Rolling - AP Physics 1

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

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

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

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

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

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

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

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

5. What is the moment of inertia of a solid disk or solid cylinder about its center?

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

6. What is the direction of static friction on a driven wheel that is powered to accelerate forward without slipping?

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

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

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

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

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

9. What is the linear acceleration of an object rolling without slipping down an incline at angle $\theta$?

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

10. What is the formula for potential energy of a rolling object at height $h$?

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

11. What is the speed of a point on the rim at the top of a rolling wheel (relative to ground)?

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

12. What is the parallel-axis theorem for moment of inertia?

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

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

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

14. What is the formula for the moment of inertia of a solid cylinder rolling about its central axis?

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

15. What is the moment of inertia for a solid sphere rolling about its central axis?

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

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

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

17. State the formula for the total mechanical energy of a rolling object.

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

18. What is the moment of inertia for a thin spherical shell rolling about its central axis?

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

19. What is the expression for the angular displacement of a rolling object?

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

20. What is the moment of inertia of a thin spherical shell about its center?

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

21. What is the moment of inertia of a hollow cylinder rolling about its central axis?

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

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

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

23. What is the equation for the translational kinetic energy of a rolling object?

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

24. What is the condition for rolling without slipping?

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

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

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

26. What is the moment of inertia of a solid sphere about its center?

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

27. What is the formula for the work done by torque on a rolling object?

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

28. What is the direction of static friction on a freely rolling wheel going down an incline without slipping?

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

29. What is the formula for the centripetal force on a rolling object?

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

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

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

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

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

32. What is the moment of inertia of a hoop (thin-walled cylinder) about its center?

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

33. What is the total kinetic energy of a rigid object rolling without slipping?

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

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