Rotational Kinetic Energy, Torque & Work (College Board AP® Physics 1: Algebra-Based): Exam Questions

Figure 1

A group of students have a wheel mounted on a horizontal axle and a small block of mass attached to one end of a light string. The other end of the string is attached to the wheel's rim and wrapped around it several times, as shown in Figure 1. When the block is released from rest and begins to fall, the wheel begins to rotate with negligible friction.

The students are asked to determine the rotational inertia of the wheel. The students measure the angular velocity of the wheel as the block falls a distance and determine the translational kinetic energy of the block immediately before it reaches the floor.

The student's measurements for different falling distances are shown in the following table.

i) Indicate two quantities that could be plotted to yield a linear graph whose slope could be used to calculate an experimental value for the rotational inertia of the wheel. Use the blank columns in the table to list any calculated quantities you will graph other than the data provided.

Vertical Axis: ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ Horizontal Axis: ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

ii) Plot the data points for the quantities indicated in part c)i) on the graph provided. Clearly scale and label all axes, including units, as appropriate.

iii) Draw a best-fit line to the data graphed in part c)ii).

Calculate an experimental value for the rotational inertia of the wheel using the best-fit line that you drew in part c)iii.

Figure 1

A system consists of a small sphere of mass and radius at rest on a horizontal surface and a uniform rod of mass and length attached at one end to a pivot with negligible friction, where . There is negligible friction between the surface and the sphere. The rod is held horizontally as shown in Figure 1, and then is released from rest. The total rotational inertia of the rod about the pivot is . After the rod is released, the rod swings down and strikes the sphere head-on.

Starting with conservation of energy, derive an expression for the angular speed of the rod just before striking the sphere in terms of the length and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 1

A group of students would like to determine the spring constant of a spring. One end of the spring is attached to a wall, and the other end is attached to a string, as shown in Figure 1. The string is wrapped around a horizontal cylinder of known rotational inertia . The cylinder is free to rotate about a fixed horizontal axle with negligible friction.

The students want to take measurements using the setup shown in Figure 1. They do not have access to any devices that can measure force or mass.

Describe an experimental procedure the students could use to collect the data needed to determine by creating a linear graph. Provide enough detail so that the experiment could be replicated, including any steps necessary to reduce experimental uncertainty.

Describe how the data collected in part a) could be plotted to create a linear graph and how that graph would be analyzed to determine .

Figure 2

The students would like to verify the value of the rotational inertia of the cylinder. With the cylinder removed from the axle shown in Figure 1, the students measured its diameter and its mass .

In a series of experimental trials, the cylinder was released from rest on a ramp from various heights , as shown in Figure 2. In each trial, the cylinder rolled without slipping down the ramp and onto a horizontal surface. The students then measured the time it took the cylinder to travel a distance along the horizontal surface. The results of each experimental trial are shown in the table.

i) For a graph that has on the horizontal axis, indicate which measured or calculated quantity could be plotted on the vertical axis to yield a linear graph whose slope could be used to calculate an experimental value for . Use the blank columns in the table to list any calculated quantities you will graph other than the data provided.

Horizontal Axis: ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ Vertical Axis: ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

ii) On the grid in Figure 3, plot the data points for the quantities indicated in part c)i) that can be used to determine . Clearly scale and label all axes, including units, as appropriate.

Figure 3

iii) Draw a best-fit line to the data graphed in part c)ii).

Calculate an experimental value for using the best-fit line that you drew in part c)iii).

Figure 1

A horizontal circular platform with rotational inertia rotates freely without friction on a vertical axis. A small motor-driven wheel that is used to rotate the platform is mounted under the platform and touches it. The wheel has radius and touches the platform a distance from the vertical axis of the platform, as shown in Figure 1. The platform starts at rest, and the wheel exerts a constant horizontal force of magnitude tangent to the wheel until the platform reaches an angular speed after time . During time , the wheel stays in contact with the platform without slipping.

i) On the axes shown in Figure 2, sketch and label a graph of the magnitude of the net torque exerted on the platform as a function of angular position during time .

Figure 2

Express your answers for a)ii), a)iii), and a)iv) in terms of , , , , , and physical constants as appropriate.

ii) Derive an expression for the angular speed of the platform. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

iii) Determine an expression for the kinetic energy of the platform at the moment it reaches angular speed .

iv) Derive an expression for the angular speed of the wheel when the platform has reached angular speed . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 3

A student holds a disk directly above and concentric with the platform, as shown in Figure 3. The disk has the same rotational inertia as the platform. When the platform is spinning at angular speed , the student releases the disk from rest, and the disk falls onto the platform. The motor-driven wheel keeps the disk and platform rotating together at the same constant angular speed .

In order to keep the system rotating with constant angular speed , is the motor doing positive work, negative work, or no work on the rotating system?

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ Positive work‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ Negative work ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ No work

Justify your answer.