Uniform Circular Motion

Part I. The Centripetal Impulse

The “centripetal impulse” was Sir Isaac Newton’ favorite force.

The Polygon Approximation. Newton made a business of analyzing the motion of bodies in circular orbits, or on any curved path, as motion on a polygon. Straight lines are easier to handle than circular arcs. The following pictures show how an inscribed or circumscribed polygon approximates a circular path. The approximation gets better as the number of sides of the polygon increases.

To keep a body moving along a circular path at constant speed, a force of constant magnitude that is always directed toward the center of the circle must be applied to the body at all times. To keep the body moving on a polygon at constant speed, a sequence of impulses (quick hits), each directed toward the center, must be applied to the body only at those points on the path where there is a bend in the straight- line motion.

The force Fc causing uniform circular motion and the force Fp causing uniform polygonal motion are both centripetal forces: “center seeking” forces that change the direction of the velocity but not its magnitude.

A graph of the magnitudes of Fc and Fp as a function of time would look as follows:

Force F p causing Polygon Motion Force

Force F c causing Circular Motion

time

In essence, Fp is the “digital version ” of Fc . Imagine applying an “on-off force” to a ball rolling on the table − one hit every nanosecond – in a direction perpendicular to the motion. By moving around one billion small bends (polygon corners) per second, the ball will appear to move on perfect circle.

Applying a Centripetal Impulse

Here you will get a kinesthetic sense of Newton’s favorite force. Your team will learn what it “feels like”

to apply the force Fp that causes a ball to move on a polygon. The bent-line polygonal path pictured below and the circular arc it approximates appear on a sheet of paper at your table.

DO NOT tape this sheet to the table. USE CAUTION so heavy ball does not roll off table onto floor/foot.

For a ball moving at constant speed along the left line in the picture, how do you get the ball to suddenly turn the corner or “round the bend” and end up moving at the same constant speed on the right line?

• Let the ball roll down the ramp onto the arrow (initial velocity vector) that leads into the polygon segment (bold straight line). Assume the speed of the ball is constant along the line, i.e. neglect friction. • When the ball reaches the bend, use the force-probe tip to apply one “quick hit” to the ball so that the ball successfully rounds the bend − but does not noticeably change its speed. The direction of your hit is crucial – this special direction is a huge part of the physics of circular motion! Do not record the force. The probe is used here simply as a pushing device. Learning how to push is your quest. • The speed of the ball should not be too fast (so that it is difficult to push) or too slow (so that friction noticeably slows the ball). Try different speeds by varying the initial location of the ball on the ramp until you find a speed that is just right – the “optimal speed”. Remember this optimal initial location because you will use it in future parts of this lab. • Practice the push (polygonal impulse) several times until you get a feel for the magnitude , the duration , and the direction of the push. All team members should give it a try.

After your team has discovered the optimal speed and the correct push, summarize your findings in the space below. Draw a picture that shows the path of the ball, the force vector , and the acceleration vector of the ball. Estimate the magnitude (maximum value) and the duration (how long it lasts) of your push − this is a rough estimate since it is based solely on what you felt while pushing.

Magnitude of ≈ _____ N. Duration of ≈ _____ s.

*Before you move on to the next part, ask your lab instructor to check your pushing technique*

Part II. Measuring Centripetal Impulse and Momentum Change

Impulse-Momentum Theorem. Newton’s law of motion can be written as or =

= − .

This relation says , where impulse is and momentum is . Impulse measures = ∆ ≡ ≡ both the strength and the duration of the force. Note that , where is the average of F = ∆ over the time interval ∆t during which F acts. The impulse-momentum theorem reveals a wonderful relation between “force × time area” and “momentum change”:

F(t)

= ∆ = ∆ Area under F (t) Curve = Change in Momentum

t ∆t Measuring the Impulse Jp

1. Open Logger Pro file “Force Probe”. Go to Experiment, Data Collection, change the “Sampling Rate” to 200 samples per second. ZERO the force probe while it is in the horizontal position.

2. Start the ball from its optimal location on the ramp – the place that gets you the best speed. Let the ball roll down the ramp and onto the arrow that leads into the polygon segment.

3. Apply Fp so that the ball turns the corner but does not noticeably change its speed.

4. Change the scales on the Force vs Time graph so that the shape (height and width) of the “triangular force spike” is clearly displayed.

5. Compute the impulse Jp of your force-probe hit Fp by highlighting the force spike on you graph and

clicking on the area icon . Record the value of Jp in the table below.

6. Find the average force (use the STATS button) and the duration time ∆tp. Enter their values in the table. Their product should be equal to Jp.

7. Do four more trials. Start with the ball at the same initial location on the ramp each time so that the speed of the ball is the same each trial. Average your Jp values. Find the uncertainty (half-width spread).

Trial 1 2 3 4 5 Average Force Duration Time ∆tp

Impulse Jp

Jp = ______± ______Ns .

Measuring the Momentum Change ∆∆∆p

The velocity vectors of the ball along the two polygon lines are drawn below. Note that is also the velocity of the ball when it enters the circular arc while is the velocity when it exits. y

θ θ x

θ θ

1. Use this diagram to calculate the change in the velocity vector: . ∆ ≡ −

Hint: Express each velocity vector in terms of its x and y components and then subtract the vectors. Show your vector algebra in the space above. Express your answer in terms of the variables v and θ, where v is the constant speed of the ball: , and θ is the angle defined in the diagram. = || =

______. − =

2. How is the direction of related to the direction of ? ∆

3. Measure the mass m of the ball and the angle θ: m = ______kg. θ = ______.

4. Measure the speed v of the ball as follows. Place the ball at its optimal initial position on the ramp. Place the short ruler near the end of the ramp so that when the ball rolls off the ramp onto the table, the ball moves alongside the ruler. Measure the time it takes the ball to cover the length of the ruler. Repeat this measurement to make sure you are getting a consistent value for the speed of the ball. Display your data in the space below.

v = ______m/s .

5. Compute the momentum change. Include the direction.

______kg m/s . ∆ =

Does your Experimental Data “Prove” the Theorem ? = ∆

You have measured with a force probe. You have measured with a ruler, protractor, stopwatch, and ∆ mass scale. Compare the directions of and . Compare their magnitudes. Does ∆p fall within the ∆ uncertainty range of Jp ?

Part III. Centripetal Force Fc

“Polygon Formula” for F c . The force acts continuously over the whole circular arc, while the force acts momentarily only at the bend in the polygon.

Fc Fc Fc Fc Fc Fc Fc F Fc Fc c Fc ∆ Fc Fc tp Fc Fc F c ∆tc

The change in the velocity of the ball is the same along the c-path and the p-path because of the way the polygon fits the circle. Thus according to Newton’s law, the relation between Fc and Fp is

∆ = ∆

In words, a small force acting over the long time ∆tc causes the same momentum change − as a large force acting over the short time ∆tp .

Comment: This relation is approximate. The exact relation between F c and F p is = − = . The impulse reduces to Fc ∆tc × sin θ/θ (directed downward). The angular factor (sin θ/θ) arises because the direction of along the circular arc does not always point downward, like it does for at the polygon bend. We neglect this factor since its value is near unity for the short arcs in this lab.

Thus we arrive at an interesting formula for the centripetal force in terms of a centripetal (polygonal) impulse and a circle time:

∆ = = . ∆ ∆

Use this impulse-time formula to calculate the value of the centripetal force Fc . You have already

measured the centripetal impulse Jp . You can find the time ∆tc it takes the ball to transit the circular arc from your measured value of the speed v of the ball and the value of the length ∆Lc of the circular arc (use a piece of string).

∆Lc = ______m. ∆tc = ______s.

= = ______N . ∆

“Circle Formula” for F c . The standard formula for the centripetal force Fc that causes a mass m to move at a uniform speed v around a circle of radius R is

= . Use this theoretical formula and your measured values of m, v, and Fc to calculate the theoretical value of the radius R of your circular arc. Show your calculation:

R (theory ) = ______m.

Radius of Arc. Measure the radius of your arc. Compare R(measured ) and R (theory ) .

Part IV. Weighing an Airplane Without Using a Scale

You will determine the mass of a toy airplane by observing its motion. The airplane of mass m is suspended from a cord of length L and travels in a horizontal circle of radius R:

m R

The Theory

Write down an expression for the velocity of the orbiting plane in terms of the radius R and period P of the orbit.

Based on your answer above, what is the plane’s centripetal acceleration? Express your answer in terms of R and P.

In a free body diagram of the plane, you would use a dot to represent the plane. Indicate where this “center of mass” dot is located in the picture below.

Draw a free body diagram of the plane when it is in flight. Let T denote the tension in the cord.

Derive the formula for the magnitude of the net force Fnet acting on the plane in terms of R, L , and T. Carefully show the direction of the net force.

Fnet = ______

Now that you have expressions for a and F, write Newton’s Second Law for the airplane in flight. Let m denote the mass of the plane.

Derive the formula that gives the mass m of the plane in terms of T, L, and P.

m =

Ask your instructor to check your theoretical work before you move on to the experimental phase.

The Experiment

Read this paragraph before you turn on the plane’s propeller. To calculate m using your theoretical formula, you need to measure T, L, and P. The spring scale will read the tension T in the cord when the plane is in flight. The cord length L is the distance from the bottom of the plastic tube where the cord pivots to the “center of mass” of the plane – when the plane is in flight! To measure this in- flight L, you will have to stop the plane and pull on the cord until the spring scale reads the in-flight tension. Use the meter stick to measure this L. Use the stopwatch to determine the period P of the plane’s motion. For accuracy, measure the time it takes the plane to complete ten revolutions and then divide by ten to obtain the period P.

Turn on the plane’s propeller. CAUTION : Rotating Propeller Blade! Give the plane a push to launch it into a circular path. Allow the plane to circle several times so that its motion can stabilize before you take any measurements – but don’t wait too long because the string becomes twisted (shortens) and the batteries become weaker. Also use CAUTION when stopping the plane – stay away from the propeller – DO NOT use your hands. Ask your instructor for safe stopping techniques.

You will make two separate determinations of the mass of the plane using two different cord lengths (Trial 1 and Trial 2). Fill in the data table. Change the length of the cord by sliding the spring scale up or down the pole. If you must stand on the table to do this, then please be careful when climbing on and off the table. DO NOT stand on a stool!

m L T Time for ten P Calculated Trial Length of cord Tension revolutions Period of orbit mass of plane (m) (N) (seconds) (seconds) (kg)

Your two values of m should agree to within 10%. If not, consult your instructor. Report the average value of your two masses:

Calculated m (from analyzing circular motion) = ______kg.

Take your plane off the cord and weigh it on the scale at the back of the room.

Measured m (read directly from kilogram scale) = ______kg .

% difference between Calculated m and Measured m is ______%.

Are there any force(s) that you neglected in your theoretical analysis that could account for the discrepancy between the calculated m and the measured m? Explain.

When the plane is at rest, hanging vertically, what does the spring scale read? Is this value of tension consistent with your results for m above? Hint: Draw a free body diagram of the resting plane. Show all calculations.

T (rest) = ______N.