AP Physics 1 Quiz: Rotational Kinetic Energy

What this quiz covers

This quiz focuses on Rotational Kinetic Energy, 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. It becomes half because rotational kinetic energy is proportional to $\omega$.
2. It becomes one-fourth because rotational kinetic energy is proportional to $\omega^2$. (correct answer)
3. It becomes twice as large because the hoop has the same mass.
4. It is unchanged because the axis of rotation did not change.

Explanation: This question assesses understanding of rotational kinetic energy. The formula ( K_{\text{rot}} = \frac{1}{2} I \omega^2 ) indicates that energy scales with the square of angular speed ( \omega ), while moment of inertia ( I ) remains fixed if mass distribution doesn't change. Halving ( \omega ) reduces the energy to one-fourth because ( (\frac{1}{2}\omega)^2 = \frac{1}{4}\omega^2 ). This quadratic relationship amplifies reductions in speed more than linear changes would. Choice D is a distractor that incorrectly states the energy is unchanged because the axis is the same, ignoring the speed variation. A transferable approach is to compute the ratio of squared speeds when ( I ) is constant to find the energy change factor.

Question 2

1. Object Y has greater rotational kinetic energy because it is easier to spin.
2. They have equal rotational kinetic energy because $\omega$ is the same.
3. Object X has greater rotational kinetic energy because $K_{\text{rot}}\propto I$ for fixed $\omega$. (correct answer)
4. Their rotational kinetic energies depend only on mass, not on $I$.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is given by K_rot = ½Iω², showing linear dependence on moment of inertia I and quadratic dependence on angular speed ω. Since both objects rotate at the same angular speed ω, their rotational kinetic energies differ only due to their different moments of inertia. Object X has larger I, so K_rot,X = ½I_Xω² > K_rot,Y = ½I_Yω², meaning Object X has greater rotational kinetic energy. Choice A incorrectly associates "easier to spin" (lower I) with greater kinetic energy, when the opposite is true for fixed ω. When comparing rotational kinetic energies at equal angular speeds, the object with larger moment of inertia has more rotational kinetic energy.

Question 3

1. They have equal rotational kinetic energy because their masses are equal.
2. The second object has greater rotational kinetic energy because smaller $I$ means larger energy.
3. The first object has greater rotational kinetic energy because $K_{\text{rot}}\propto I$ at fixed $\omega$. (correct answer)
4. Their rotational kinetic energies depend only on $v$ of the center of mass.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is calculated as K_rot = ½Iω², where I is the moment of inertia and ω is the angular speed. When two objects with equal mass rotate at the same angular speed ω but have different moments of inertia, the object with larger I has greater rotational kinetic energy. Since the first object has larger I than the second object, it has greater K_rot. Choice B incorrectly suggests that smaller I means larger energy at fixed ω, reversing the actual relationship. The key principle is that rotational kinetic energy is directly proportional to moment of inertia when angular speed is held constant.

Question 4

1. Only the wheel’s total mass increases.
2. Only the wheel’s radius decreases.
3. Only the angular speed $\omega$ increases. (correct answer)
4. Only the linear speed of the axle increases.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is K_rot = ½Iω², depending on moment of inertia I and angular speed ω squared. The problem states that mass distribution is unchanged, meaning I remains constant. If angular speed ω increases while I stays constant, then K_rot must increase due to the ω² dependence. Choice A (increasing mass) would change I, contradicting the given constraint, while choice D (linear speed of axle) is irrelevant to rotational kinetic energy about the axle. When a rigid body's angular speed increases with constant moment of inertia, its rotational kinetic energy must increase quadratically.

Question 5

1. Wheel 2 has 3 times the rotational kinetic energy of Wheel 1.
2. Wheel 2 has 9 times the rotational kinetic energy of Wheel 1. (correct answer)
3. They have equal rotational kinetic energy because their masses are equal.
4. Wheel 1 has greater rotational kinetic energy because it has less linear speed at the rim.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy depends on both moment of inertia and angular speed according to K_rot = ½Iω². Since the wheels are identical and rotate about the same axis, they have equal moments of inertia I. Wheel 2 rotates at 3ω while Wheel 1 rotates at ω, so Wheel 2's rotational kinetic energy is K_rot,2 = ½I(3ω)² = 9(½Iω²) = 9K_rot,1. Choice C incorrectly assumes that equal mass means equal rotational kinetic energy, ignoring the crucial role of angular speed. When comparing rotational kinetic energies, remember that K_rot scales with the square of angular speed when I is constant.

Question 6

1. It decreases because the masses move farther from the axis.
2. It stays the same because $\omega$ is unchanged.
3. It increases because $K_{\text{rot}}=\tfrac12 I\omega^2$ and $I$ increases. (correct answer)
4. It becomes equal to the object’s linear kinetic energy at the rim.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is given by K_rot = ½Iω², where it depends linearly on moment of inertia I and quadratically on angular speed ω. When masses move outward along the rotor, they increase their distance from the axis, which increases the system's moment of inertia I. Since angular speed ω is kept constant, the rotational kinetic energy must increase proportionally with I. Choice B incorrectly assumes that constant ω means constant K_rot, ignoring the role of changing I. When analyzing rotational kinetic energy changes, consider both factors: changes in I (mass distribution) and changes in ω (rotation rate).

Question 7

1. Its linear kinetic energy increases, but its rotational kinetic energy stays the same.
2. Its rotational kinetic energy increases, even though the center of mass does not translate. (correct answer)
3. Its total kinetic energy stays zero because it is not moving linearly.
4. Its rotational kinetic energy depends only on mass, so it cannot change.

Explanation: This question assesses the concept of rotational kinetic energy in AP Physics 1. Rotational kinetic energy depends on moment of inertia I, which is constant for a given object and axis. It increases with the square of angular speed ω, independent of linear motion. Qualitatively, even with zero translational speed of the center of mass, increasing ω boosts K_rot as rotational motion accelerates. A common distractor is choice A, which incorrectly states rotational kinetic energy stays the same, confusing it with linear kinetic energy. To distinguish kinetic energy types, separate rotational contributions (depending on ω and I) from translational ones (depending on v_cm and M).

Question 8

1. It triples because $K_{\text{rot}}\propto\omega$.
2. It increases by a factor of $9$ because $K_{\text{rot}}\propto\omega^2$. (correct answer)
3. It is unchanged because the wheel’s moment of inertia is constant.
4. It increases by a factor of $3$ because linear kinetic energy increases by $3$.

Explanation: This question assesses understanding of rotational kinetic energy. Rotational kinetic energy relies on moment of inertia ( I ), which remains constant for the same object, and quadratically on angular speed ( \omega ), so tripling ( \omega ) multiplies the energy by nine due to ( (3\omega)^2 = 9\omega^2 ). The dependence on ( \omega^2 ) means speed changes have a squared impact, unlike linear kinetic energy's ( v^2 ) but adapted for rotation. With ( I ) unchanged, the energy scales directly with this factor. Choice C is a distractor that wrongly states the energy is unchanged because ( I ) is constant, ignoring the quadratic role of ( \omega ). A useful strategy for these problems is to isolate the changing variable in the formula and compute the ratio of new to old energy values.

Question 9

1. $K_{\text{rot},X}=K_{\text{rot},Y}$ because the wheels are identical.
2. $K_{\text{rot},X}>K_{\text{rot},Y}$ because the faster angular speed gives larger $\omega^2$. (correct answer)
3. $K_{\text{rot},X}<K_{\text{rot},Y}$ because faster rotation reduces energy per revolution.
4. $K_{\text{rot},X}>K_{\text{rot},Y}$ only if wheel X also has greater mass.

Explanation: This question assesses understanding of rotational kinetic energy. Rotational kinetic energy is ( K_{\text{rot}} = \frac{1}{2} I \omega^2 ), where for identical wheels, ( I ) is the same, but differences in ( \omega ) affect energy quadratically. A faster ( \omega ) for wheel X means its energy is larger due to the higher ( \omega^2 ) value. Since mass distributions are identical, ( I ) doesn't vary, isolating the effect to speed. Distractor A incorrectly suggests equal energies because the wheels are identical, but overlooks the differing speeds. For analogous questions, compare energies by calculating ratios based on the squared angular speeds when ( I ) is the same.

Question 10

1. The disk, because its mass is more concentrated near the axis.
2. The hoop, because it has a larger moment of inertia. (correct answer)
3. They are equal, because mass and radius are the same.
4. They are equal, because angular speed is the same.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is $K_{rot} = \frac{1}{2}I\omega^2$, where both objects have the same angular speed $\omega$. For objects of mass $m$ and radius $r$, the moment of inertia of a hoop is $I_{hoop} = mr^2$ while for a solid disk it's $I_{disk} = \frac{1}{2}mr^2$. Since $I_{hoop} > I_{disk}$ and both rotate at the same $\omega$, the hoop has greater rotational kinetic energy. Choice A incorrectly relates mass concentration to kinetic energy, confusing the effect on moment of inertia. To compare rotational kinetic energies at the same angular speed, compare the moments of inertia directly.

Question 11

1. smaller, because larger moment of inertia means less kinetic energy at the same angular speed.
2. the same, because the mass is the same.
3. larger, because $K_{rot}=\tfrac12 I\omega^2$ increases with $I$ when omega is constant. (correct answer)
4. the same, because kinetic energy depends only on linear speed.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is given by $K_{rot} = \frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular speed. When the wheel is replaced with one having the same mass but larger moment of inertia, while keeping angular speed $\omega$ constant, the rotational kinetic energy increases proportionally with $I$. This is because $K_{rot}$ is directly proportional to $I$ when $\omega$ is held constant. Choice A incorrectly suggests an inverse relationship between moment of inertia and kinetic energy at constant angular speed. To analyze rotational kinetic energy changes, identify which variables ($I$ or $\omega$) change and apply the formula $K_{rot} = \frac{1}{2}I\omega^2$ directly.

Question 12

1. It increases, because rotational kinetic energy depends on $\omega$ (correct answer)
2. It decreases, because increasing rotation reduces translational motion
3. It stays the same, because the rod’s mass is unchanged
4. It stays the same, because the axis is through the center

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is K_rot = ½Iω², depending on both moment of inertia I and angular speed ω squared. The rod's mass distribution (and thus I) stays unchanged, but ω increases. Since K_rot is proportional to ω², the rotational kinetic energy must increase. Choice C incorrectly assumes energy depends only on mass, not on how fast the object rotates. To analyze rotational energy changes, check whether I changes (mass distribution) and whether ω changes (rotation rate), then apply the formula.

Question 13

1. greater (correct answer)
2. smaller
3. the same, because mass and $\omega$ are unchanged
4. the same, because rotational kinetic energy depends only on $\omega$

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is K_rot = ½Iω², depending directly on moment of inertia I when angular speed ω is constant. The new object has the same mass and ω but larger I than the original disk. Since K_rot is proportional to I (when ω is fixed), the new object must have greater rotational kinetic energy. Choice C incorrectly assumes mass alone determines energy, ignoring how that mass is distributed (which affects I). When comparing rotating objects, remember that same mass can yield different I values depending on mass distribution.

Question 14

1. Object 1 definitely has greater rotational kinetic energy because it spins faster
2. Object 2 definitely has greater rotational kinetic energy because it has larger $I$
3. They must have equal rotational kinetic energy because both rotate about their centers
4. It cannot be determined without knowing both $I$ and $\omega$ (correct answer)

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is K_rot = ½Iω², depending on both moment of inertia I and angular speed ω squared. Object 1 has smaller I but larger ω, while Object 2 has larger I but smaller ω. Since K_rot depends on both I and ω², and these factors work in opposite directions for the two objects, we cannot determine which has greater energy without specific values. Choice A incorrectly assumes larger ω always means more energy, ignoring I. When comparing rotational energies without numbers, check if the competing factors (I and ω²) can be definitively ranked.

Question 15

1. Object Y has greater rotational kinetic energy because it is easier to spin
2. Object X has greater rotational kinetic energy because $K_\text{rot}=\tfrac12 I\omega^2$ (correct answer)
3. They have the same rotational kinetic energy because $\omega$ is the same
4. They have the same rotational kinetic energy because they rotate about the same axis

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is given by K_rot = ½Iω², showing it depends on both moment of inertia I and angular speed ω. Since both objects have the same ω but Object X has larger I, Object X has greater rotational kinetic energy. The formula directly shows that K_rot is proportional to I when ω is constant. Choice A incorrectly relates ease of spinning (which involves torque) to kinetic energy. To compare rotational kinetic energies, identify which quantities in K_rot = ½Iω² are the same and which differ.

Question 16

1. Cylinder 2 has 9 times the rotational kinetic energy of cylinder 1. (correct answer)
2. Cylinder 2 has 3 times the rotational kinetic energy of cylinder 1.
3. They have the same rotational kinetic energy because their masses are equal.
4. Cylinder 2 has 3 times the kinetic energy because $K=\tfrac12 mv^2$.

Explanation: This question tests understanding of rotational kinetic energy. For rotating objects, rotational kinetic energy is K_rot = ½Iω², where I is the moment of inertia and ω is the angular speed. Since the cylinders are identical, they have the same moment of inertia I. Cylinder 2 rotates at 3ω compared to cylinder 1's ω, so its kinetic energy is K_rot,2 = ½I(3ω)² = 9(½Iω²) = 9K_rot,1. Choice D incorrectly applies the translational kinetic energy formula to a rotation problem, missing that we need the rotational formula. To solve rotational energy problems, always use K_rot = ½Iω² and remember that energy scales with the square of angular speed.

Question 17

1. It increases because smaller objects always have more kinetic energy.
2. It decreases because $K_{\text{rot}}=\tfrac12 I\omega^2$ and $I$ is smaller. (correct answer)
3. It stays the same because mass is the only factor affecting rotational kinetic energy.
4. It changes sign because the axis is fixed.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is given by K_rot = ½Iω², showing direct proportionality to moment of inertia I when angular speed ω is constant. When the object is replaced by one with the same mass but smaller I (mass closer to the axis), while keeping ω unchanged, the rotational kinetic energy decreases proportionally with the decrease in I. Choice C incorrectly claims that only mass matters, ignoring how mass distribution (reflected in I) affects rotational kinetic energy. To analyze rotational kinetic energy changes, consider both factors separately: I (mass distribution relative to axis) and ω (rotation rate).

Question 18

1. It increases because $K_{\text{rot}}=\tfrac12 I\omega^2$ and $\omega$ is unchanged. (correct answer)
2. It decreases because added mass makes the wheel rotate more slowly.
3. It stays the same because only $\omega$ determines rotational kinetic energy.
4. It stays the same because the masses are added symmetrically.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is given by K_rot = ½Iω², where I is the moment of inertia and ω is the angular speed. When masses are added symmetrically near the rim while maintaining constant angular speed, the moment of inertia I increases (mass farther from axis contributes more to I). With ω constant and I increased, the rotational kinetic energy K_rot = ½Iω² must increase. Choice C incorrectly claims that only ω determines rotational kinetic energy, ignoring the critical role of moment of inertia. The key insight is that both I and ω² contribute to rotational kinetic energy, so increasing either while holding the other constant increases the total energy.

Question 19

1. $K_{\text{rot},A}=2K_{\text{rot},B}$ because $K_{\text{rot}}\propto \omega$.
2. $K_{\text{rot},A}=4K_{\text{rot},B}$ because $K_{\text{rot}}\propto \omega^2$. (correct answer)
3. $K_{\text{rot},A}=\tfrac14 K_{\text{rot},B}$ because $\omega$ is larger.
4. $K_{\text{rot},A}=K_{\text{rot},B}$ because the disks have equal mass.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy follows K_rot = ½Iω², where I is the moment of inertia and ω is the angular speed. For identical disks with the same I, disk A rotating at ω has kinetic energy K_rot,A = ½Iω², while disk B rotating at ½ω has K_rot,B = ½I(½ω)² = ¼(½Iω²). Therefore, K_rot,A = 4K_rot,B, meaning disk A has four times the kinetic energy of disk B. Choice A incorrectly assumes a linear relationship between K_rot and ω, when the actual relationship is quadratic. The strategy is to remember that rotational kinetic energy depends on the square of angular speed, so doubling ω quadruples the energy.

Question 20

1. The hoop, because more mass is farther from the axis, increasing $I$. (correct answer)
2. The cylinder, because it has greater mass density near the axis.
3. They are equal because both have the same mass, radius, and $\omega$.
4. They are equal because rotational kinetic energy depends only on mass.

Explanation: This question tests understanding of rotational kinetic energy. Rotational kinetic energy is K_rot = ½Iω², depending on both moment of inertia I and angular speed ω. For objects with the same mass and radius, the hoop has all its mass at distance R from the axis (I_hoop = MR²), while the solid cylinder has mass distributed throughout (I_cylinder = ½MR²). Since both rotate at the same ω, the hoop has greater rotational kinetic energy because its larger I dominates: K_hoop = ½(MR²)ω² > K_cylinder = ½(½MR²)ω². Choice C incorrectly assumes equal mass and radius guarantee equal rotational kinetic energy, missing that mass distribution affects I. To compare rotational kinetic energies at equal ω, focus on which object has mass farther from the rotation axis.