Forces Laboratory. After Completing This Lab and the Associated

1.- Forces laboratory.

Lab 4: Forces Name ______

1. Introduction

In this laboratory we look at the forces that are commonly encountered as we go about daily life: gravitational forces (weight), normal forces, tension forces, and friction forces. In order to find the net force acting on an object, we need to decide which of these forces are acting and to determine their direction. It is also useful to be able to reason qualitatively about these forces, and to understand what factors influence (and don’t influence) them. We will look at each of these forces in turn.

Newton’s second law states that the net force acting on a body is equal to its mass times its r r acceleration: Σ F = ma . The acceleration of an object is a vector quantity, and we must take the direction of this quantity into consideration as well as the magnitude. Likewise, force is also a vector quantity. The free-body diagram is a very useful tool in helping us identify and keep track of the forces acting on a body, and of the directions in which these forces are acting so that we can find the net force. Free-body diagrams serve as a guide for adding these forces to find the net force. In this lab, we will practice drawing free-body diagrams for situations involving the forces identified above. We will use a convention for labeling these forces that serves to minimize common errors and that will also help to identify Newton’s 3 rd law pairs.

1.1 Lab Objectives. After completing this lab and the associated homework, you should be able to:

1. Identify the forces acting on an object for typical situations encountered in mechanics. 2. Draw a free-body diagram that includes identification of the type of force, the object on which the force is acting, and the object exerting the force. 3. Relate the mass of an object to its weight 4. Predict the direction of tension forces and the effect of pulleys on tension forces. 5. Identify normal forces acting on an object. 6. Correctly identify friction forces acting on an object.

1 1.2 Outline of Laboratory Approximate sequence of the lab and homework:

1. Practice identifying forces and labeling them in a systematic manner. 2. Relate the mass and weight of objects. 3. Use springs to observe the magnitudes of forces along a string. 4. Observe the effect of pulleys on forces. 5. Practice correctly identifying the direction of the normal force. 6. Find out which factors do and do not affect the magnitude of the friction force.

2. Free-body diagrams Two people are attempting to move a large block. The block, however, does not move. Chris is pushing on the block. Pam is pulling on a rope attached to the block.

Copy your group’s sketch here after discussion.

Chris Pam

2.1 Draw a large dot on your large sheet of paper to represent the block. Draw vectors with their “tails” on the dot to show the forces exerted on the block. Label each vector and write a brief description of that force next to the vector.

In Newtonian physics, all forces arise from an interaction between two objects. Forces are specified by identifying the object on which the force is exerted, and the object that is exerting the force. For example, in the situation above, a gravitational force is exerted on the block by the earth.

2.2 Describe the remaining forces you have indicated above in a similar fashion.

The diagram you have drawn is called a free-body diagram . A free-body diagram should show only the forces exerted on the object or system of interest, that is, in this case, on the block. Check your free-body diagram and, if necessary, modify it accordingly.

2 Sometimes a free-body diagram involves a simplified sketch of the object rather than the dot. (Your instructor will indicate the convention you are to use.) Regardless of which form is used, a proper free-body diagram should not have anything on it except a representation of the object and the (labeled) forces exerted on that object. A free-body diagram never includes (1) forces exerted by the object of interest on other objects or (2) sketches of other objects that exert forces on the object of interest

Write your observations and comments about this point of the lab.

3 2.3 All forces arise from interactions between objects, but the interactions can take different forms. Which of the forces exerted on the block require direct contact between the block and the object exerting the force?

Which of the forces exerted on the block do not arise from direct contact between the block and the object exerting the force?

We will call forces that depend on contact between two objects contact forces. We will call forces that do not arise from contact between two objects non-contactr forces. r 2.3.1 There are manyr different types of forces,r including: friction ( ƒ ), tensionr (T ), magnetic forces (F mag ), normal forces (N ), and the gravitational force ( W, for weight). Categorize these forces according to whether they are contact or non-contact forces.

Contact forces Non-contact forces

2.4 Consider the following discussion between two students.

Student 1: “I think the free-body diagram for the block should have a force by Chris, a force by the rope, and a force by Pam.”

Student 2: “I don’t think the diagram should show a force by Pam. People can’t exert forces on blocks without touching them.”

With which student, if either, do you agree? Explain your reasoning.

It is often useful to label forces in a way that makes clear (1) the type of force, (2) the object on which the force is exerted, and (3) the object exerting rthe force. For example, the gravitational force exerted on the block by the earth might be labeled WBE . Your instructor will indicate the notation that you are to use.

Label each of the forces on your free-body diagram in part 2.1 in the manner described above.

Discuss your answers above with your laboratory instructor before continuing.

4 ➪ 3. Types of Forces In this section we will be looking at the forces that are most commonly encountered in mechanics: weight, tension forces, normal forces, and friction forces.

3.1: Weight Hold the force probe vertically (so that the hook is hanging down). Your lab instructor will show you how to ‘zero’ the force probe so that it reads zero when there is no force on the hook. (The force probe will read negative forces when the hook is pulled away from the body of the probe and will read positive values when it is pushed toward the body of the probe.)

3.1.1 Start recording data and push and pull on the hook to observe the effect of exerting forces on the hook.

When we hang an object by the hook on the force probe so that it is at rest, the net force on that object is zero. Only two forces act on this object: Force probe the weight of the object and the force on the object by the hook. These two forces must have the same magnitude and act in opposite directions Slotted masses in order for the net force to be zero. For this 50-g hanger reason, we can use the force probe to tell us about the weight of an object suspended from it.

3.1.2 Hang a 50-gram mass hanger from the force probe as shown. After starting to record data, add slotted masses to the hanger until the total mass of the hanger and the slotted masses is 500 grams.

3.1.3 What is the relationship between the suspended mass and the force required to support it? In other words, if you suspended a mass m from the force probe, how would you determine the force F that the force probe would read?

3.1.4 Is this relationship consistent with the mathematical relationship between force and weight (on the surface of the earth) that you have used in the lecture portion of your physics course?

3.2: Tension forces, strings, and pulleys

3.2.1. Replace the hook on the force probe. Connect a string between the mass and the force probe and repeat the experiment in part 3.1.

3.2.2. What is the effect (if any) of the string on the force measured by the force probe? Force probe

3.2.3 Suppose instead that the mass were connected to force probe by a 1-kg chain. Would you expect the force measured by the probe to change? Explain.

Under what condition(s) can we ignore the mass of a string (thread, rope, wire, or chain) when measuring tension forces? Slotted 3.2.4 Would the force measured by the probe change if 50-g hanger masses you were to increase the length ⋅ Of a light string?

⋅ Of a heavy chain?

3.2.5 We can obtain a rough measure of the tension in a string by replacing portions of the string with springs. The amount that the spring stretches increases as the tension increases. Rod Hang a string with three springs from a hook on a rod as shown in the diagram at right. Hang the hook without any slotted masses from the lower loop on the string. Spring 1 When slotted masses are added to the string, the springs will stretch. Predict which spring will stretch the most, and which will stretch the least. Explain the basis for your Spring prediction. 2

Spring 3 Test your prediction by adding approximately 250g of slotted masses to the hook. (Since the springs are not perfectly identical, you should ignore any differences that Hook are smaller than about one-half centimeter in this and all subsequent exercises.) Resolve any inconsistencies.

Suppose that we replaced these springs with springs that each had a mass of 300 grams. Rank, from greatest to least, the amount of stretch you would expect in the three springs.

3.2.6 At right a string with springs is held against a pulley as shown in the ‘initial’ situation shown at right. For this situation, predict how the stretch of the upper spring will compare with the stretch of the other springs.

Test your prediction and resolve any inconsistencies. Spring 1

Pulley Pulley

Spring Spring Spring 2 2 1

Spring Spring 3 3

Hook and Hook and slotted masses slotted masses

Initial Final

3.2.7 Without moving the pulley or changing the height of the suspended masses, try varying the angle between the horizontal and the portion of the string above the pulley. (Start with this portion of the string almost vertical, and vary the angle until this portion points vertically downward as in the drawing below right.) Observe the springs as you do this. Based on your observations: Does the tension in a string change significantly when the angle changes?

For this exercise, does a pulley affect ⋅ The magnitude of the force exerted by a string?

⋅ The direction of the force exerted by the string on the probe?

Discuss your answers above with your laboratory instructor before continuing.

3.3: Normal Forces

7 In mathematics a normal vector or normal is Normal defined as a vector pointing perpendicular to vector a surface (or a small portion of a curved surface as shown in the diagram). Surface In general, when the surfaces of two objects are in contact with each other, these objects will exert forces on each other that are perpendicular to the surface of contact. In keeping with the mathematical definition of normal, these forces are called normal forces.

3.3.1. For each of the following situations, identify (1) any normal forces acting on the identified object, (2) the objects that are exerting these normal forces, and (3) the direction of these Block normal forces:

i. A block sliding down an incline. θ

ii. A marble resting in a v-shaped wedge. Marble

iii. A ladder leaning against a wall. Ladder

3.3.2 Is the normal force always in the opposite direction to the weight? Explain.

Ask your lab instructor to exchange the hook on the force probe for a platform. Rezero the force probe with the platform attached and the probe held vertically. 3.4.1 Predict what value the force probe will measure with a 200 gram mass placed on the platform.

3.4.2 Draw a free-body diagram for the mass. What force is exerted on the mass by the platform?

3.4.3 Now push down on the mass with your finger. Is the value measured by the force probe equal to the weight of the mass?

Mass 3.4.4 Draw a free-body diagram for the mass with the finger Platform pushing down on it. Based on this free-body diagram and the force measured by the force probe, what is the magnitude Force of the force exerted by the finger? Explain how you can tell. probe

Mass 3.4.5 Remove your finger, and tilt the force probe with the weight Platform on it as shown. (Be sure not to tilt the platform so far that the mass falls.) In this case, is the force exerted on the mass by the platform the same as the weight of the mass? Force probe

3.4.6 Based on your answers above, under what conditions is the normal force exerted on an object equal to the weight of the object?

➪ Discuss your answers above with your laboratory instructor before continuing. The normal force is a measure of how tightly two objects are squeezed together. As you saw when you pushed with your finger on the mass, the normal force between the cube and the platform increased – suggesting that the mass and the platform were squeezed more tightly together.

3.4: Friction Forces Cart with masses Remove the platform from the force probe and replace the hook. Zero the force probe when it is held horizontally. Track 3.4.1 Attach the force probe to the string tied to the force cart with the cork bottom surface. Place masses totaling 500 grams into the cart. Using the force probe, pull the string horizontally so that the cart moves along the track at a constant speed. Record the magnitude of the force measured by the probe.

3.4.2 Draw a single free-body diagram representing the forces acting on a system consisting of the cart and the masses in the cart. 3.4.3 Which force in your free-body diagram is the force probe measuring? Which other force in your free-body diagram has the same magnitude? Explain how you can tell.

3.4.4 Repeat exercise 3.4.1 for the force carts with plastic and black fabric bottom surfaces with the same mass inside the cart. Is the friction force the same for all 3 carts?

3.4.5 Add an additional 500 grams to the cart with the cork bottom, and again use the force probe to determine the friction force on the cart. Identify the forces in your free-body diagram that are different from those in exercise 3.4.1 and those that are the same.

3.4.6 Remove the additional mass from the cart so that there are again 500 grams total inside the cart. While the cart is pulled at a constant speed, push down on the cart with a finger. (Try to exert only a vertical force on the cart with your finger.) Use the force probe to determine the friction force.

3.4.7 Draw a single free-body diagram representing the forces acting on a system consisting of the cart and the masses in the cart for this situation. 3.4.8 Two students are discussing the friction force: Abel: The amount of friction acting on an object depends on the weight of the object. All other things being equal, it’s harder to push a heavier object. Beth: That’s true, but only because increasing the weight of an object increases how much the surfaces of the objects are squeezed together. Friction depends on the normal force.

Based on your answers to exercises 3.4.5, 3.4.6, and 3.4.7, do you agree with Abel or Beth? Explain.

Does the frictional force depend on the area of contact between the surfaces? (You will want to use your comparison in 3.4.5 to answer this question.) Explain.

In introductory physics, we usually use a model for friction that is given by the equation fAB = µNAB , where µ is called the coefficient of friction and N AB is the normal force acting on object A by object B. The coefficient of friction µ is an experimentally

10 determined quantity that depends on the surface of objects A and B that are in contact with each other.

In exercise 3.4.1, what is surface A? What is surface B? What is the value of µ between these surfaces? Show how you determined your answer.

Use your answers to the exercises above to rank the coefficients of friction between the surface of the track and the surfaces of the cork, fabric, and plastic carts. Explain how you determined your ranking. Generally, there are two coefficients given, one for static friction µS, and one for kinetic friction µK. You may or may not have seen a drop in force at the instant that the brick started to move. This drop suggests a difference between the maximum frictional force that may be exerted in a static situation and the frictional force that acts when there is motion. Comparing exercises 3.4.2, 3.4.3, and 3.4.4, what force from your free-body diagram does the frictional force depend on? What evidence do you have for your answer?

3.5: Free-body diagrams again

Now that we have looked at each of the forces (weight, tension, normal, and friction) individually, we finish this lab with a more challenging free-body diagram. (This will serve as preparation for the homework.)

Consider a climber in a ‘chimney’ as shown in the diagram. Draw a free-body diagram of the climber in the space below. Label each force that you have included in your free-body diagram to indicate (1) the type of force, (2) the object on which the force is exerted, and (3) the object exerting the force.

2.- Addition of Forces laboratory

Lab 5: Addition of Forces Name ______

1. Introduction

In this laboratory we look at forces in equilibrium. Newton’s second law states that the net force r r acting on a body is equal to its mass times its acceleration: Σ F = ma A complete understanding of Newton’s second law requires that you be able to find the net force. Forces add as vectors, and in this lab we will practice adding forces. We will make inferences about the directions and magnitudes of individual forces in the special case where the net force acting on an object is zero.

When an object is moving at a constant speed and in a constant direction, the acceleration of that object is zero. In this case, we can make an inference from Newton’s second law that the net force acting on that object is zero. That is, if we add all of the individual forces acting on that object as vectors the resultant must be zero. Since an object at rest can be considered to be moving at a constant speed (zero!) Newton’s second law requires that if we add all of the forces acting on an object that is at rest, the resultant will be zero. 1.1 Lab Objectives. After completing this lab and the associated homework, you should be able to:

7. Add vectors together to determine the resultant. 8. Determine the magnitude and direction of the resultant. 9. Draw a vector sum based on the forces in a free-body diagram. 10. Make inferences about the magnitudes and directions of unknown forces in cases where the net force acting is zero.

1.2 Outline of Laboratory Approximate sequence of the lab and homework:

7. Practice adding vectors to find the resultant. 8. Practice adding the forces from a free-body diagram. 9. Make inferences about the magnitudes of forces based on a vector sum. 10. Predict the relative magnitudes of tensions for three forces acting on a ring. 11. Use a force probe to determine the third force for an object at rest. 12. Use addition of forces to predict an unknown force.

2. Adding vectors. We begin this lab by considering vector addition without worrying about what those vectors represent.

12 2.1 Recall that vectors have both a magnitude (size) and a direction. When we add or subtract vectors, it is important that we do not change the vectors we are adding – that is, we do not want to change the magnitude or direction of the vector we are adding. When you add vectorsr r togetherr graphically, it is a good idea to add themr in a different location. To add the vectors Ar , B , and C together, we start by redrawing vector A in a newr location. We then redraw vector B rwith its tail placed at the positionr of the head of vector A . In the same way, the tail of vector C is placed at the head of vector B . The vector obtained when you add two or more vectors together is called the resultant. The resultant vector from the addition of the three vectors abover is the vector whose tail is at the position of the tail of the first vectorr (in this case, vector A ) andr whose head is at the position of the last vector (in this case, vector C ). The resultant is labeled R in the drawing below.

r r A A r B r r C B r r r r R=A+B+C r r r C 2.1.1 Show the sum of vectors D (a vector of magnitude 5) and E (a vector of magnitude 3) below.

r E r D 3 5

2.1.2 Three students discussing this vector sum make the following contentions: Student 1: “Once we find the resultant vector we can measure it to determine the magnitude of the resultant. That’s why we have to draw the vectors to scale.”

Student 2: “We don’t have to do that. Since we are adding a vector of magnitude 3 to a vector of magnitude 5, the vector sum will have a magnitude 8.”

Student 3: “No, that won’t work because they are not scalars. The resultant will be one side of a triangle, and we need to use the Pythagorean theorem to find the magnitude of the sum. I calculate a sum of magnitude 5.83.”

With which (if any) of these students do you agree? Explain.

r r r r 2.1.3 Show the vector F that satisfies the equation D + E + F = 0. What is the relationship between this vector and the resultant in section 2.1.1?

r r r 2.2 Use vectors G , H , and I at right to show that vector r addition is commutative (i.e., show that you get the same G resultant no matter what order you add the vectors in).

r r I H

2.2.1 Show how it is possible for two vectors to add to zero.

2.2.2 Show how it is possible for three vectors all of the same magnitude to add to zero. What is the angle between any two of these vectors? (Recall that the angle between two vectors is found by placing them tail-to-tail.)

Have your lab instructor check your answers to the questions above before proceeding.

3. Adding forces In the previous lab you practiced drawing free-body diagrams. Here we add the forces that act on a body as represented by the free-body diagram to find the resultant, called the net force. Newton’s second law relates the net force acting on a body to the acceleration

14 of that body. In this lab we investigate the special case of zero acceleration, and therefore zero net force. An object has zero acceleration if it is at rest or if it is moving at a constant speed and is not changing direction.

3.1 Consider a suitcase sliding at a r constant speed down a ramp that N r SR f makes an angle of 45° with the SR horizontal. From the drawing and Suitcase description, we can draw a free-body diagram: r WSE

We know that the weight points towards the center of the earth, the normal force is perpendicular to the ramp, and the friction force is parallel to the ramp opposite to the direction of motion.

We add the forces on the free-body diagram in the same way that we added the vectors in the previous section, except that in this case we don’t know the lengths of these vectors (i.e., we don’t know the magnitudes of the forces). However, we know that the vector sum must be zero since the suitcase is not accelerating. Based on this, we can construct a r N r SR NSR fSR 45° 45° 45° WSE fSR r 45° WSE vector sum:

The three vectors in the vector sum form a triangle, and we can use this triangle to reason about the relative sizes of the individual forces. In this case, suppose the mass of the suitcase is 20 kg.

3.1.1 What is the magnitude of the weight?

3.1.2 What is the magnitude of the normal force? Explain how you determined your answer.

3.1.3 What is the magnitude of the friction force? Explain how you determined your answer.

3.1.4 What is the coefficient of kinetic friction? Explain how you determined your answer.

Suppose that the ramp from section 3.1 was at an angle that was less than 45°, and that a different suitcase was sliding at a constant speed. Suitcase

3.2.1 Draw a free-body diagram for this situation, and from this free-body diagram construct a vector sum.

3.2.2 Rank the magnitudes of the forces in your free-body diagram. Explain how you used the vector sum to determine your answer.

3.3.1 A mass is suspended from two strings as shown. Draw a free-body diagram, and then use the free-body diagram to construct a vector sum.

String A

String B

3.3.2 Rank the magnitudes of the forces in your free-body diagram. Explain how you used the vector sum to determine your answer.

3.3.3 If the mass M is 300 grams, use your vector sum to find the approximate value of the tensions in strings A and B.

Have your lab instructor check your answers to the questions above before proceeding.

Rod

120° Spring Spring 1 Ring 2

4. Adding force vectors Spring 4.1 For the situation shown at right, draw a free-body 3 diagram for the ring. (You can ignore the weight of the Rod ring.) What is the net force on the ring? Explain how you Hook and can tell. Spring Springslotted masses 1 2

Ring

4.2 Show a vector sum of the forces on your free-body diagram. Spring Based on this vector sum, predict the relative lengths of the three 3 springs. Explain how you determined your prediction.

Hook and slotted masses

Rod Test your prediction using about 400 grams of mass on the hook (450 grams total), and resolve any inconsistencies. Measure the lengths of the 3 springs in this case, and record Spring Spring them here. 60° 1 2 4.3 Again draw a free-body diagram for the situation shown at right. Again show a vector sum of the forces on your free- Ring body diagram. (When you draw this vector sum, be careful that you do not change the direction of any of the vectors!) Spring 3

Hook and 4.4 Based on this vector sum, predict the relative lengths of the slotted masses three springs. Explain how you determined your prediction.

Would you expect the length of spring 1 in this case to be greater than, less than, or equal to the length of spring 1 in the situation shown in exercise 4.1 above? Explain your reasoning.

Move the suspension hooks outward so that when about 450 grams total is suspended, the angle between the upper strings is 60° as shown. Measure the lengths of the 3 springs and record them here. Were your answers above correct?

4.5 Now draw a free-body diagram for the case where the angle between the upper strings is 120°. Predict the relative lengths of the three springs. Use a drawing of the vector sum of the forces on the ring to explain the basis for your prediction. (Hint: What kind of triangle is formed by the vectors of the vector sum you have drawn?)

Move the suspension hooks outward so that with 250 grams total suspended, the angle between the upper strings is 120° as shown. Measure the lengths of the 3 springs and record them here. Were your answers above correct?

4.6 For the situation shown at right, the angle α is Rod greater than the angle β. Predict the relative lengths of the springs. Explain the basis for your predictions. Spring Spring β α 2 1

Ring α > β

Spring 3 18 Hook and slotted masses

4.7 Test your prediction, and resolve any inconsistencies. Then discuss your answers above with your laboratory instructor before continuing.

5. Addition of scaled vectors 5.1 We will now use a force table to observe the forces acting on a small ring. When the ring is not accelerating, it is said to be in equilibrium. ( Later this semester we will add second condition to the motion of an object in equilibrium having to do with rotation.) Place mass hangers on two of the strings that are attached to the ring. Hook the force probe to a third string. Add different masses (between 200g and 500g) to the two mass hangers.

Ring

Force probe

Force table

Hold the force probe horizontally and re-zero it (with no mass attached). Then hold the force probe so that the ring is no longer touching the pin and the ring is centered on the force table. Record the direction and magnitude of each horizontal force acting on the ring. In the space to the right, draw a free-body diagram of these forces on the ring. 5.2 On the next page we will add the forces (as vectors) recorded above graphically and to scale. Choose a scale for the vectors that you will use to represent the forces on the ring (for example,4 cm = 1N). Do not choose a scaling factor such that your vectors are too small to work with! ( Note: For simplicity, you may approximate g ≅ 10m/s 2.)

Ring

By constructing a scale drawing using a ruler and protractor, find the vector sum (or resultant) of the 3 tension forces on the ring by the tip-to-tail or polygon method. (That is, draw the tip of each vector at the tail of the succeeding vector.) The resultant vector is drawn from the tail of the first vector to the tip of the last vector. What would you expect the magnitude of the resultant vector to be in this case?

Is the resultant vector that you actually obtained graphically consistent with what you expected? 5.3 Your instructor will give you an unknown mass. Hang this mass from one of the pulleys. Hang a 300 gram mass from a second pulley that is placed at an angle of between 110° and 150° from the pulley that has the unknown mass.

Use the force probe to find the magnitude of the tension in a third string attached to the ring, and the direction of this third force acting on the ring.

Use a scaled vector sum to determine the unknown mass.

Once you have made a prediction based on your scaled diagram, use the scale to measure your unknown mass.

Ring

Force probe

Force table

? 300 g Unknown mass

Before you leave the lab, show your lab instructor your scaled vector sum.