AP Physics 1 Flashcards: Rolling

What this deck covers

This deck focuses on Rolling, giving you a quick way to review the definitions, rules, and examples that matter most for AP Physics 1.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the instantaneous speed of the contact point on a rolling wheel (relative to ground)?

Answer: $v_{contact}=0$. Contact point is instantaneous center of rotation for pure rolling.

Flashcard 2: Find the rolling distance given angular displacement $\theta$ and radius $r$.

Answer: $s = r \theta$. Rolling distance equals radius times angular displacement.

Flashcard 3: Calculate the moment of inertia for a ring rolling about its central axis.

Answer: $I = m r^2$. For a thin ring, all mass is concentrated at radius $r$.

Flashcard 4: What is the formula for potential energy of a rolling object at height $h$?

Answer: $PE = mgh$. Gravitational potential energy depends on mass, gravity, and height.

Flashcard 5: What is the static friction magnitude for rolling down an incline at angle $\theta$ (using $\beta=\frac{I_{cm}}{mR^2}$)?

Answer: $f_s=\frac{\beta}{1+\beta}mg\sin\theta$. Friction provides torque to maintain rolling constraint on incline.

Flashcard 6: What is the moment of inertia of a solid disk or solid cylinder about its center?

Answer: $I_{cm}=\frac{1}{2}mR^2$. Mass distributed uniformly throughout solid disk or cylinder.

Flashcard 7: Calculate the moment of inertia for a ring rolling about its central axis.

Answer: $I = m r^2$. For a thin ring, all mass is concentrated at radius $r$.

Flashcard 8: What is the direction of static friction on a driven wheel that is powered to accelerate forward without slipping?

Answer: Static friction points backward. Friction opposes slipping tendency; powered wheel would slip forward.

Flashcard 9: What is the direction of static friction on a freely rolling wheel pulled forward at its axle (no applied torque)?

Answer: Static friction points backward. Friction provides torque to accelerate rotation when pulled at axle.

Flashcard 10: State the formula for the total mechanical energy of a rolling object.

Answer: $E = KE_{\text{trans}} + KE_{\text{rot}}$. Total energy is the sum of translational and rotational kinetic energies.

Flashcard 11: What is the formula for the work done by torque on a rolling object?

Answer: $W = \tau \theta$. Work equals torque times angular displacement.

Flashcard 12: What is the moment of inertia for a solid sphere rolling about its central axis?

Answer: $I = \frac{2}{5} m r^2$. For a solid sphere, $I$ is two-fifths of $mr^2$.

Flashcard 13: What is the moment of inertia of a hollow cylinder rolling about its central axis?

Answer: $I = m r^2$. For a thin-walled cylinder, all mass is at radius $r$.

Flashcard 14: Find $\omega$ for rolling without slipping if $v_{cm}=6\,\text{m/s}$ and $R=0.50\,\text{m}$.

Answer: $\omega=12\,\text{rad/s}$. Apply $\omega = \frac{v_{cm}}{R} = \frac{6}{0.5} = 12$ rad/s.

Flashcard 15: What is the linear acceleration of an object rolling without slipping down an incline at angle $\theta$?

Answer: $a=\frac{g\sin\theta}{1+\frac{I_{cm}}{mR^2}}$. Derived from Newton's second law with rolling constraint.

Flashcard 16: What is the formula for the centripetal force on a rolling object?

Answer: $F_c = \frac{mv^2}{r}$. Centripetal force equals mass times velocity squared over radius.

Flashcard 17: What is the formula for potential energy of a rolling object at height $h$?

Answer: $PE = mgh$. Gravitational potential energy depends on mass, gravity, and height.

Flashcard 18: What is the speed of a point on the rim at the top of a rolling wheel (relative to ground)?

Answer: $v_{top}=2v_{cm}$. Top point moves at $v_{cm}$ plus rim speed $\omega R = v_{cm}$.

Flashcard 19: What is the parallel-axis theorem for moment of inertia?

Answer: $I=I_{cm}+md^2$. Relates moment of inertia about any axis to that about center of mass.

Flashcard 20: Find $a_{cm}$ for rolling without slipping if $\alpha=8\,\text{rad/s}^2$ and $R=0.25\,\text{m}$.

Answer: $a_{cm}=2.0\,\text{m/s}^2$. Apply $a_{cm} = \alpha R = 8 \times 0.25 = 2.0$ m/s².

Flashcard 21: What is the formula for the moment of inertia of a solid cylinder rolling about its central axis?

Answer: $I = \frac{1}{2} m r^2$. For a solid cylinder, moment of inertia equals half mass times radius squared.

Flashcard 22: What is the moment of inertia for a solid sphere rolling about its central axis?

Answer: $I = \frac{2}{5} m r^2$. For a solid sphere, $I$ is two-fifths of $mr^2$.

Flashcard 23: What is the rolling-without-slipping condition relating $v_{cm}$ and $\omega$ for radius $R$?

Answer: $v_{cm}=\omega R$. Center of mass velocity equals angular velocity times radius for pure rolling.

Flashcard 24: What is the condition for rolling without slipping?

Answer: $v = r \theta$. For no slipping, contact point velocity equals zero.

Flashcard 25: State the formula for the total mechanical energy of a rolling object.

Answer: $E = KE_{\text{trans}} + KE_{\text{rot}}$. Total energy is the sum of translational and rotational kinetic energies.

Flashcard 26: What is the moment of inertia for a thin spherical shell rolling about its central axis?

Answer: $I = \frac{2}{3} m r^2$. For a hollow sphere, $I$ is two-thirds of $mr^2$.

Flashcard 27: Find the rolling distance given angular displacement $\theta$ and radius $r$.

Answer: $s = r \theta$. Rolling distance equals radius times angular displacement.

Flashcard 28: What is the expression for the angular displacement of a rolling object?

Answer: $\theta = \frac{s}{r}$. Angular displacement equals arc length divided by radius.

Flashcard 29: What is the moment of inertia of a thin spherical shell about its center?

Answer: $I_{cm}=\frac{2}{3}mR^2$. All mass concentrated at surface of hollow sphere.

Flashcard 30: What is the moment of inertia of a hollow cylinder rolling about its central axis?

Answer: $I = m r^2$. For a thin-walled cylinder, all mass is at radius $r$.

Flashcard 31: What is the minimum coefficient $\mu_s$ needed to roll without slipping down an incline (using $\beta=\frac{I_{cm}}{mR^2}$)?

Answer: $\mu_s\ge\frac{\beta}{1+\beta}\tan\theta$. Ensures friction force doesn't exceed maximum static friction.

Flashcard 32: What is the expression for the angular displacement of a rolling object?

Answer: $\theta = \frac{s}{r}$. Angular displacement equals arc length divided by radius.

Flashcard 33: What is the equation for the translational kinetic energy of a rolling object?

Answer: $KE_{\text{trans}} = \frac{1}{2} mv^2$. Translational KE depends on mass and linear velocity squared.

Flashcard 34: What is the condition for rolling without slipping?

Answer: $v = r \theta$. For no slipping, contact point velocity equals zero.

Flashcard 35: What is the moment of inertia for a thin spherical shell rolling about its central axis?

Answer: $I = \frac{2}{3} m r^2$. For a hollow sphere, $I$ is two-thirds of $mr^2$.

Flashcard 36: What is the rolling-without-slipping condition relating $a_{cm}$ and $\alpha$ for radius $R$?

Answer: $a_{cm}=\alpha R$. Center of mass acceleration equals angular acceleration times radius for pure rolling.

Flashcard 37: What is the equation for the translational kinetic energy of a rolling object?

Answer: $KE_{\text{trans}} = \frac{1}{2} mv^2$. Translational KE depends on mass and linear velocity squared.

Flashcard 38: What is the moment of inertia of a solid sphere about its center?

Answer: $I_{cm}=\frac{2}{5}mR^2$. Mass distributed uniformly throughout solid sphere.

Flashcard 39: What is the formula for the work done by torque on a rolling object?

Answer: $W = \tau \theta$. Work equals torque times angular displacement.

Flashcard 40: What is the direction of static friction on a freely rolling wheel going down an incline without slipping?

Answer: Static friction points up the incline. Friction prevents wheel from sliding down faster than it rolls.

Flashcard 41: What is the formula for the centripetal force on a rolling object?

Answer: $F_c = \frac{mv^2}{r}$. Centripetal force equals mass times velocity squared over radius.

Flashcard 42: Identify the correct $a$ down a ramp: disk vs hoop, same $m$ and $R$, rolling without slipping.

Answer: Disk has larger $a$ than hoop. Disk has $\beta=0.5$ vs hoop's $\beta=1$, giving larger $a$.

Flashcard 43: Identify which reaches the bottom first (no slipping): hoop or solid sphere, same $R$ and same ramp.

Answer: Solid sphere. Sphere has smaller $\beta=\frac{I_{cm}}{mR^2}$, so larger acceleration.

Flashcard 44: What is the formula for the moment of inertia of a solid cylinder rolling about its central axis?

Answer: $I = \frac{1}{2} m r^2$. For a solid cylinder, moment of inertia equals half mass times radius squared.

Flashcard 45: What is the moment of inertia of a hoop (thin-walled cylinder) about its center?

Answer: $I_{cm}=mR^2$. All mass concentrated at distance $R$ from center.

Flashcard 46: What is the total kinetic energy of a rigid object rolling without slipping?

Answer: $K=\frac{1}{2}mv_{cm}^2+\frac{1}{2}I_{cm}\omega^2$. Sum of translational and rotational kinetic energies.