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

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. The total rotational inertia of the rod about the pivot is .

• At time , the rod is released from rest.

• At time , the rod strikes the sphere head-on. The rod is stopped and does not stick to the sphere. The sphere begins to slide to the left.

• At time , the sphere continues to slide to the left without rolling with constant speed .

i) On the axes shown in Figure 2, sketch a graph of the magnitude of the angular momentum of the rod-sphere system as a function of time from until .

Figure 2

ii) Starting with conservation of energy, derive an expression for the angular speed of the rod just before . 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.

iii) Starting with conservation of angular momentum, derive an expression for the translational speed of the sphere's center of mass after . 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.

Consider the case where a sphere made from a different material is used. The new sphere is identical to the original, but it sticks to the rod during the collision at time .

Indicate whether the angular momentum of the new rod-sphere system at is greater than, less than, or equal to the angular momentum of the original rod-sphere system at . Justify your response.

A person of mass is standing on a horizontal disk-shaped platform of mass and radius that can rotate with negligible friction about an axis at the disk's center. Initially, the person stands at the center of the disk as the disk rotates counterclockwise with a constant angular speed , as shown in Figure 1. Later, the person moves to the edge of the disk, and the disk rotates counterclockwise with a constant angular speed , as shown in Figure 2.

Indicate whether is greater than, less than, or equal to .

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your answer using physical principles, without deriving or manipulating equations.

The rotational inertia of the disk about its center is .

Derive an equation for in terms of , , , , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Indicate whether the equation you derived in part b) is consistent with your claim from part a). Justify your answer.

Figure 1

A uniform bar of length can rotate with negligible friction about a vertical axle at the bar's center, as shown in the top view of Figure 1. Two darts, each of mass , are traveling horizontally and perpendicular to the bar with the same speed . The bar is initially at rest and then rotates after the darts hit and stick to it at the same instant. Dart 1 hits the bar at a distance from the axle, and Dart 2 hits at the end of the bar on the opposite side of the axle, as shown. The bar has mass and rotational inertia about its center.

Figure 2 is a top view of the bar during the brief time interval that the darts are colliding with the bar. The dots in the figure represent the locations where the darts hit the bar and the bar's center.

On Figure 2, draw and label the horizontal forces, or force components, that act on the bar during the collisions. Each force must be represented by a distinct arrow starting on, and pointing away from, the point at which the force is exerted on the bar.

Figure 2

After the darts collide and stick to the bar, the bar–darts system rotates, and the Dart 1 moves in a circle with a final speed .

Starting with the conservation of angular momentum, derive an expression for in terms of , , , , and physical constants, as appropriate. Begin your derivation by writing either a fundamental physics principle or an equation from the reference book.

Figure 3

Figure 3 shows a top view of two nonuniform bars, Bar X and Bar Y, that can each rotate with negligible friction on vertical axles at their centers, similar to the original bar in Figure 1. Bars X and Y are both initially at rest and have the same mass. However, the mass is distributed differently in the two bars. Bar X has more of its mass concentrated near the center of the bar, while Bar Y has more of its mass located near the ends, as shown in Figure 3.

Identical darts are launched toward each bar, with a dart hitting each bar at the same velocity and at the same distance from that bar’s axle. The darts collide and stick to the bars. After the collisions, the Bar X–dart system rotates with angular speed , and the Bar Y–dart system rotates with angular speed .

Indicate whether is greater than, less than, or equal to .

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning.

Figure 1

Two satellites, of masses and , respectively, are in the same circular orbit about the Earth’s center, as shown in Figure 1. The Earth has mass and radius . In this orbit, which has a radius of , the satellites initially move with the same orbital speed but in opposite directions. The satellites collide head-on and stick together. After the collision, the satellites move with orbital speed .

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

i) Derive an expression for the orbital speed of the satellites before the collision. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

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

iii) Determine an expression for the total angular momentum of the system.