen crust. He could talk and did not appear injured; nevertheless, he was air evacuated to a hospital. His only injuries were an incomplete fracture of a clavicle, a chip fracture of L-2, and a few bruises. He was released from the hospital in time to return south with his unit." (Alaska, 1955)

"[O]n the frigid 23rd of March, 1944, ... Flight Sergeant Nicholas Alkemade, an RAF rear gunner [had his bomber set afire] by a German night fighter on a raid over Hamburg. [H]e found that he was unable to reach his parachute, stowed forward in the flaming fuselage. Deciding he didn't care to burn alive, he jumped without a parachute just as the aircraft exploded above him. His altitude was 18, 000 feet. Falling at a terminal velocity of about 120 mph during this 3 1/2 mile fall (which lasted about 90 seconds), he struck the snowy branches of a pine forest and then landed in less than 18 inches of snow, only twenty yards from the bare open ground. Incredibly, his only reported injuries consisted of superficial scratches and bruises, and burns received prior to the jump." From "Terminal Velocity Impacts Into Snow", R. G. Snyder. Military Medicine 131, 1290-1298 (1966).

112 Impulse in a Collision

In this experiment, you will measure forces exerted on a cart during the short time interval it takes to bounce off an obstacle. Your job is to figure out the pre- and post-collision velocities of the cart, calculate the change in its momentum, and compare that to the impulse. Equipment. Laptop, track, cart+rubber bumper, force sensor, LabPro, mass bars, massive object.

Set up

Start up the laptop, go to My Computer > Student Shares on svphy01 > University Physics Students > Team Physics 311 > Lab Pro and copy the cart_impulse file to the desktop. Double-click it to start LoggerPro.  Connect the force sensor to the CH1 port on the LoggerPro interface.  Calibrate the force sensor on its +/- 50 N setting, and zero it. Now, set up the track so that one end is slightly higher, at an angle of 3–4°. Sketch the setup and record your calculations for the angle. Remove the hook from the force sensor and replace it with the rubber bumper. Place the force sensor at the lower end of the ramp. Put two mass bars or other heavy weights into the force sensor to hold it in place (you might even tape it into place on the track). Make sure that a heavy item sits next to the bottom end of the track to prevent the track from sliding across the table.

Cart with rubber bumper Force sensor with rubber bumper+mass bars

h Heavy object

 Calculate the uncertainty in the tilt angle Now arrange the cart so that its spring-loaded bumper is extended and will strike the force sensor's bumper head-on when it rolls down the track  Measure and record the mass of your cart Place the cart so that the end of the bumper is 45 cm from the sensor. This is your starting point, from which you will release the cart.  Use your expertise in 1-d kinematics to calculate the speed of the cart at the instant it bumps into the force sensor (ignore friction). This is the initial velocity. Record this value. (g = 9.8 m s-2).

113 Part 1: Predict the impulse

Minimize the LoggerPro window — we are not going to use the sensor yet. Now measure the distance the cart “recoils” back up the ramp after it collides with the force sensor. Release the cart from its starting point. The cart should roll down, collide with the force sensor, bounce off and roll back up the ramp. Watch carefully and note where it comes momentarily to rest (i.e. the measure the maximum distance it recoils from the force sensor).  Make three trial runs releasing the cart from the same starting point and neatly record your results in a table.  Using the measured recoil distance, calculate the recoil velocity of the cart just after the collision (ignoring friction). This is the final velocity; calculate its value for each trial.  Draw a series of three pictures showing the cart and sensor just before, during, and just after they collide  Using the initial and final velocities, determine the change in the cart's momentum for each trial. Be careful to get all your signs correct.  Using the average change in momentum, predict the impulse given to the cart by the bumper.  Estimate the uncertainty in the distance the cart rolled down the ramp.  Estimate the uncertainty in the recoil distance from the spread in the measurements for the three trials.  Combine the uncertainties on the distance with the uncertainty in the tilt angle to compute the uncertainties in the cart's initial and final velocities.  Use those uncertainties to compute the uncertainty in the change of momentum of the cart.

When you have reached this point, get an instructor to check your calculations.

Part 2: Measure the impulse

Maximize the LoggerPro window. Place the cart at the same starting point you used earlier. Press the Collect button on LoggerPro and release the cart. The cart will roll down, bounce off the bumper, and recoil back up the ramp. Measure the final position of the cart and record it. Check that it is consistent with the previous trials. Examine the force vs time graph in LoggerPro. It should read zero, except for the fraction of a second when the collision took place. Zooming in on that small time interval, the trace might look something like this:

(see next page)

114  Use LabPro's Integrate function to find the impulse by integrating the area under the force-time curve. Select just the portion of the data around the impact. Print out your graph and attach it to your report.

Compare the impulse you measured from the force versus time graph to your prediction (based on the cart’s velocity before and after the collision). Do the computed impulse values agree with the measured impulse value(s) within the uncertainty?

Report

One report is required form each team. It should include:

 Cover page with the names of all members who participated in the activity

 A diagram of your setup, with your calculation for the tilt angle and its uncertainty.

 Your calculation for the initial velocity.

 Your results for the recoil distance trials and calculation for the final velocity.

 Your calculation of the average change in momentum (predicted impulse).

 Your uncertainty calculations for the predicted impulse.

 Your force vs time graph and the value of the impulse determined from this graph.

 A brief statement of your conclusion.

115 Impulse on a tennis ball

A tennis ball of mass m = 0.03 kg flies towards a wall with velocity v = 20 m s-1 west. It hits the wall, stops momentarily, and then bounces back with the same speed in the opposite direction.

(a) What is the ball’s initial momentum?

  v v

m m m   Before collision During collision After collision

(b) What ball’s final momentum?

(c) What is the impulse?

(d) The ball is in contact with the wall for t=0.03 s. What is the average force on the ball during the collision?

Car-truck collisions

116 Momentum turns out to be enormously useful in solving problems that involve collisions, even when you don't know the details. Why? Because under very common circumstances, the total momentum of a set of colliding objects will be the same before and after they collide. In other words, momentum is conserved.

So, considering just 1-D motion for now, we can say

  m A,1vA,1 mB,1vB,1  mX,1vX,1  mA,2vA,2  mB,2vB,2   mX,2vX,2

where subscripts A, B, C etc label different objects and 1 and 2 refer to times immediately before and immediately after the collision respectively.  Now consider the following cases:

1. Car crashes into stationary truck, they stick together

v v 2 1

Before After

-1 The car has mass m = 2000 kg and velocity v1 = 20 m s . The truck is initially stationary and has mass M =8000 kg.

What is the velocity, v2, of the tangled wreckage immediately after the collision?

Is this an elastic, or inelastic collision (is kinetic energy conserved)?

117 2. Car rolls into stationary truck, and bounces off

v v v 2 T,2 1

Before After

-1 The car has mass m = 2000 kg and velocity v1 = 5 m s . The truck is initially stationary and has mass M =15000 kg. The car rolls slowly into the truck and bounces off with velocity v2. The truck rolls forward with velocity vT,2.

What are the velocities of the car, v2, and truck vT,2, immediately after the collision?

Can you solve this problem using conservation of momentum alone?

If not, what if I tell you that the collision is elastic (kinetic energy is conserved). Can you solve it now?

118 Hockey puck – brick collision

Conservation of momentum also works in two or more dimensions. But now we equate the vector sums of the momenta before and after the collision.   p (t )  p (t ) sys 1 sys 2

Essentially this boils down to resolving the velocity vectors into components and applying conservation of momentum separately to each direction.  1. Strike 1. v 2

Before After v 1   m M V 2

A hockey puck of mass m = 2 kg slides due East at v1 = 15 m/s. It bumps into a stationary brick of mass M = 5 kg. Afterwards, the puck slides away to the northeast with velocity v2 = 8 m/s at θ = 10° degrees, and the brick slides to the southeast at V2 at  = 11 degrees,

What is the final speed of the brick?

Is this an elastic collision (does it conserve kinetic energy)?

119 2. Strike 2.

A hockey puck of mass m = 2 kg slides due East at v1 = 15 m/s. It bumps into the brick as before. -1 Afterwards, the puck slides away to the northeast with velocity v2 = 10 m s at θ = 30°, and the brick slides to the southeast at V2 at angle .

What is the final velocity of the brick (we need a direction here)?

Is this an elastic collision (does it conserve kinetic energy)?

120 Energy and Conservation of Momentum

Suppose I have a system of two pucks on an air hockey table, a 5.00 kg block puck initially moving at 3.00 m/s to the right toward a 3.00 kg puck moving initially to the left at 1.00 m/s. Consider different situations with the pucks having the same mass and initial velocities in each situation, but with the pucks being made of different materials in each case. (a) After the collision the velocity of the 5 kg puck is 1.50 m/s to the right. (b) After the collision the velocity of the 5 kg puck is 0.90 m/s to the right. (c) After the collision the velocity of the 5 kg puck is stationary. (d) After the collision the velocity of the 5 kg puck is 1.50 m/s to the left. Find the final velocity of the 3 kg ball and complete the table below.

Case Initial v of 5 Final v of 5 Initial v of 3 Final v of 3 Total Total Final kg kg kg kg Initial KE KE (J) (J) (a)

Collisions where the total kinetic energy before and after the collision are equal are called “elastic collisions.” Are any of your collisions elastic? Collisions that are not elastic are called “inelastic.”

Collisions in Two Dimensions The same two pucks are used but now after the collision the 5.00 kg puck is deflected and leaves with a ˆ ˆ velocity of v 5 f  2 . 7 0 i  0 . 7 2 j m / s . Using Conservation of Momentum, the final velocity of the 3.00  kg puck is found to be v  0 . 5 1 iˆ  1 . 2 1 jˆ m / s . Is this collision elastic or inelastic? Justify your 3 f   answer.

121 Problems Using Momentum and Impulse 1. A one dimensional collision occurs between a cart of mass 11.0 kg moving to the right at 3.0 m/s and a block of mass 5.0 kg moving B e f o r e C o l l i s i o n to the left at 12.0 m/s. After the collision, the block moves to the right at 4.0 m/s. (a) What is the velocity of the cart after the collision? (b) If the collision lasts 0.020 s, find the average force on the cart. (c) Find the average force on the block.

2. A board is standing on the floor, and two different experiments are done. In one a piece of clay is thrown toward the top of the board, it collides and sticks to the board. In the other experiment, a superball is thrown that bounces off the top of the board. The masses of the board, the clay and the ball are the same, and the initial velocities are the same. In one experiment the board falls over, and in the other it wobbles but stays standing. Which object, clay or ball, made the board fall over? Explain.

3. Two pucks move on a horizontal air-hockey table. One puck has mass 3.0 kg and 3 initially moves east at 4.0 m/s. The other puck has a mass of 6.0 kg and moves at 5.0 m/s at 60° north of east, as shown. The two pucks collide and stick together. (a) Find the velocity after the collision of the combined pucks. 6 0 ° (b) If the collision lasts 35 ms, find the average force on the 3.0 kg puck. 6

4 The same two pucks as in problem 3 now collide and separate. After the collision the 3.0 kg puck moves north at 1.0 m/s. (a) Find the velocity of the 6.0 kg puck after the collision. (b) If the collision lasts 35 ms, find the average force on the 3.0 kg puck.

5. A ball with mass 2.5 kg is moving in outer space with a velocity of 6.0 m/s horizontally, and a box of mass 4.5 kg is moving with a velocity of 4.0 m/s at an angle of 120° from the horizontal. The two collide and stick together. Find the final velocity of the pair.

6. Consider a rail car of mass 500 kg coasting along a horizontal frictionless track at 3.0 m/s. I drop a 100 kg box from a height of 60 cm into the car. (a) Find the final velocity of the car+box. (b) If the collision between the box and the car lasts 12 ms, find the average normal force that acts on the box during the collision.

122 Week #10

123 Ballistic Pendulum – Part I

In this experiment, you will use a ballistic pendulum to measure the initial velocity of a projectile fired from a spring-loaded gun. Equipment. Ballistic pendulum device. Rule.

Important: make a note of the serial number of your device.

Theory

Your first task is to derive an equation relating the initial speed of the ball (v1) to quantities we can measure directly. Break the problem into two steps.

1 A ball of mass m, moving at speed v1 slams into the pendulum arm of mass M. Since the ball sticks in the cradle, this is a completely inelastic collision. Use conservation of momentum to write an equation relating v1 to the velocity of the arm-plus-ball, v2, immediately after the collision. 2 After the collision, the pendulum arm, which has now has a mass M+m and an initial speed

v2, swings up until it reaches a high point at an angle . Use conservation of energy to write an equation relating v2 to the distance h between the initial and final positions of the center of mass of the arm. Now use trigonometry to figure out the relationship between h,  and the length, L, of the pendulum arm measured between its pivot point and its center of mass (including the ball). When you have reached this point, get an instructor to check your equations.

124 Measurements

You will use the ballistic pendulum to measure the angle . But first, look at your equation for v1 and decide what other quantities you need to determine/measure (there are three, in addition to the angle). You may need to detach the pendulum arm for this purpose — please be very careful with this delicate piece of equipment. Make these measurements and record the value and your estimated uncertainty for each quantity. When you are ready to measure the angle, reattach the pendulum arm. Then  use the stuffing rods to push the ball into the catapult;  fire 3 shots using the medium-range setting of the spring-loaded gun;  use the inclinometer to measure the swing angle of the arm for each shot. Watch the device carefully when you fire the gun. Make sure it does not slide across the table. If it does, your measurements will be inaccurate: some of the momentum goes into the motion of the whole device, which you cannot measure. You should end up with one table showing all your measurements, and the uncertainty associated with each measurement. Use the data to calculate the value of v1, the velocity of the ball when it leaves the muzzle of the gun.

Bonus points may be given for a correct analysis of the uncertainty on v1.

Report

One report is required from each team. It should include:

 Cover page with the names of all members who participated in the activity.

 A page showing your derivation of the equations relating v1 to the measured quantities.

 Your table of measurements and estimated uncertainty.

 Your final value for v1.

 Your calculations for the uncertainty on v1

125 Ballistic Pendulum — Part II

Previously, we used a ballistic pendulum to measure the initial velocity of a projectile (steel ball) fired from a spring-loaded gun. Now we’ll test that result by predicting and then measuring the range of a ball fired from the gun.

Equipment. Ballistic pendulum device. Make sure you use the same device that you used in Part I. Rule. Target box, carbon paper.  v 0 h

 H

Theory

First make your prediction for L, given the muzzle velocity v0 you determined earlier, using the ballistic pendulum. You should also work out a formula for the uncertainty in L. 1. Use your knowledge of projectile motion to find an equation for L in terms of (H+h), the initial height of the ball above the floor and the muzzle velocity v0.

2. Given estimated uncertainties H, h and v0, find an equation for the uncertainty in L.

When you have done this, have your work checked. If it looks good, you can go ahead and set up your equipment.

126 Measurements

1. First, set up your ballistic pendulum (make sure you use the same apparatus as you did in part 1!). Carefully remove the pendulum arm. 2. Next, measure the distance H from the floor to the top of your table, and the distance h from the tabletop to the ball as it sits in the spring-gun’s muzzle. Note the uncertainty in each quantity (H, h). 3. Determine the total vertical distance between the ball and the floor (H+h), and its uncertainty. 4. Use you equation to calculate the distance the ball should fly horizontally (L), using this vertical distance and the initial velocity of the ball you determined previously. Also calculate the uncertainty L. 5. Obtain a cardboard box. Measure out your predicted distance (L) from the gun to the landing spot and the place the box on the floor so that it is centered on that spot. 6. Place a sheet of carbon paper in the box, and cover it with a blank sheet of ordinary paper. 7. Tape the paper into the box, and the box onto the floor 8. On the sheet of paper, mark your predicted landing spot, and also the range of distances given by your uncertainty analysis (i.e. L±L). 9. Fire your gun three times, then remove the sheet of white paper from the cardboard box and check whether your shots landed within the predicted range.

Report

One report is required from each team. It should include:  Cover page with the names of all members who participated in the activity.  A page showing your derivation of the equations L and L.  A page showing your measurements of H, h, your predicted value of L and the calculated uncertainty.  Your target sheet, showing your prediction, uncertainty and the impacts of your shots.

127 Uncertainties in the ballistic pendulum experiment

Here are some tips, which may help you do the calculation. Break the problem up into a series of steps.

1. First calculate the fractional uncertainty in h (h/h). This will involve combining the estimated uncertainties in L and in a term that looks like (1-cos). Note 1: to convert an estimated uncertainty in  into an uncertainty in (1-cos), evaluate the 1 function for + and -. Then (1  cos)  1  cos(  ) 1  cos(  ) 2 Q x y Note 2: if Q=Cxy or Q = C(x/y), where C=constant,   Q x y  2. Use the fractional uncertainty in h to determine the fractional uncertainty in v2 Q x Note 3: if Q  C x n , then  n Q x 

3. Determine the fractional uncertainties in the mass of the bullet and the mass of the pendulum arm plus bullet  Note 4: if Q=x+y, then Q=x+y.

4. Combine these with the fractional uncertainty in v2 to calculate the fractional uncertainty in v1. Note 5: see Note 2.

5. Calculate the absolute uncertainty v1.

128 Appendix

Useful reference material for:

Equations Significant Figures Uncertainties Error Propagation Graphing

129 Equations for University Physics I (Vector quantities are in bold.) Uncertainties xavg =  xi / N

x = (‌‌|xi - xavg|)‌ / N z = x + y z / z = x / x + y / y

Vectors

A = Axi + Ayj + Azk 2 2 2 1/2 |A| = A = (Ax + Ay + Az )

Ax = A cos

Ay = A sin -1  = tan (Ay/Ax)

A • B = A B cosA,B

A • B = AxBx + AyBy + AzBz

|A x B| = A B sinA,B

A x B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k n = A / A n = cosi + cosj + cosk

Kinematics v = dr/dt a = dv/dt 2 x = xo + vox t + (1/2) ax t vx avg = (vx + vox) / 2 vx = vox + ax t 2 2 vx = vox + 2 ax (x - xo) x = xo + vx avg t 2 ac = v / r Ax2 + Bx + C = 0 ; x = [ -B +/- sqrt(B2 – 4 A C)] / 2 A 2 R = (vo / g) sin(2o) 2 2 y = yo + (tano) (x-xo) - [ g / (2(vocoso) )] (x-xo) r = R + r' v = V + v'

130 Dynamics  F = m a

F12 = - F21 fs = s N fk = k N W = m g 2 F12 = - [ G m1 m2 / r ] runit P2 = k a3

Work and Energy K = (1/2) m v2 W =  F • dr W = F • r

Wnet = K P = dW/dt P = F • v

Fx = - dU/dx E = K + U K = -U

Ef = Ei + Wnc

Systems of Particles rcm = ( miri )/  mi rcm =  r dm /  dm

 Fext = m acm p = m v  F = dp/dt

P =  pi

 Fext = dP/dt P = constant J =  F dt J = p

131 Significant Figures

The rules for propagation of errors hold true for cases when we are in the lab, but doing propagation of errors is time consuming. The rules for significant figures allow a much quicker method to get results that are approximately correct even when we have no uncertainty values. A significant figure is any digit 1 to 9 and any zero which is not a place holder. Thus, in 1.350 there are 4 significant figures since the zero is not needed to make sense of the number. In a number like 0.00320 there are 3 significant figures --the first three zeros are just place holders. However the number 1350 is ambiguous. You cannot tell if there are 3 significant figures --the 0 is only used to hold the units place --or if there are 4 significant figures and the zero in the units place was actually measured to be zero. How do we resolve ambiguities that arise with zeros when we need to use zero as a place holder as well as a significant figure? Suppose we measure a length to three significant figures as 8000 cm. Written this way we cannot tell if there are 1, 2, 3, or 4 significant figures. To make the number of significant figures apparent we use scientific notation, 8 x 103 cm (which has one significant figure), or 8.00 x 103 cm (which has three significant figures), or whatever is correct under the circumstances. We start then with numbers each with their own number of significant figures and compute a new quantity. How many significant figures should be in the final answer? In doing running computations we maintain numbers to many figures, but we must report the answer only to the proper number of significant figures. In the case of addition and subtraction we can best explain with an example. Suppose one object is measured to have a mass of 9.9 gm and a second object is measured on a different balance to have a mass of 0.3163 gm. What is the total mass? We write the numbers with question marks at places where we lack information. Thus 9.9???? gm and 0.3163? gm. Adding them with the decimal points lined up we see 9.9???? 0.3163? 10.2???? = 10.2 gm. In the case of multiplication or division we can use the same idea of unknown digits. Thus the product of 3.413? and 2.3? can be written in long hand as 3.413? 2.3? ????? 10239? 6826? 7.8????? = 7.8 The short rule for multiplication and division is that the answer will contain a number of significant figures equal to the number of significant figures in the entering number having the least number of significant figures. In the above example 2.3 had 2 significant figures while 3.413 had 4, so the answer is given to 2 significant figures. It is important to keep these concepts in mind as you use calculators with 8 or 10 digit displays if you are to avoid mistakes in your answers and to avoid the wrath of physics instructors everywhere. A good procedure to use is to use use all digits (significant or not) throughout calculations, and only round off the answers to appropriate "sig fig."

Glossary of Important Terms

Term Brief Definition The actual error in a quantity, having the same units as the quantity. Thus if Absolute error c = (2.95 ± 0.07) m/s, the absolute error is 0.07 m/s. See Relative Error. Δ c How close a measurement is to being correct. For gravitational Accuracy acceleration near the earth, g = 9.7 m/s2 is more accurate than g = 9.532706 m/s2. See Precision. When several measurements of a quantity are made, the sum of the Average measurements divided by the number of measurements. Average The average of the absolute value of the differences between each Deviation measurement and the average. See Standard Deviation. The fraction of measurements that can be expected to lie within a given range. Thus if m = (15.34 ± 0.18) g, at 67% confidence level, Confidence 67% of the measurements lie within (15.34 - 0.18) g and (15.34 + Level 0.18) g. If we use 2 deviations (±0.36 here) we have a 95% confidence level. A measure of range of measurements from the average. Also called Deviation error or uncertainty. Δ x A measure of range of measurements from the average. Also called Error deviation or uncertainty. An uncertainty estimated by the observer based on his or her Estimated knowledge of the experiment and the equipment. This is in contrast Uncertainty to ILE, standard deviation or average deviation. The familiar bell-shaped distribution. Simple statistics assumes that Gaussian random errors are distributed in this distribution. Also called Normal Distribution Distribution. Changing the value of one variable has no effect on any of the other Independent variables. Propagation of errors assumes that all variables are Variables independent. Instrument The smallest reading that an observer can make from an instrument. Limit This is generally smaller than the Least Count. of Error (ILE) The size of the smallest division on a scale. Typically the ILE equals Least Count the least count or 1/2 or 1/5 of the least count. The familiar bell-shaped distribution. Simple statistics assumes that Normal random errors are distributed in this distribution. Also called Distribution Gaussian Distribution. Precision The number of significant figures in a measurement. For gravitational

133 acceleration near the earth, g = 9.532706 m/s2 is more precise than g = 9.7 m/s2. Greater precision does not mean greater accuracy! See Accuracy. Propagation of Given independent variables each with an uncertainty, the method of Errors determining an uncertainty in a function of these variables. Deviations from the "true value" can be equally likely to be higher or Random Error lower than the true value. See Systematic Error. Range of Measurements give an average value, and an uncertainty, Δ x. At Possible the 67% confidence level the range of possible true values is from True Values - Δ x to + Δ x. See Confidence Level . The ratio of absolute error to the average, x/x. This may also be Relative Error Δ called percentage error or fractional uncertainty. See Absolute Error. Significant All non-zero digits plus zeros that do not just hold a place before or Figures after a decimal point. Standard The statistical measure of uncertainty. See Average Deviation Deviation.Symbol is σ Standard Error An advanced statistical measure of the effect of large numbers of measurements on the range of values expected for the average (or in the Mean mean). Systematic A situation where all measurements fall above or below the "true Error value". Recognizing and correcting systematic errors is very difficult. A measure of range of measurements from the average. Also called Uncertainty deviation or error. Δ x

134 Uncertainties and Statistics Reference: < http://www.rit.edu/~uphysics/uncertainties/Uncertainties.html > Variations in measurements Suppose that you make several measurements of a physical quantity such as the period of a pendulum. You are not likely to get the same answer each time. Two main reasons for this are 1. The quantity being measured is somewhat variable. (E.g your height will vary depending on whether you stand straight and it is different when you get out of bed than when you have been standing all day, your mass varies as you breath in and out.) 2. The methods used for the measurement introduce some variation. (E.g. in measuring the pendulum period you rely on your visual observation to decide when to start and when to stop the watch.) The measurement of the variations leads to the field of statistics. We will try to take many measurements of a quantity and use the average as a best estimate, but we also want an idea of how far the measurements were from the average. The terms error, standard deviation, and uncertainty are all used to describe the variation. Random versus Systematic error Suppose I measure the length of a wood block using a metal ruler in the following locations: Bogota, Mexico City, Nashville, Rochester, and Nome. I might get results that suggest that the wood block is longer when the country speaks English than when it speaks Spanish. In fact there is a systematic error relating to the temperature and the change in the length of the ruler in the different locations. Systematic errors occur when an uncontrolled (unmeasured) variable affects the data so that the values are always too large or too small. A famous example is the newspaper that polled voters and declared that Dewey had beaten Truman in the 1948 presidential election. What was the systematic error? We will not discuss systematic errors any further.

Random errors cause the measurements to be centered on the “average” with equal numbers above and below. If we plot a histogram of our measurements we get the well known Gaussian curve, also called a normal curve, or a bell-shaped curve. Precision and Accuracy Accuracy deals with how well the center of the curve matches the “real value” of what we are measuring. Accurate measurements have no systematic error. Suppose that the thickness of a piece of foam is supposed to be 4.48 cm. The diagram below shows two measurements of the foam, and in both cases the Gaussian curves are centered on 4.45 cm, meaning that the measurements are quite accurate.

The top curve represents measurements with a wider spread of values than the bottom curve. We say that the bottom curve is a more precise measurement of the thickness.

135 Standard Deviation The Gaussian is asymptotic to the axis (infinitely wide). We need a method to specify the relative widths of the curves. The standard deviation, sx or , is a measure of the width of the curves. The horizontal line shows the standard deviation (from the center to where the line crosses the curve.)

We can represent the curve by a shaded box. It is darkest in the middle where most measurements occur, and fades out to zero as we go away from the center.

Approximately 2/3 of the measurements lie within 1 standard deviation of the center. If we go two standard deviations out, 95% of the measurements are accounted for.

0 1 2 3 4 5 6

Comparing numbers. Suppose we have four measurements: A (3.8 ± 0.5) cm, B (5.1 ± 0.5) cm, C (3.8 ± 0.2) cm, D (5.1 ± 0.3) A cm. These are shown in the diagram to the right. B If we don’t have the standard deviations we can only say that the measurements are close. C D With standard deviations we can say that A and B are equal within uncertainties. That is within 2, A ranges from 2.8 to 4.8 cm 3 4 5 6 overlapping B which varies from 4.1 to 6.1 cm.

However within 2, C and D are not equal, that is C = 3.4 to 4.2 cm does not overlap D, 4.5 to 5.7 cm.

136 Rules for propagating uncertainties

Suppose that we need to calculate some quantity Q, which depends on the measured variables x and y. There might also be a constant, C, involved. How do we find the uncertainty on Q (Q), given experimental uncertainties on x and y (x and y, respectively)?

Addition or subtraction Q  x  y  Add absolute uncertainties:  Q  x  y Q  x  y 

Multiplication or Division Q  x y  Q x y Add fractional uncertainties:     Q  x y  Q x y

Powers Multiply fractional uncertainty by  Q x absolute value of power: Q  x n  n Q x

Multiplication by a constant Q x No effect on fractional uncertainty:  Q  C x  Q x Scales absolute uncertainty: Q Cx Q Cx

  Q x y Often you will need to combine several rules: Q  C xy n   n Q x y



Basic layout of a graph

Certain conventions are used when plotting graphs. Refer to Figure 2 for a general description and Figures 5, 7, and 8 for examples.

a) The horizontal axis is called the abscissa and the vertical axis is called the ordinate. You can use the terms horizontal and vertical just as well. b) The graph must have a title which clearly states the purpose of the graph. This should be located on a clear space near the top of the graph. A possible title for a graph would be "Figure 1. Variation of Displacement With Elapsed Time for a Freely Falling Ball." The title should uniquely identify the graph --you should not have three graphs with the same title. You may wish to elaborate on the title with a brief caption. Do not just repeat the labels for the axes! Example Poor choices of titles: The title should be in words and should not just repeat the "y vs t" symbols on the axes! "Displacement This title is in words, but just repeats the names on the axes. versus time" The title should add information. Again, this adds minimal information. It may be useful to "Data from Table 1" include this information, but tell what the graph is and what it means. c) Normally you plot the independent variable (the one over which you have control) on the abscissa (horizontal), and the dependent variable (the one you read) on the ordinate (vertical). If for example you measure the position of a falling ball at each of several chosen times, you plot the position on the ordinate (vertical) and the time on the abscissa (horizontal.) In speaking of a graph you say "I plotted vertical versus horizontal or ordinate versus abscissa". If you are told to plot current versus voltage, voltage goes on the abscissa (horizontal).

138 d) The scale should be chosen so that it is easy to read, and so that it makes the data occupy more than half of the paper. Good choices of units to place next to major divisions on the paper are multiples of 1, 2, and 5. This makes reading subdivisions easy. Avoid other numbers, especially 3, 6, 7, 9, since you will likely make errors in plotting and in reading values from the graph. The zero of a scale does not need to appear on the graph. Computer plotting packages should allow you control over the minimum and maximum values on the axis, as well as the size of major and minor divisions. The packages should allow you to include a grid on the plot to make it look more like real graph paper. e) Tick marks should be made next to the lines for major divisions and subdivisions. Look at the sample graphs to see examples. Logarithmic scales are pre-printed with tick marks. f) Axis label. The axes should be labeled with words and with units clearly indicated. The words describe what is plotted, and perhaps its symbol. The units are generally in parentheses. An example would be Example Displacement, y, of ball (cm) On the horizontal axis (abscissa) the label is oriented normally, as are the numbers for the major divisions. The numbers for the major divisions on the vertical axis are also oriented normally. The vertical axis label is rotated so that it reads normally when the graph paper is rotated 1/4 turn clockwise. See the sample graphs for examples. Avoid saying Diameter in meters (x 10-4) since this confuses the reader. (Do I multiply the value by 10- 4or was the true value multiplied by this before plotting?) Instead state Diameter (x 10-4meters) or use standard prefixes like kilo or micro so that the exponent is not needed: "Diameter (mm)". g) Data should be plotted as precisely as possible, with a sharp pencil and a small dot. In order to see the dot after it has been plotted, put a circle or box around the dot. If you plot more than one set of data on the same axes use a circle for one, a box for the second, etc., as shown in Figure 3.

139 140 141 Uncertainties and Graphs: Error Bars (a) Error Bars Data that you plot on a graph will have experimental uncertainties. These are shown on a graph with error bars, and used to find uncertainties in the slope and intercept. In this discussion we will describe simple means for finding uncertainties in slope and intercept; a full statistical discussion would begin with "Least Squares Fitting." Consider a point with coordinates X ± ΔX and Y ± ΔY. (a) Plot a point, circled, at the point (X,Y). (b) Draw lines from the circle to X + ΔX, X - ΔX, Y + ΔY, and Y - ΔY and put bars on the lines, as shown in Figure 6(a). These are called error bars. (c) The true value of the point is likely to lie somewhere in the oval whose dimension is two deviations, i.e. twice the size of the error bars.

The oval shown in Figure 6(c) shows the uncertainty region (at 95% confidence--this is statistics speak). It is not usually drawn on graphs. Often the error bars may be visible only for the ordinate (vertical), as Figure 6(b). Draw the best error bars that you can! If they cannot be seen, make a note to that effect on the graph. (b) Uncertainties in Slope and Intercept Using Error Bars Once the graph is drawn and the slope and intercept are determined we wish to find uncertainties in the slope and intercept. Refer to Table 2 and Figure 5(a) to see the procedure. Data are plotted on this graph with error bars shown. The uncertainty in time is so small that no horizontal error bars are visible. A solid line is shown which best fits the data and has a slope of (2.09 cm/s) and an intercept of (- 0.68 cm) (on the vertical axis). Using the error bars as a guide we have drawn dashed lines which conceivably fit the data, although they are too steep or too shallow to be considered best fits. This is a judgment call on your part. The slopes of the dashed lines are 2.32 cm/s and 1.79 cm/s. Half the difference of these is 0.27 cm/s which we take as the uncertainty in the slope of the best line. We round off the uncertainty to the proper number of significant figures, and round the slope to match, resulting in slope = (2.1 ± 0.3) cm/s .

142 The differences between the best slope and either of the extreme slopes should equal the uncertainty in the slope. Here the differences are (2.09 - 1.79) = 0.30 cm/s and (2.32 - 2.09) = 0.23 cm/s, which are basically the same as the ±0.3 cm/s above. We try to make the three lines cross in the middle of our data. If we draw them this way we can determine the uncertainty in the intercept. The dashed lines have intercepts of -1.52 cm and +0.20 cm and half of the difference between these is 0.86 cm which we use as the uncertainty in the intercept. Intercept = (-0.7 ± 0.9) cm. It is more difficult to do this on the computer graph, but we can try as is suggested on Figure 5(b). On the Excel download I show lines and calculations resulting in an uncertainty in slope of +/- 0.4 cm/s and uncertainty in intercept of +/-1.2 cm. Also on the Excel spreadsheet I show a statistical analysis of the line resulting in a standard error of 0.13 cm/s in slope and 0.55 cm in intercept. Doubling these to get to 95% confidence results in values close to what we get graphically. (c) Uncertainties in Slope and Intercept When There Are No Error Bars Even if we lack error bars we use the same approach to find the errors in slope and intercept. Using this method it is possible to get good estimates of uncertainty in the slope and intercept. Generally you will have less confidence in the intercept uncertainty. (d) What is being done in statistical terms The process described in parts (b) and (c) above estimates the statistical procedure of finding standard errors in the slope and intercept. Statistics programs will allow this to be done automatically (in Excel see the LINEST function). The values of uncertainties you get by visual estimation will be similar to the values obtained by a full regression analysis.

143 Straight line graphs on linear graph paper.

Suppose that we have plotted a graph with Y on the ordinate and X on the abscissa and the result is a straight line. We know that the general equation for a straight line is Y = M X + B where M is the slope and B is the intercept on the Y-axis (or Y-intercept). The capital forms of Y and X are chosen to represent any arbitrary variables we choose to plot. For example we may choose to plot position, x, on the Y-axis versus mass, m, on the X-axis, so we need different symbols for our general case. Refer to Figure 4 to see what is being done.

We choose two points, (X1,Y1) and (X2,Y2), from the straight line that are not data points and that lie near opposite ends of the line so that a precise slope can be calculated. (Y2-Y1) is called the rise of the line, while (X2-X1) is the run. The slope is

Slope has units and these must be included in your answer! The point where the line crosses the vertical axis is called the intercept (or the Y-intercept). The intercept has the same units as the vertical axis. The equation of the straight line with Y on the vertical axis and X on the horizontal axis is

The line can be extended to cross the horizontal axis as well. The value of X where this happens is called the X-intercept, with the same units as variable X, and will be used only rarely. If the line goes directly through the origin, with intercepts of zero, we say that Y is directly proportional to X. The word proportional implies that not only is there a linear (straight line) relation between Y and X, but also that the intercept is zero.

144 How can one estimate the uncertainty of a slope on a graph? If one has more than a few points on a graph, one should calculate the uncertainty in the slope as follows. In the picture below, the data points are shown by small, filled, black circles; each datum has error bars to indicate the uncertainty in each measurement. It appears that current is measured to +/- 2.5 milliamps, and voltage to about +/- 0.1 volts. The hollow triangles represent points used to calculate slopes. Notice how I picked points near the ends of the lines to calculate the slopes!

1. Draw the "best" line through all the points, taking into account the error bars. Measure the slope of this line.

2. Draw the "min" line -- the one with as small a slope as you think reasonable (taking into account error bars), while still doing a fair job of representing all the data. Measure the slope of this line.

3. Draw the "max" line -- the one with as large a slope as you think reasonable (taking into account error bars), while still doing a fair job of representing all the data. Measure the slope of this line.

4. Calculate the uncertainty in the slope as one-half of the difference between max and min slopes.

145 In the example above, I find 147 mA - 107 mA mA "best" slope = ------= 7.27 ---- 10 V - 4.5 V V

145 mA - 115 mA mA "min" slope = ------= 5.45 ---- 10.5 V - 5.0 V V

152 mA - 106 mA mA "max" slope = ------= 9.20 ---- 10 V - 5.0 V V

Uncertainty in slope is 0.5 * (9.20 - 5.45)mA/V = 1.875 mA/V

There are at most two significant digits in the slope, based on the uncertainty. So, I would say the graph shows slope = 7.3 +/- 1.9 mA/V

Adapted from M. Richmond.

146 Curve Fitting

We are free to make many plots from a given set of data. For instance if we have position (x) as a function of time (t) we can make plots of x versus t, x versus , log(x) versus t, or any number of any choices. If possible, we choose our plot so that it will produce a straight line. A straight line is easy to draw, we can quickly determine slope and intercept of a straight line, and we can quickly detect deviations from the straight line.

If we have the guidance of a theory we can choose our plot variables accordingly. If we are using data for which we have no theory we can empirically try different plots until we arrive at a straight line. Some common functions are listed in Table 1 along with plots which yield straight lines.

Table 1. Different graphs for different functions. This summarizes some of the most common mathematical relations and the graphing techniques needed to find slopes and intercepts. FORM PLOT (to yield a straight line) SLOPE Y-INTERCEPT y = a x + b y versus x on linear graph paper a b y2 = c x + d y2 versus x on linear graph paper c d log y versus log x on linear paper log a y = a xm or m * a (at x = 1) y versus x on log-log paper x y = K y versus (1/x) on linear paper K 0 ln y versus x on linear paper or ln a y = a ebx b* y (on log scale) versus x on semi- a log paper

* Special techniques are needed when using logarithmic graph paper. These will be discussed in a later section.