AP®PhysicsCollege BoardPhysics 1: Algebra-BasedExam QuestionsUnit 5: Torque & Rotational DynamicsRotational Inertia

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

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Rotational Inertia

1
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2 marks

A light, inextensible string connects two blocks of unequal mass over a pulley that rotates about a frictionless axle. In one experiment, the pulley is modeled as having negligible mass. In a second experiment, the pulley is modeled as having significant rotational inertia.

Indicate whether the magnitude of the acceleration a subscript 2 of the blocks in the second experiment is greater than, less than, or equal to the acceleration a subscript 1 of the blocks in the first experiment by writing one of the following:

  • a subscript 2 greater than a subscript 1

  • a subscript 2 equals a subscript 1

  • a subscript 2 less than a subscript 1

Justify your answer. In your justification, include qualitative reasoning beyond mathematical derivations or expressions.

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2
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4 marks
Diagram of a pulley system with two masses. Object 1 mass is \( m_0 \), Object 2 mass is \( 1.5m_0 \). Pulley radii labelled \( r_0 \) and \( 2r_0 \).

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 m subscript 0 and hangs from a light string wrapped around the larger pulley of radius 2 r subscript 0, while Object 2 has mass 1.5 m subscript 0 and hangs from another light string wrapped around the smaller pulley of radius r subscript 0, as shown in Figure 1. The rotational inertia of a pulley of mass M and radius R is I space equals space 1 half M R squared. The total mass of the two pulleys is M subscript P. 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 M subscript P, m subscript 0, r subscript 0, and physical constants as appropriate. Begin your derivation by writing either a fundamental physics principle or an equation from the reference book.

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3
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3 marks
A horizontal beam of length L hinged to a wall at its left end with a string at angle θ, supporting a block at its right end.

Figure 1

The left end of a uniform beam of mass M and length L 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 m is also attached to the right end of the beam. The beam remains horizontal. The rotational inertia of a beam about its center is 1 over 12 M L squared.

Derive an expression for the rotational inertia of the beam-block system about the hinge. Express your answer in terms of M, m, L, and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

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4a
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3 marks
A top view of a horizontal bar of length L with three darts approaching perpendicularly. Dart 1 approaches the bar at a distance r to the left of the axle, Dart 2 approaches the axle of the bar, and Dart 3 approaches the right end of the bar in the opposite direction to the other two.

Figure 1

A uniform bar of length L 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 m subscript D, 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 r space equals space L over 3 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 m subscript B and rotational inertia I subscript B space equals space 1 over 12 m subscript B L squared about its center.

Derive an expression for the rotational inertia of the bar-darts system in terms of L, m subscript B, m subscript D, and physical constants, as appropriate. Begin your derivation by writing either a fundamental physics principle or an equation from the reference book.

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4b
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3 marks
The same bar-darts system as in Figure 1, but with the axle moved to the right end of the bar.

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 by writing one of the following:

  • I_{n e w} > I_{o r i g i n a l}

  • I_{n e w} = I_{o r i g i n a l}

  • I_{n e w} < I_{o r i g i n a l}

Justify your response.

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5
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4 marks
A sphere attached to a rod on an axle via a rod. The center of mass of the system is located just below the center of the sphere.

Figure 1

A rod with a sphere attached to the end is connected to a horizontal mounted axle. The mass of the rod is m subscript r and the mass of the sphere is m subscript s space equals space 5 m subscript r. The center of mass of the rod-sphere system is indicated with a circled times, as shown in Figure 1.

A rod-sphere system hanging vertically downward with respect to the axle. Two lengths are indicated, length "L" represents the distance from axle to the bottom of the sphere, and length "3/4 L" represents the distance from the axle to the center of mass.

Figure 2

The rod-sphere system has mass M and length L, the sphere has radius R space equals space 1 over 6 L, and the center of mass of the rod-sphere system is located a distance 3 over 4 L from the axle, as shown in Figure 2.

The rotational inertia of a rod of mass m and length l about one of its ends is 1 third m l squared. The rotational inertia of a solid sphere of mass m and radius r about its center is 2 over 5 m r squared.

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

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