AP Physics 1 Quiz: Rotational Inertia

What this quiz covers

This quiz focuses on Rotational Inertia, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Physics 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Disk A, because rotational inertia depends only on total mass
2. They are equal because the axes are both perpendicular to the disk
3. Disk B, because its mass is on average farther from the axis (correct answer)
4. Disk A, because Disk B experiences greater torque from any applied force

Explanation: This question assesses rotational inertia, measuring opposition to angular acceleration via mass placement. Rotational inertia is larger when mass distribution features greater perpendicular distances from the axis. Disk A's central axis symmetrizes mass, yielding I = (1/2)MR². Disk B's offset axis (at R/2) shifts mass farther on average, increasing I to (1/2)MR² + M(R/2)² = (3/4)MR². Choice B distracts by suggesting equality from perpendicular axes, ignoring position. For offset axes in uniform objects, employ the parallel axis theorem to compute and compare inertias reliably.

Question 2

1. Sphere A (solid), because it contains more mass overall.
2. Sphere B (hollow shell), because more mass is farther from the axis. (correct answer)
3. They are equal because both have the same $M$ and $R$.
4. Sphere A, because it would require greater torque to stop.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. Sphere B's hollow shell places all mass at radius R, increasing inertia over Sphere A's solid distribution with mass closer to the center. A common distractor is choice C, which falsely equates inertia based on total mass and radius alone, ignoring internal distribution. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 3

1. Dumbbell 1, because concentrating mass near the axis increases inertia.
2. Dumbbell 2, because placing mass farther from the axis increases inertia. (correct answer)
3. They are equal because the dumbbells have the same total mass.
4. They are equal because the same torque always produces the same angular acceleration.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. Dumbbell 2's masses at greater distances from the midpoint result in higher inertia than Dumbbell 1's closer placement. A common distractor is choice C, which incorrectly assumes equal inertia from total mass, disregarding positional differences. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 4

1. Frame A, because rotating about the center always gives the largest inertia.
2. Frame B, because more of the frame’s mass is farther from the axis. (correct answer)
3. They are equal because both frames have the same mass $M$.
4. Frame A, because it would require more torque to maintain constant angular velocity.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. Frame B's axis through the midpoint of a side shifts the distribution, placing more mass farther away than Frame A's central axis, resulting in higher inertia. A common distractor is choice C, which incorrectly assumes equal inertia from matching mass, neglecting axis location effects. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 5

1. Object 1, because smaller radius means larger inertia.
2. Object 2, because its mass is farther from the axis. (correct answer)
3. They are equal because both have mass $M$.
4. They are equal because both experience the same torque due to gravity.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. Object 2's larger radius of 2R places its mass farther out, significantly increasing inertia due to the r² dependence compared to Object 1. A common distractor is choice C, which mistakenly ties inertia only to mass, overlooking the radius's squared impact. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 6

1. Dumbbell A, because the axis passes through the center of mass
2. They are equal because both dumbbells have the same total mass
3. Dumbbell B, because more mass is at larger distance from its axis (correct answer)
4. Dumbbell A, because it experiences less torque for the same force

Explanation: This question tests comprehension of rotational inertia, quantifying resistance to angular acceleration based on mass placement relative to the rotation axis. Rotational inertia increases when more mass is located at larger perpendicular distances from the axis, as per I = Σ m r² for point masses. For Dumbbell A, the central axis places both masses at equal distances, giving a moderate I. For Dumbbell B, the axis through one mass sets its r = 0 and positions the other at maximum distance, yielding a higher I overall. Choice B distracts by claiming equality due to same total mass, overlooking how axis position alters effective distances. When analyzing systems of point masses, calculate each component's contribution separately to compare total inertias effectively.

Question 7

1. Object 1, because rotational inertia depends only on total mass
2. Object 2, because more mass is farther from the axis (correct answer)
3. They are equal, because the axle location is the same
4. Object 1, because the torque needed to start it rotating is smaller

Explanation: This question tests understanding of rotational inertia and how mass distribution affects it. Rotational inertia depends not just on the total mass of an object, but critically on how that mass is distributed relative to the axis of rotation. For Object 2, the dense caps at both ends mean more mass is located farther from the axis at the pivot end, increasing the rotational inertia according to I = Σmr². Since both objects have the same total mass M, but Object 2 has mass shifted outward, it will have greater rotational inertia. Choice A incorrectly assumes rotational inertia depends only on total mass, ignoring the crucial role of mass distribution. The key strategy is to identify where mass is concentrated relative to the rotation axis—mass farther from the axis contributes more to rotational inertia.

Question 8

1. Sphere 1, because uniform density maximizes rotational inertia
2. Sphere 2, because more mass is located farther from the axis (correct answer)
3. They are equal because both have the same $M$ and $R$
4. Cannot be determined without knowing the torque needed to spin them

Explanation: This question tests understanding of rotational inertia and density distribution effects. Rotational inertia depends on how mass is distributed relative to the rotation axis, with mass farther from the axis contributing more (proportional to r²). Sphere 1 has uniform density throughout, while Sphere 2 has higher density near the surface and lower density near the center, meaning more of its mass is located at larger radii. Since both spheres have the same total mass M and radius R, but Sphere 2 has more mass concentrated farther from the axis, Sphere 2 has greater rotational inertia. Choice A incorrectly claims uniform density maximizes rotational inertia, when actually concentrating mass far from the axis maximizes it. When comparing objects with the same mass and size but different density distributions, the one with more mass farther from the rotation axis has greater rotational inertia.

Question 9

1. Object B, because the masses are closer so it spins faster
2. Object A, because the masses are farther from the axis (correct answer)
3. They are equal, because total mass is $2m$ for both
4. Object B, because the torque needed to start rotating is larger

Explanation: This question tests understanding of rotational inertia. Rotational inertia is calculated as I = Σmr², where r is the distance from each mass to the axis. For Object A, each mass m is at distance L/2 from the axis, giving I = 2m(L/2)² = mL²/2. For Object B, each mass m is at distance L/4 from the axis, giving I = 2m(L/4)² = mL²/8. Object A has four times the rotational inertia of Object B because its masses are twice as far from the axis, and rotational inertia depends on the square of the distance. Choice C incorrectly focuses only on total mass, ignoring the critical role of mass distribution. To compare rotational inertias, calculate the distance of each mass element from the axis and remember that distance is squared in the calculation.

Question 10

1. Sphere 1, because solid objects always have larger inertia
2. Sphere 2, because more mass is farther from the axis (correct answer)
3. They are equal, because both have the same $M$ and $R$
4. Sphere 1, because it requires more torque to stop

Explanation: This question tests understanding of rotational inertia for solid versus hollow spheres. Rotational inertia depends on mass distribution according to I = Σmr², where r is the distance from the rotation axis. For the spherical shell (Sphere 2), all mass M is concentrated at radius R, giving I = ⅔MR². For the solid sphere (Sphere 1), mass is distributed throughout from r = 0 to r = R, resulting in I = ⅖MR². Since the shell has all its mass at the maximum possible distance R while the solid sphere has mass averaging closer to the center, the shell has greater rotational inertia. Choice A incorrectly claims solid objects always have larger inertia, when actually hollow objects of the same mass and size have more. The key insight is that hollow objects concentrate mass farther from the axis than solid ones.

Question 11

1. Dumbbell 1, because the masses are closer to the axis
2. They are equal, because each has the same total mass
3. Dumbbell 2, because the masses are farther from the axis (correct answer)
4. Dumbbell 1, because it experiences less torque from gravity

Explanation: This question tests understanding of rotational inertia for a simple dumbbell system. Rotational inertia depends on mass distribution according to I = Σmr², where r is the perpendicular distance from each mass to the rotation axis. In Dumbbell 1, each mass is at distance d/2 from the central axis, while in Dumbbell 2, each mass is at distance d from the axis. Since rotational inertia increases with the square of distance, Dumbbell 2 has I₂ = 2m(d)² = 2md², while Dumbbell 1 has I₁ = 2m(d/2)² = md²/2, making Dumbbell 2's rotational inertia four times larger. Choice B incorrectly assumes equal mass means equal rotational inertia, missing the critical role of distance. The strategy is to calculate or compare mr² terms for each mass element—doubling the distance quadruples the contribution to rotational inertia.

Question 12

1. Ring A, because smaller radius means larger rotational inertia
2. Ring B, because more mass is farther from the axis (correct answer)
3. They are equal, because the masses are the same
4. Ring A, because it requires more torque to maintain constant angular speed

Explanation: This question tests understanding of rotational inertia. For a uniform ring rotating about its central axis, all mass is at a constant distance from the axis, giving I = MR². Ring A has rotational inertia I_A = MR². Ring B, with radius 2R, has rotational inertia I_B = M(2R)² = 4MR². Ring B has four times the rotational inertia of Ring A because rotational inertia depends on the square of the distance from the axis. Choice A incorrectly suggests that smaller radius means larger rotational inertia, which contradicts the fundamental relationship I = Σmr². When comparing objects with different sizes but same mass, the one with mass distributed farther from the axis always has greater rotational inertia.

Question 13

1. Rod A, because its axis is farther from the center of mass. (correct answer)
2. Rod B, because symmetry about the center increases inertia.
3. They are equal because both rods have the same $M$ and $L$.
4. Rod B, because it would feel a larger torque for the same angular acceleration.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. Rod A's axis, offset from the center, distributes more mass farther away compared to Rod B's central axis, yielding higher inertia for Rod A. A common distractor is choice C, which wrongly suggests equal inertia from identical mass and length, ignoring axis position. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 14

1. System 1, because the masses are closer to the axis.
2. System 2, because the masses are farther from the axis. (correct answer)
3. They are equal because both systems have total mass $2m$.
4. They are equal because both would have the same angular acceleration under the same torque.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. System 2's masses at 2r contribute more due to the r² term, yielding higher inertia than System 1's at r. A common distractor is choice C, which erroneously links inertia only to total mass, neglecting the distance factor. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 15

1. Rod 1, because its mass is closer to the axis.
2. Rod 2, because more of its mass is farther from the axis. (correct answer)
3. They are equal because both rods have the same mass $M$.
4. They are equal because the same torque would spin either rod the same way.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. In this case, Rod 2 has more mass farther from its axis at the end compared to Rod 1's central axis, resulting in higher inertia for Rod 2. A common distractor is choice C, which incorrectly assumes rotational inertia depends only on total mass, ignoring the distribution relative to the axis. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 16

1. The solid cylinder, because its mass is spread throughout the volume.
2. They are equal because both have the same mass $M$ and radius $R$.
3. The thin hoop, because more of its mass is at larger radius from the axis. (correct answer)
4. The solid cylinder, because it needs more torque to keep rotating at constant speed.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. The thin hoop has all its mass at radius R from the axis, while the solid cylinder has mass distributed inward, resulting in higher inertia for the hoop. A common distractor is choice B, which incorrectly assumes equal inertia due to matching mass and radius, ignoring distribution differences. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 17

1. Disk A, because uniform mass distribution increases rotational inertia.
2. Disk B, because more mass is farther from the axis. (correct answer)
3. They are equal because both have mass $M$ and radius $R$.
4. Disk A, because it would require less torque to start rotating.

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion. Rotational inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the rotational inertia, as it follows the formula I = ∫ r² dm where r is the distance from the axis. Here, Disk B has more mass concentrated at larger radii near the rim, increasing its inertia compared to the uniform distribution in Disk A. A common distractor is choice C, which wrongly equates inertia solely to total mass and radius, disregarding the specific mass distribution. To compare rotational inertias, always consider the average squared distance of mass elements from the axis of rotation.

Question 18

1. The thin spherical shell, because more mass is farther from the axis (correct answer)
2. The solid sphere, because it contains more mass inside
3. They are equal because both have the same $M$ and $R$
4. The solid sphere, because it requires more torque to balance

Explanation: This question tests rotational inertia, which quantifies rotational resistance based on mass distribution from the axis. Rotational inertia grows with mass farther from the axis, emphasizing outer concentration. The solid sphere distributes mass inward, resulting in I = (2/5)MR² with lower average r². The thin shell places all mass at r = R, yielding I = (2/3)MR², which is greater. Choice C incorrectly posits equality from same M and R, ignoring internal structure. Compare hollow versus solid objects by using standard formulas to highlight how outer mass distribution increases inertia.

Question 19

1. Case 2, because more of the stick’s mass is farther from the axis (correct answer)
2. Case 1, because the stick is balanced about the axis
3. They are equal because the mass is the same in both cases
4. Case 2, because the torque from gravity is larger

Explanation: This question evaluates knowledge of rotational inertia, which describes how mass distribution affects an object's rotational dynamics around a specified axis. Rotational inertia is calculated as the integral of r² dm, where r is the distance from the axis, emphasizing that mass farther from the axis contributes more significantly. In case 1, the axis through the center yields I = (1/12)ML², with mass evenly distributed on both sides. In case 2, the end axis shifts the distribution, placing more mass at greater distances, resulting in I = (1/3)ML², which is larger. Choice C is a distractor that mistakenly equates rotational inertia solely to total mass, disregarding the axis location's impact on distance. A useful strategy is to use known formulas for common shapes and the parallel axis theorem to quantify how axis shifts increase inertia.

Question 20

1. Disk X, because rotational inertia depends only on mass
2. Disk Y, because more of its mass is farther from the axis (correct answer)
3. Disk X, because it would require more torque to start rotating
4. They are equal because the disks have the same $M$ and $R$

Explanation: This question assesses understanding of rotational inertia, which measures an object's resistance to changes in rotational motion about a given axis. Rotational inertia depends on the distribution of mass relative to the axis, specifically the sum of each mass element times its squared perpendicular distance from the axis. For Disk X, the axis is through the center, so mass is distributed symmetrically with an average squared distance leading to I = (1/2)MR². For Disk Y, the axis is at the rim, shifting the mass distribution such that more mass is farther away on average, resulting in a larger I = (3/2)MR² by the parallel axis theorem. A common distractor is choice D, which incorrectly assumes rotational inertia depends only on total mass and radius, ignoring the axis position's effect on mass distribution. To compare rotational inertias for similar objects, apply the parallel axis theorem when axes are offset from the center of mass.