# Rotational Motion and Angular Momentum

## Inquiry Lab Kit for AP® Physics 1

### Materials Included In Kit

Pulley cord, 13.7 meters
Table pulleys, 6
Wheel and axles, 4-pulley, 6

(for each lab group)
Balance, 0.01-g precision (may be shared)
Hanging mass, 10-g
Scissors
Slotted masses, 1.0-g, 2-g, 5-g, 10-g
Support stands, 2
Support stand rods and clamps, 2
Tape measure or meter stick
Timer or clock with secondhand

### Safety Precautions

Wear safety glasses while performing this experiment.

### Disposal

All materials may be saved and stored for future use.

### Lab Hints

• This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Prelab Questions may be completed before lab begins the first day.
• Commercial slotted masses are recommended owing to their ease of use and premarked mass. Metric weights may also be used. In fact, any kind of mass, such as paperclips, can be used as the load for this experiment. If the mass is not marked, make sure students measure the mass of the object using a balance and record this in their data table.
• The wheel and axle resembles a hollow cone. However, the wheel and axle has an uneven distribution of mass. When looking at the underside of the wheels, there is more mass concentrated at the first wheel. The moment of inertia changes between the different wheels because the mass is unevenly distributed.
• For the Guided-Inquiry Design and Procedure, students may need to use small objects, such as washers and paper clips, to match the calculated masses for each wheel.

### Teacher Tips

• The guided-inquiry design and procedure section was developed to guide students through the experimental design process. You may decide to provide less or more information than is included—so long as the students are challenged to think critically.

### Further Extensions

Opportunities for Inquiry

Follow the Procedure in the Introductory Activity to determine the pulley radius and fall distance required for a 5-g mass to strike the floor after 5 seconds.

Alignment to the Curriculum Framework for AP® Physics 1

Enduring Understandings and Essential Knowledge
A force exerted on an object can cause a torque on that object.(3F)
3F1: Only the force component perpendicular to the line connecting the axis of rotation and the point of application of the force results in a torque about that axis.
3F2: The presence of a net torque along any axis will cause a rigid system to change its rotational motion or an object to change its rotational motion about that axis.

A net torque exerted on a system by other objects or systems will change the angular momentum of the system. (4D)
4D1: Torque, angular velocity, angular acceleration and angular momentum are vectors and can be characterized as positive or negative depending upon whether they give rise to or correspond to counterclockwise or clockwise rotation with respect to an axis.
4D2: The angular momentum of a system may change due to interactions with other objects or systems.
4D3: The change in angular momentum is given by the product of the average torque and the time interval during which the torque is exerted.

Learning Objectives
3F1.1: The student is able to use representations of the relationship between force and torque.
3F1.2: The student is able to compare the torques on an object caused by various forces.
3F1.3: The student is able to estimate the torque on an object caused by various forces in comparison to other situations.
3F2.1: The student is able to make predictions about the change in the angular velocity about an axis for an object when forces exerted on the object cause a torque about that axis.
3F2.2: The student is able to plan data collection and analysis strategies designed to test the relationship between a torque exerted on an object and the change in angular velocity of that object about an axis.
3F3.3: The student is able to plan data collection and analysis strategies designed to test the relationship between torques exerted on an object and the change in angular momentum of that object.
4D1.2: The student is able to plan data collection strategies designed to establish that torque, angular velocity, angular acceleration, and angular momentum can be predicted accurately when the variables are treated as being clockwise or counterclockwise with respect to a well-defined axis of rotation, and refine the research question based on the examination of data.
4D2.2: The student is able to plan a data collection and analysis strategy to determine the change in angular momentum of a system and relate it to interactions with other objects and systems.
4D3.1 The student is able to use appropriate mathematical routines to calculate values for initial or final angular momentum, or change in angular momentum of a system, or average torque or time during which the torque is exerted in analyzing a situation involving torque and angular momentum.
4D3.2 The student is able to plan a data collection strategy designed to test the relationship between the change in angular momentum of a system and the product of the average torque applied to the system and the time interval during which the torque is exerted.

Science Practices
2.2 The student can apply mathematical routines to quantities that describe natural phenomena.
4.1 The student can justify the selection of the kind of data needed to answer a particular scientific question.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
4.3 The student can collect data to answer a particular scientific question.
4.4 The student can evaluate sources of data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
5.3 The student can evaluate the evidence provided by data sets in relation to a particular scientific question.
6.1 The student can justify claims with evidence.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.

### Science & Engineering Practices

Developing and using models
Planning and carrying out investigations
Analyzing and interpreting data
Using mathematics and computational thinking
Constructing explanations and designing solutions

### Disciplinary Core Ideas

HS-PS2.A: Forces and Motion
HS-PS3.A: Definitions of Energy
HS-PS3.B: Conservation of Energy and Energy Transfer

### Crosscutting Concepts

Patterns
Cause and effect
Scale, proportion, and quantity
Energy and matter

### Performance Expectations

HS-PS2-1. Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.
HS-PS2-2. Use mathematical representations to support the claim that the total momentum of a system of objects is conserved when there is no net force on the system.
HS-PS3-2. Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motion of particles (objects) and energy associated with the relative position of particles (objects).
HS-PS2-3. Apply scientific and engineering ideas to design, evaluate, and refine a device that minimizes the force on a macroscopic object during a collision.
HS-PS3-1. Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.

1. A block (m = 150 g) is held 100 cm above the floor. Calculate the time, in seconds, it will take for the object to strike the ground once it is released.

F = mg
d = ½gt2

2. Consider the same block hanging vertically and attached by a cord and pulley to a 100-g frictionless cart sitting on a table. Calculate the time, in seconds, it takes for the object to strike the ground once it is released.

d = ½at2

3. Consider the same block hanging vertically and, this time, attached by a cord and pulley to a wheel having a uniform mass of 100 g and a radius of 10 cm. Calculate the time, in seconds, it takes for the object to strike the ground once it is released. Hint: Keep the moment of inertia as the variable I throughout the calculations for simplicity.

For the forces on the block:

mh = 150 g = 0.15 kg
Fnet = mha = Fg – FT
Fg = mhg
mha = mhg – FT

For the forces on the disk:

r = 10 cm = 0.10 m
md = 100 g = 0.10 kg
I = ½md r2 = 0.5 x 0.10 kg x (0.10 m)2
I = 5.0 x 10–4 kg∙m2
T = r x FT = rFT
T = Iα
rFT = Iα
α = a/r
rFT = Ia/r
FT = Ia/r2

mha = mhg – I a/r2
mha + I a/r2 = mhg
a(mh + I/r2) = mhg

a = 7.36 m/s2

t = 0.52 sec

### Sample Data

Introductory Activity

Hanging Mass = 10.0 g = 0.0100 kg
Fall Distance = 93.0 cm = 0.930 m

{13791_Data_Table_1}

Calculations for Wheel 1

a = 2d/t2
{13791_Data_Equation_1}
a = 0.246 m/s2
{13791_Data_Equation_2}
{13791_Data_Equation_3}
T = FT x r = I x α
{13791_Data_Equation_4}
{13791_Data_Equation_5}
{13791_Data_Equation_6}
{13791_Data_Equation_7}
I = 1.19 x 10–4 kg∙m2
ΔL = IαΔt
ΔL = (1.19 x 10–4 kg∙m2)(14.1 rad/s2)(2.75 s)
ΔL = 0.00460 kg∙m2/s

ω = ωo + αt

θ = θo + ωot + ½αt2
θ = ½αt2

Guided-Inquiry Design and Procedure
{13791_Data_Table_1}
Percent Error of Fall Time

Percent Error = |Experimental – Theoretical|/Theoretical x 100
For Wheel 1:

Percent Error = |1.85 s – 2.00 s|/2.00 s x 100
Percent Error = 7.50%

Error Analysis
Overall, the experiment was successful. Each wheel had fall times that were close to the required 2.00 seconds. Random errors due to timing may have contributed to the larger percent errors of the larger wheels. The masses required for the larger wheels were small washers and it was difficult to determine when the washers hit the floor. It was easier to hear the masses for the smaller wheels because larger washers were used. Photogates with automated timekeeping could be used to lower the number of timing errors due to human reaction time.

Guided-Inquiry Design and Procedure

1. Examine the collected free-fall time data. What is the relationship between the radius of the wheel and the free-fall time?

As the radius of the wheel was increased and the hanging mass and fall distance remained constant, less time was required for the mass to fall the prescribed distance. When the radius was 1.75 cm (0.0175 m), the average fall time was 2.78 seconds. When the radius was 5.21 cm (0.0521 m), the average fall time was 0.96 seconds. Note to teachers: When working with ideal objects, there is no relationship between the free-fall time and the radius of the object. The wheel and axle used in the Introductory Activity has an uneven distribution of mass giving rise to different moments of inertia for the different wheel sizes.

2. The wheel and axle resembles a hollowed-stepped cone. The moment of inertia of a cone is: Icone = ½ mcR2. Explain why a cone can be used as an analogous object for the wheel and axle.

When a force is applied to a cone at some radius, the entire mass of the cone rotates. This is similar to what was observed with the wheel and axle. When a force was applied to the smallest radius, the whole wheel and axle rotated. Unlike a hollowed cone, when the radius increased, the wheel still rotated and the moment of inertia changed. On the smallest radius, the moment of inertia was 1.22 x 10–4 kg∙m2; on the largest radius the moment of inertia was 1.05 x 10–4 kg∙m2. This inverse relationship between radius and moment of inertia is unlike the cone because as the radius increases on a cone, the moment increases (direct relationship).

3. A cone has a mass of 100.0 g and is 15.0 cm in radius at the base. Its axis of rotation runs vertically from the base to the top point. A 25.0 g mass is attached to a string and wound around the cone where the radius is 5.00 cm.
1. Find the time it takes for the mass to fall 1.00 m.

F = mha = Fg – FT
FT = Fg – mha = mh(g–a)

Icone = ½ x mcR2
I = (0.5)(0.1000 kg)(0.150 m)2
I = 1.13 x 10–3 kg∙m2

T = FT x r = I x α

mhgr2 – mhr2a = Ia
mhgr2 = a(I + mhr2)
a = (mhgr2)/(I + mhr2)

t = 1.97 s

2. Find the angular momentum of the cone when the mass hits the ground.

L = Txt = Iαt

L = 2.29 x 10–2 kg∙m2/s

4. Using the cone described in the previous question and keeping the radius, fall distance, and hanging mass constant, calculate the following.
1. What mass would be required for the fall time to be 1.50 seconds?

a = 2d/t2

a = 0.889 m/s2
T = FT x r = I x α

mh = 0.0450 kg = 45.0 g

2. What is the angular momentum of the cone when the mass hits the ground?

L = Txt = Iαt

L = 3.01 x 10–2 kg∙m2/s

5. Without performing a calculation, predict the amount of mass required to free-fall for 0.50 seconds. Is the mass greater than, less than, or equal to the amounts found in the previous two questions? Explain in terms of torque, angular acceleration and moment of inertia.

When a 25.0 g mass was attached to the cone, the fall time was approximately 1.97 seconds. In order to decrease the fall time to 1.50 seconds, a mass of approximately 45.0 g is needed. When the mass was increased, the fall time decreased. A larger mass produces a greater torque on the cone. The moment of inertia is constant for the cone. A greater torque results in a greater angular acceleration. A large angular acceleration means the hanging mass has a greater acceleration towards the ground.

To lengthen the fall time, a mass smaller than 25.0 g would be needed. A smaller mass will produce a smaller torque on the wheel, thereby producing a smaller angular acceleration. With less angular acceleration, the hanging mass will accelerate more slowly towards the ground.

6. Without performing a calculation, predict the angular momentum of the cone if the hanging mass were to fall to the ground in 0.50 seconds. Is the momentum greater than, less than or equal to the values found in the previous questions? Explain in terms of torque and angular acceleration.

The change in angular moment of the cone when the free-fall time was 1.97 seconds is 2.29 x 10–2 kg∙m2/s. The change in angular moment when the free-fall time was 1.50 seconds is 3.01 x 10–2 kg∙m2/s. As the free-fall time decreased, the change in angular moment increased. In order to have a shorter fall time, a greater hanging mass is needed to produce a larger torque and angular acceleration on the wheel. Although the fall time decreases, the increase in torque and angular acceleration are to a greater degree causing the angular momentum to increase overall.

7. Based on the previous questions, predict how changing the hanging mass would affect the free-fall time and angular momentum of the wheel and axle. For example, if the string were used on the largest wheel and the mass were increased, what effect would the mass have on the free fall time and angular momentum?

As seen with the cone, decreasing the hanging mass while keeping the radius the same causes the fall-time to increase. If the hanging mass were decreased on the large wheel, the fall time would increase and the change in angular momentum would decrease. If the hanging mass were increased, the fall time would decrease and the change in angular momentum would increase.

8. Using the apparatus set up in the Introductory Activity, design an experiment so that each wheel results in the mass falling over a 2.00 second time. Identify the independent and dependent variables, as well as any other factors that should remain constant throughout the experiment.

In this experiment, the hanging mass must be changed in order to achieve a 2.00 second free-fall time. The actual time will be measured. The fall distance, string, and wheel and axle should be kept the same. When the string is moved between the different wheels, the supporting pulley should be adjusted so the string remains parallel to the table. The table below shows the mass required for each wheel to have a 2.00 second free-fall time. A combination of small objects will be used to match the predicted mass as closely as possible.

Calculations for Wheel 1
{13791_Data_Equation_8}
{13791_Data_Equation_9}
a = 0.465 m/s2
T = FT x r = I x α
{13791_Data_Equation_10}
{13791_Data_Equation_11}
{13791_Data_Equation_12}
{13791_Data_Equation_13}
mh = 0.0198 kg = 19.8 g

Answers to the Review Questions for AP® Physics 1

A merry-go-round of mass 100.0 kg and radius 3.00 m is at rest. The merry-go-round resembles a disk and is free to rotate about its central point on frictionless bearings. It is set into rotational motion by a person holding a handle on the outside edge and running counterclockwise for 13.0 seconds. When the handle is released, the angular velocity of the merry-go-round is 45.5 rad/s.

1. What is the angular momentum of the merry-go-round when the person releases the handle?

L = Iω
I = ½mr2
I = ½(100.0 kg )(3.00 m)2
I = 450 kg∙m2
L = 450 kg∙m2 x 45.5 rad/s
L = 20475 kg∙m2/s

2. What is the angular acceleration of the merry-go-round immediately before the handle is released?

ω = ω0 + αt
α = ω/t = (45.5 rad/s)/13.0 s

3. What is the angular acceleration immediately after the handle is released? Explain your answer.

The angular acceleration immediately after the handle is released is zero. When the handle is released, a torque (force) is no longer being applied to the merry-go-round. If the net torque on the merry-go-round is zero, then the angular acceleration is also zero.

4. What amount of torque was applied to the merry-go-round?

T = Iα
T = 1575 N∙m
In order to stop the merry-go-round, a person pushes a “brake” button. The merry-go-round comes to a stop in 2.50 seconds.

5. What is the final angular momentum of the merry-go-round?

The final angular momentum of the merry-go-round is 0 kg∙m2/s because it is no longer rotating.

6. What amount of torque is applied by the brakes?

ΔL = TΔt
T = ΔLt
T = (0 – 20475 kg∙m2/s)/2.5 s
T = – 8190 N∙m

7. What is the angular acceleration of the merry-go-round over the 2.50 seconds?

ω = ω0 + αt
0 = ω0 + αt
α = –ω0/t

8. What do the signs on the torque and angular acceleration values mean?

Both the torque and angular acceleration have negative signs. The negative sign indicates that the torque and angular acceleration are opposite to the initial direction (counterclockwise) of the merry-go-round. By opposing the motion, torque and angular acceleration were slowing the merry-go-round to a stop.

### References

AP Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014.

# Rotational Motion and Angular Momentum

### Introduction

What would be more difficult to stop: a small, solid wheel spinning fast or a Ferris wheel spinning slowly? What equations and variables describe the motion and properties of these spinning objects? Discover the relationship between rotational motion and angular momentum.

### Concepts

• Torque
• Angular momentum
• Moment of inertia
• Acceleration

### Background

A rigid body sitting on a frictionless surface follows Newton’s second law of motion. If a constant force, F, is applied (see Figure 1), and the body is initially at rest, the equations of motion are as follows for acceleration, velocity and distance where m is the total mass of the rigid body:

{13791_Background_Figure_1}
{13791_Background_Equation_1}
{13791_Background_Equation_2}
{13791_Background_Equation_3}
A rigid body that rotates around a fixed axis has equivalent rotational equations to the translational (linear) equations above. The rotational equivalents are: angular displacement, θ, measured in radians (rad); angular velocity, ω, measured in rad/sec; and angular acceleration, α, measured in rad/s2. Rotational motion with constant angular acceleration can be described by the following equations:
{13791_Background_Equation_4}
{13791_Background_Equation_5}
{13791_Background_Equation_6}
Rotational motion is related to linear motion by considering how far the point of interest is from the axis of rotation. This distance is typically the radius, r, of the object.
{13791_Background_Equation_7}
{13791_Background_Equation_8}
{13791_Background_Equation_9}
For a rotating object, the equivalent to mass is called the moment of inertia, I. For a symmetrical disk of uniform density and a radius of R (see Figure 2):
{13791_Background_Equation_10}
{13791_Background_Figure_2}
To cause rotation, a force must be applied at a radial distance, r, from the fixed, rotational axis. Torque, T, is the rotational equivalent to force in a linear system and is equal to the product of the radial distance from the center to where the force is applied, the magnitude of the force, and the angle between the two. In the case of rotating objects, the angle between the force and radial distance is 90°. See Figure 2 and Equation 11:
{13791_Background_Equation_11}
Because torque is an external force on the disk, it influences the angular acceleration of the disk:
{13791_Background_Equation_12}
If the external torque were to be removed from the disk, it would continue to rotate. Like a box sliding across the floor, both objects have momentum. For the disk, this is called angular momentum and is the product of the angular velocity and moment of inertia of the object. Angular momentum is measured in kg∙m2/s.
{13791_Background_Equation_13}
If the disk starts out at rest, the change in angular momentum can be calculated by multiplying the net torque and the duration of time the torque is applied to the disk.
{13791_Background_Equation_14}
The sign convention for rotational motion, including torque, angular acceleration and angular momentum, is based on the direction of the rotation. If the rotation is counterclockwise, the value is positive. If the rotation is clockwise, it is a negative value.

### Experiment Overview

The purpose of this advanced inquiry investigation is to verify the connection between Newton’s second laws of linear and rotational motion. The lab begins with an Introductory Activity to determine the moment of inertia for a solid disk having four pulley arrangements. Then, the Guided-Inquiry Design and Procedure section presents a challenge to use the data from the Introductory Activity to calculate, and then verify, the correct mass to attach to each flywheel, so that it takes 2.00 seconds to fall from the bench height to the floor.

### Materials

Balance, 0.01-g precision
Mass, hooked, 10-g
Pulley, with clamp
Pulley cord, 190 cm
Slotted mass set, 1.0-g, 2-g, 5-g, 10-g
Support stands, 2
Support stand rods and clamps, 2
Tape measure or meter stick
Timer
Wheel and axle, 4-pulley

### Prelab Questions

1. A block (m = 150 g) is held 100 cm above the floor. Calculate the time, in seconds, it will take for the object to strike the ground once it is released.
2. Consider the same block hanging vertically and attached by a cord and pulley to a 100-g frictionless cart sitting on a table. Calculate the time, in seconds, it takes for the object to strike the ground once it is released.
3. Consider the same block hanging vertically and, this time, attached by a cord and pulley to a wheel having a uniform mass of 100 g and a radius of 10 cm. Calculate the time, in seconds, it will take for the object to strike the ground once it is released. Hint: Keep the moment of inertia as the variable I throughout the calculations for simplicity.

### Safety Precautions

Students should wear safety glasses while performing this experiment.

### Procedure

Introductory Activity

Falling Mass and Rotational Motion

1. Measure the distance, in centimeters, from the tabletop to the floor and record this value in a data table.
2. Set up the equipment to determine the moment of inertia, I, acceleration of the mass, a, and the rotational acceleration, α. Cut a length of pulley cord that is twice as long as the distance from the tabletop to the floor.
3. Thread the cord through the opening on the smallest pulley of the wheel and axle (see Figure 3). Tie a knot on the cord inside the wheel and axle.
{13791_Procedure_Figure_3}
4. Attach the wheel and axle to the support stand and rod using the clamp holders (see Figure 4). Note: The wheel should be parallel to the tabletop.
{13791_Procedure_Figure_4}
5. Attach the free pulley to the second support stand and support block (see Figure 5).
{13791_Procedure_Figure_5}
6. Align the free pulley with the wheel and axle so that the pulley cord, when placed in the pulley, is parallel to the tabletop and is 90° tangentially to the occupied wheel. See Figures 6a and 6b.
{13791_Procedure_Figure_6a}
{13791_Procedure_Figure_6b}
7. Make a loop on the free end of the pulley cord (see Figure 7).
{13791_Procedure_Figure_7}
8. Hang the 10-g mass from the loop, then wind the pulley cord clockwise around the first wheel until the line is taut and the mass is at the top edge of the table.
9. Start the timer while simultaneously releasing the 10-gram weight.
10. Stop the timer when the mass strikes the floor. Record the time.
11. Repeat steps 8 through 10 four more times, recording each acceleration time in the data table.
12. Remove the pulley cord from the first pulley, and attach it to the second wheel through the opening (see Figure 8).
{13791_Procedure_Figure_8}
13. Repeat steps 8 through 10 five times for the second pulley, recording each acceleration time in the data table.
14. Continue the procedure for each of the remaining two wheels.
15. Place the mass on the support stand base for use in the next section.

Analyze the Results

Present the data collected in the Introductory Activity in an appropriate table. Calculate the following for each radii of the wheel: acceleration of the free falling mass, angular acceleration of the the wheel, moment of inertia of the wheel, and angular momentum of the wheel. Remember to consider the sign of torque, angular acceleration and angular momentum based on the direction of rotation. Also calculate the final angular velocity, ω, and the angular displacement, θ, of the wheel at each radius.

Guided-Inquiry Design and Procedure

Form a working group with other students and discuss the following questions.

1. Examine the collected free-fall time data. What is the relationship between the radius of the wheel and the free-fall time?
2. The wheel and axle resembles a hollowed-stepped cone. The moment of inertia of a cone is: Icone = ½mc R2. Explain why a cone can be used as an analogous object for the wheel and axle.
3. A cone has a mass of 100.0 g and is 15.0 cm in radius at the base. Its axis of rotation runs vertically from the base to the top point. A 25.0 g mass is attached to a string and wound around the cone where the radius is 5.00 cm.
1. Find the time it takes for the mass to fall 1.00 m.
2. Find the angular momentum of the cone when the mass hits the ground.
4. Using the cone described in the previous question and keeping the radius, fall distance, and hanging mass constant, calculate the following.
1. What mass would be required for the fall time to be 1.50 seconds?
2. What is the angular momentum of the cone when the mass hits the ground?
5. Without performing a calculation, predict the amount of mass required to free-fall for 0.50 seconds. Is the mass greater than, less than or equal to the amounts found in the previous two questions? Explain in terms of torque, angular acceleration and moment of inertia.
6. Without performing a calculation, predict the angular momentum of the cone if the hanging mass were to fall to the ground in 0.50 seconds. Is the momentum greater than, less than or equal to the values found in the previous questions? Explain in terms of torque and angular acceleration.
7. Based on the previous questions, predict how changing the hanging mass would affect the free-fall time and angular momentum of the wheel and axle. For example, if the string were used on the largest wheel and the mass were increased, what effect would the mass have on the free fall time and angular momentum?
8. Using the apparatus set up in the Introductory Activity, design an experiment so that each wheel results in the mass falling over 2.00 seconds. Identify the independent and dependent variables, as well as any other factors that should remain constant throughout the experiment.

Analyze the Results

Present the actual hanging mass values and free fall times in an appropriate data table. Calculate the following for each radii of the wheel: acceleration of the free falling mass, angular acceleration of the wheel and the change in angular momentum of the wheel. Remember to consider the sign of torque, angular acceleration and angular momentum based on the direction of rotation. Calculate the percent error in the fall times for each wheel and discuss the reliability of the experiment and methods to improve the experiment.

### Student Worksheet PDF

13791_Student1.pdf

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