Rotational Kinetic Energy
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AP Physics 1 › Rotational Kinetic Energy
A rigid hoop rotates about a fixed axis through its center. The hoop’s angular speed is reduced to half its original value, with no change in mass distribution. How does $K_{\text{rot}}$ change?
It becomes half because rotational kinetic energy is proportional to $\omega$.
It is unchanged because the axis of rotation did not change.
It becomes one-fourth because rotational kinetic energy is proportional to $\omega^2$.
It becomes twice as large because the hoop has the same mass.
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.
Two objects rotate about fixed axes with the same angular speed $\omega$. Object X has a larger moment of inertia than Object Y about its axis. Which statement about rotational kinetic energy is correct?
Their rotational kinetic energies depend only on mass, not on $I$.
They have equal rotational kinetic energy because $\omega$ is the same.
Object X has greater rotational kinetic energy because $K_{\text{rot}}\propto I$ for fixed $\omega$.
Object Y has greater rotational kinetic energy because it is easier to spin.
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.
A rigid object rotates about a fixed axis with angular speed $\omega$. A second object has the same mass but smaller moment of inertia about its axis and rotates at the same $\omega$. Which statement is correct?
The first object has greater rotational kinetic energy because $K_{\text{rot}}\propto I$ at fixed $\omega$.
The second object has greater rotational kinetic energy because smaller $I$ means larger energy.
They have equal rotational kinetic energy because their masses are equal.
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.
A wheel rotates about a fixed axle. Its angular speed increases, but the wheel’s mass distribution about the axle is unchanged. Which quantity is sufficient to conclude that $K_{\text{rot}}$ increases?
Only the wheel’s radius decreases.
Only the angular speed $\omega$ increases.
Only the linear speed of the axle increases.
Only the wheel’s total mass 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.
Two identical wheels rotate about the same fixed axle. Wheel 1 spins with angular speed $\omega$, and Wheel 2 spins with angular speed $3\omega$. How do their rotational kinetic energies compare?
Wheel 1 has greater rotational kinetic energy because it has less linear speed at the rim.
Wheel 2 has 3 times the rotational kinetic energy of Wheel 1.
They have equal rotational kinetic energy because their masses are equal.
Wheel 2 has 9 times the rotational kinetic energy of Wheel 1.
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.
A rigid rotor spins about a fixed axis. A student moves small masses outward along the rotor, increasing its moment of inertia, while keeping angular speed $\omega$ the same. What happens to rotational kinetic energy?
It increases because $K_{\text{rot}}=\tfrac12 I\omega^2$ and $I$ increases.
It decreases because the masses move farther from the axis.
It stays the same because $\omega$ is unchanged.
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).
A solid cylinder rotates about its symmetry axis. The cylinders angular speed increases while its translational speed of the center of mass remains zero. Which statement is correct?
Its rotational kinetic energy depends only on mass, so it cannot change.
Its rotational kinetic energy increases, even though the center of mass does not translate.
Its linear kinetic energy increases, but its rotational kinetic energy stays the same.
Its total kinetic energy stays zero because it is not moving linearly.
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).
A wheel rotates about its axle. A student compares two cases: (1) angular speed $\omega$ and (2) angular speed $3\omega$, with the same wheel. How does rotational kinetic energy change?
It triples because $K_{\text{rot}}\propto\omega$.
It increases by a factor of $9$ because $K_{\text{rot}}\propto\omega^2$.
It increases by a factor of $3$ because linear kinetic energy increases by $3$.
It is unchanged because the wheel’s moment of inertia is constant.
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.
Two identical wheels rotate about identical axles. Wheel X spins faster than wheel Y, but both have the same mass distribution. Which comparison of rotational kinetic energies is correct?
$K_{\text{rot},X}>K_{\text{rot},Y}$ because the faster angular speed gives larger $\omega^2$.
$K_{\text{rot},X}>K_{\text{rot},Y}$ only if wheel X also has greater mass.
$K_{\text{rot},X}=K_{\text{rot},Y}$ because the wheels are identical.
$K_{\text{rot},X}<K_{\text{rot},Y}$ because faster rotation reduces energy per revolution.
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.
Two objects rotate about the same axis with the same angular speed $ omega$: a solid disk and a hoop of equal mass and radius. Which has greater rotational kinetic energy?
They are equal, because mass and radius are the same.
The disk, because its mass is more concentrated near the axis.
They are equal, because angular speed is the same.
The hoop, because it has a larger moment of inertia.
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.