Rotational Inertia
Help Questions
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?
Disk B, because its mass is on average farther from the axis
Disk A, because Disk B experiences greater torque from any applied force
Disk A, because rotational inertia depends only on total mass
They are equal because the axes are both perpendicular to the disk
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 rods each have mass $M$ and length $L$ and rotate about axes perpendicular to the rod. Rod A rotates about an axis through a point one-quarter of the length from an end. Rod B rotates about an axis through its center. Which rod has the greater rotational inertia about its axis?
They are equal because both rods have the same $M$ and $L$.
Rod B, because it would feel a larger torque for the same angular acceleration.
Rod A, because its axis is farther from the center of mass.
Rod B, because symmetry about the center 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. 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.
A thin hoop and a solid cylinder each have mass $M$ and radius $R$. Both rotate about an axis through their centers, along the symmetry axis. Which object has the greater rotational inertia about the axis?
The solid cylinder, because it needs more torque to keep rotating at constant speed.
They are equal because both have the same mass $M$ and radius $R$.
The thin hoop, because more of its mass is at larger radius from the axis.
The solid cylinder, because its mass is spread throughout the volume.
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.
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?
They are equal because both have the same $M$ and $R$.
Sphere A, because it would require greater torque to stop.
Sphere A (solid), because it contains more mass overall.
Sphere B (hollow shell), because more 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. 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?
Dumbbell 2, because placing mass farther from the axis increases inertia.
They are equal because the same torque always produces the same angular acceleration.
They are equal because the dumbbells have the same total mass.
Dumbbell 1, because concentrating mass near 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.
Frame A, because it would require more torque to maintain constant angular velocity.
Frame A, because rotating about the center always gives the largest inertia.
They are equal because both frames have the same mass $M$.
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 point-mass systems rotate about the same fixed axis through point $O$. System 1 has two masses $m$ each at distance $r$ from $O$. System 2 has two masses $m$ each at distance $2r$ from $O$. Which system has the greater rotational inertia about $O$?
System 1, because the masses are closer to the axis.
They are equal because both would have the same angular acceleration under the same torque.
System 2, because the masses are farther from the axis.
They are equal because both systems have total mass $2m$.
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.
Two uniform rods, each of mass $M$ and length $L$, can rotate about a frictionless axle. Rod 1 rotates about an axis through its center, perpendicular to the rod. Rod 2 rotates about an axis through one end, perpendicular to the rod. Which rod has the greater rotational inertia about its stated axis?
They are equal because both rods have the same mass $M$.
They are equal because the same torque would spin either rod the same way.
Rod 2, because more of its mass is farther from the axis.
Rod 1, because its mass is closer to 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. 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.
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?
Object 1, because smaller radius means larger inertia.
Object 2, because its mass is farther from the axis.
They are equal because both have mass $M$.
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.
Two solid disks have the same mass $M$ and radius $R$ and rotate about the same type of axis: through the center and perpendicular to the disk. Disk A is uniform. Disk B has most of its mass concentrated near the rim (like a heavy outer ring). Which disk has the greater rotational inertia about the axis?
Disk A, because it would require less torque to start rotating.
Disk A, because uniform mass distribution increases rotational inertia.
Disk B, because more mass is farther from the axis.
They are equal because both have mass $M$ and radius $R$.
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. 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.