Angular Momentum & Angular Impulse (College Board AP® Physics 1: Algebra-Based): Exam Questions

Figure 1

The disk shown in Figure 1 spins about an axle at its center. During an experiment, a student discovers that, while the disk is spinning, the axle exerts a constant frictional torque on the disk. At time the disk has an initial counterclockwise (positive) angular velocity . The disk later comes to rest at time .

Starting with the equation for angular momentum, derive an equation for the rotational inertia of the disk in terms of , , , 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 has a horizontal disk that can rotate on a fixed vertical axle at its center, as shown in Figure 1. A string of negligible mass is wrapped around a groove at the disk's edge, and the diameter of the groove is measured to be .

The students notice that there is friction between the disk and the axle, and wish to experimentally determine the magnitude of the frictional torque exerted on the disk when it spins. The frictional torque can be assumed to be constant. The students are not able to remove the disk from the axle and the mounting platform, or directly measure the mass or rotational inertia of the disk. They have access to a force probe, several rulers, a meterstick, and several stopwatches.

i) Indicate quantities that could be measured by the students that would allow them to determine by applying the angular impulse-angular momentum theorem.

ii) Briefly describe a method to reduce experimental uncertainty for the measured quantities.

i) Indicate what quantities students could graph on the horizontal and vertical axes to create a linear graph that can be used to determine .

ii) Briefly describe the relationship between and a feature of the graph from part b)i). Your answer may include an equation that relates and the chosen feature of the graph.

Figure 2

Later, the students are at a playground, where there is a large rotatable disk, as shown in Figure 2. The disk has metal bars that people can hold on to while the disk spins about a vertical axle at its center. The students wish to experimentally determine the rotational inertia of the disk-bars system.

Using a force sensor and some rope attached to one of the metal bars, one of the students pulls tangentially to cause the disk to start spinning from rest, as shown in the top view of Figure 2. The students determine the change in angular momentum of the disk by recording the average force recorded by the force sensor and the amount of time that the force was applied. After the force is removed, they record the time that it takes the disk to complete 8 full revolutions. They measure the perpendicular distance from the extension of the rope to the center of the disk, and are able to determine that the disk rotates with negligible friction about its axle. Five experimental trials are conducted, and the results of the experiment are shown in Table 1.

Table 1

i) Label the vertical axis of Figure 3 with a measured or calculated quantity. Include units, as appropriate. The graphed quantities should yield a linear graph that can be used to determine an experimental value for .

ii) On the grid provided in Figure 3, create a graph of the quantities indicated in part c)i).

• Clearly label the vertical axis with a numerical scale

• Plot the corresponding data points on the grid

• Any columns added to Table 1 for scratch work will not be scored.

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 thin hoop of mass and radius has a rotational inertia around its center of . Three rods, each of mass , length , and rotational inertia , are fastened to the thin hoop, as shown in Figure 1.

The hoop-rods system is initially at rest and held in place, but is free to rotate around its center. A constant force is exerted tangentially to the hoop for a time .

i) Determine an expression for the total rotational inertia of the hoop-rods system about the center of the hoop in terms of , , and physical constants as appropriate.

ii) Derive an expression for the final angular speed of the hoop-rods system. Express your answer in terms of , , , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 1

A system consists of a small sphere of mass and radius at rest on a horizontal surface and a uniform rod attached at one end to a pivot with negligible friction. There is negligible friction between the surface and the sphere to the right of Point A and non-negligible friction to the left of Point A. The surface between Points A and B has a coefficient of kinetic friction . The rod is held horizontally as shown in Figure 1, and then is released from rest. The rotational inertia of the sphere about its center is .

After the rod is released, the rod swings down and strikes the sphere head-on. As a result of this collision, the rod is stopped, and the ball initially slides without rolling to the left across the horizontal surface. When the sphere encounters Point A, it begins rolling while slipping, and eventually begins rolling without slipping at Point B. Upon reaching Point B, the sphere has angular speed .

Indicate whether the sphere's angular momentum increases, decreases, or remains constant as it travels from Point A to Point B. Justify your answer using qualitative reasoning beyond referencing equations.

Derive an expression for the time it takes the sphere to travel from Point A to Point B in terms of , , , , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

The solid sphere is replaced with a hoop of the mass and radius . The rod is held horizontally and released from rest. After the rod strikes the hoop, it follows the same motion as the solid sphere, sliding without rolling before Point A, rolling while slipping between A and B, and rolling without slipping at Point B with angular speed .

Does the hoop reach Point B in more time, less time, or the same time as the solid sphere? Use the equation derived in part b) to justify your answer.