Rotational Inertia (College Board AP® Physics 1: Algebra-Based): Exam Questions

Figure 1

Two pulleys with different radii are attached to each other so that they rotate together about a horizontal axle through their common center. There is negligible friction in the axle. Object 1 has mass and hangs from a light string wrapped around the larger pulley of radius , while Object 2 has mass and hangs from another light string wrapped around the smaller pulley of radius , as shown in Figure 1. The rotational inertia of a pulley of mass and radius is . The total mass of the two pulleys is . The pulleys are made from the same material and can be modeled as cylinders of equal length.

Derive an expression for the rotational inertia of the objects-pulleys system about the axle 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 1

The left end of a uniform beam of mass and length is attached to a wall by a hinge, as shown in Figure 1. One end of a string with negligible mass is attached to the right end of the beam. The other end of the string is attached to the wall above the hinge. A block of mass is also attached to the right end of the beam. The beam remains horizontal. The rotational inertia of a beam about its center is .

Derive an expression for the rotational inertia of the beam-block system about the hinge. 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 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. Three darts, each of mass , are traveling horizontally and perpendicular to the bar. The darts all travel at the same speed and stick to the bar at the same instant. Dart 1 hits the bar at a distance from the axle, Dart 2 hits the central axle, and Dart 3 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.

Derive an expression for the rotational inertia of the bar-darts system 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 2

The bar is repositioned so the vertical axle is attached at its right end, as shown in Figure 2. The three darts hit the bar at the same points as before.

Indicate whether the rotational inertia of the new bar-darts system is greater than, less than, or equal to the rotational inertia of the original bar-darts system. Justify your response.

Figure 1

A rod with a sphere attached to the end is connected to a horizontal mounted axle. The mass of the rod is and the mass of the sphere is . The center of mass of the rod-sphere system is indicated with a , as shown in Figure 1.

Figure 2

The rod-sphere system has mass and length , the sphere has radius , and the center of mass of the rod-sphere system is located a distance from the axle, as shown in Figure 2.

The rotational inertia of a rod of mass and length about one of its ends is . The rotational inertia of a solid sphere of mass and radius about its center is .

Derive an expression for the rotational inertia of the rod-sphere system in terms of and . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.