Rotational Inertia
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AP Physics 1 › Rotational Inertia
Two identical uniform solid disks rotate about axes perpendicular to the disks. Disk A rotates about an axis through its center. Disk B rotates about an axis perpendicular to the disk through a point halfway from the center to the rim. Which disk has the greater rotational inertia about its stated axis?
They are equal because the axes are both perpendicular to the disk
Disk A, because Disk B experiences greater torque from any applied force
Disk B, because its mass is on average farther from the axis
Disk A, because rotational inertia depends only on total mass
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.
Two uniform spheres each have mass $M$ and radius $R$ and rotate about an axis through the center. Sphere A is solid. Sphere B is hollow with its mass concentrated in a thin shell at radius $R$. Which sphere has the greater rotational inertia about the axis?
Sphere A, because it would require greater torque to stop.
Sphere B (hollow shell), because more mass is farther from the axis.
Sphere A (solid), because it contains more mass overall.
They are equal because both 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. 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.
Two identical dumbbells rotate about a fixed axis through the midpoint, perpendicular to the connecting rod. In Dumbbell 1, the two equal masses are close to the midpoint. In Dumbbell 2, the same masses are farther from the midpoint. Which dumbbell has the greater rotational inertia about the axis?
They are equal because the dumbbells have the same total mass.
Dumbbell 1, because concentrating mass near the axis increases inertia.
They are equal because the same torque always produces the same angular acceleration.
Dumbbell 2, because placing mass farther from the axis increases inertia.
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.
Two identical square frames (each mass $M$) rotate about an axis perpendicular to the frame. Frame A rotates about an axis through its center. Frame B rotates about an axis through the midpoint of one side. Which frame has the greater rotational inertia about its axis?
Frame B, because more of the frame’s mass is farther from the axis.
They are equal because both frames have the same mass $M$.
Frame A, because rotating about the center always gives the largest inertia.
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.
Two circular objects of equal mass $M$ rotate about the same axis through the center, perpendicular to the plane. Object 1 is a thin ring of radius $R$. Object 2 is a thin ring of radius $2R$. Which object has the greater rotational inertia about the axis?
They are equal because both have mass $M$.
Object 1, because smaller radius means larger inertia.
They are equal because both experience the same torque due to gravity.
Object 2, because its mass is farther from the axis.
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.
Two identical dumbbells each consist of two small masses connected by a light rod. Dumbbell A rotates about an axis through the rod’s midpoint, perpendicular to the rod. Dumbbell B rotates about a parallel axis through one of the end masses. Which dumbbell has the greater rotational inertia about its axis?
Dumbbell A, because the axis passes through the center of mass
Dumbbell B, because more mass is at larger distance from its axis
They are equal because both dumbbells have the same total mass
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.
Two objects each have mass $M$ and length $L$ and rotate about a frictionless axle through one end, perpendicular to their length. Object 1 is a uniform thin rod. Object 2 is the same rod but with small dense caps added at both ends so mass is shifted outward while total mass stays $M$. Which object has the greater rotational inertia about the axle?
They are equal, because the axle location is the same
Object 1, because rotational inertia depends only on total mass
Object 2, because more mass is farther from the axis
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.
Two solid spheres have equal mass $M$ and equal radius $R$ and rotate about an axis through the center. Sphere 1 is uniform. Sphere 2 has density greater near its surface and smaller near its center (but total mass remains $M$). Which sphere has greater rotational inertia about the stated axis?
They are equal because both have the same $M$ and $R$
Sphere 2, because more mass is located farther from the axis
Cannot be determined without knowing the torque needed to spin them
Sphere 1, because uniform density maximizes rotational inertia
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.
Two objects each consist of two identical point masses $m$ connected by a light rod of length $L$, rotating about a fixed axis through the rod’s midpoint and perpendicular to the rod. In Object A, the masses are at the ends. In Object B, the masses are each at distance $L/4$ from the midpoint. Which object has greater rotational inertia?
Object A, because the masses are farther from the axis
They are equal, because total mass is $2m$ for both
Object B, because the torque needed to start rotating is larger
Object B, because the masses are closer so it spins faster
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.
Two spheres have equal mass $M$ and radius $R$ and rotate about an axis through their centers. Sphere 1 is a uniform solid sphere. Sphere 2 is a thin spherical shell with mass concentrated at radius $R$. Which sphere has the greater rotational inertia about the stated axis?
Sphere 1, because it requires more torque to stop
They are equal, because both have the same $M$ and $R$
Sphere 2, because more mass is farther from the axis
Sphere 1, because solid objects always have larger inertia
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.