Rolling - AP Physics C Mechanics

1. What is the condition for rolling motion in terms of angular displacement?

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

2. What is $a$ down an incline for a hoop rolling without slipping?

   $a=\frac{1}{2}g\sin\theta$. Substitute $I_{\mathrm{cm}}=MR^2$ into general formula.

3. What is the formula for the moment of inertia of a solid sphere?

   $I = \frac{2}{5} m r^2$. For a solid sphere, mass distributed throughout volume.

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

   $v_{\mathrm{cm}}=\omega R$. No slipping means the contact point is instantaneously at rest.

5. State the formula for the moment of inertia of a hoop (thin ring) about its central axis.

   $I_{\mathrm{cm}}=MR^2$. All mass is concentrated at distance $R$ from the axis.

6. State the formula for the moment of inertia of a thin spherical shell about a diameter.

   $I_{\mathrm{cm}}=\frac{2}{3}MR^2$. All mass is at distance $R$ from center, integrated over sphere surface.

7. Calculate the mechanical energy of a rolling object.

   $E_{mech} = KE_{trans} + KE_{rot} + PE$. Conservation of energy includes all energy forms.

8. What is $a$ down an incline for a solid disk rolling without slipping?

   $a=\frac{2}{3}g\sin\theta$. Substitute $I_{\mathrm{cm}}=\frac{1}{2}MR^2$ into general formula.

9. Identify which has larger $a$ down the same incline: hoop ($I=MR^2$) or solid disk ($I=\frac{1}{2}MR^2$).

   Solid disk (larger $a$). Smaller $I$ gives larger $a$ in the acceleration formula.

10. What is the acceleration of the contact point for rolling without slipping when $a_{\mathrm{cm}}=0$ and $\omega$ is constant?

    $a_{\mathrm{contact}}=\omega^2R$ (upward). Contact point has centripetal acceleration toward the center.

11. What is the equation for angular displacement in rolling?

    $\theta = \frac{s}{r}$. Arc length equals radius times angle in radians.

12. State the formula for the moment of inertia of a solid sphere about a diameter.

    $I_{\mathrm{cm}}=\frac{2}{5}MR^2$. Mass distribution in 3D gives factor $\frac{2}{5}$ for solid sphere.

13. Calculate the acceleration of a disk rolling down an incline.

    $a = \frac{2}{3} g \text{sin}\theta$. For disk, $I = \frac{1}{2}mr^2$ gives this acceleration.

14. What is the formula for the moment of inertia of a disk?

    $I = \frac{1}{2} m r^2$. For uniform disk, mass distributed over area.

15. For rolling without slipping, what is the static friction direction for a body rolling down an incline under gravity only?

    Up the incline. Friction provides torque to maintain rolling constraint.

16. State the formula for the moment of inertia of a solid cylinder (disk) about its central axis.

    $I_{\mathrm{cm}}=\frac{1}{2}MR^2$. Mass is distributed uniformly throughout the solid cylinder.

17. What is the speed of the top point of a wheel rolling without slipping, in the ground frame?

    $v_{\mathrm{top}}=2v_{\mathrm{cm}}$. Top point has both translational and rotational velocities adding up.

18. What is the formula for the moment of inertia of a hoop?

    $I = m r^2$. All mass concentrated at rim, distance $r$ from center.

19. State the equation for the translational kinetic energy of a rolling object.

    $KE_{trans} = \frac{1}{2} m v^2$. Standard kinetic energy formula using linear velocity.

20. What is the parallel-axis theorem relating $I_{\mathrm{cm}}$ and $I$ about an axis a distance $d$ away?

    $I=I_{\mathrm{cm}}+Md^2$. Adds $Md^2$ when axis is displaced by distance $d$.

21. What is $I$ for a rolling body about the instantaneous contact point (use radius $R$)?

    $I_{\mathrm{contact}}=I_{\mathrm{cm}}+MR^2$. Apply parallel-axis theorem with $d=R$.

22. Calculate the final speed of a rolling object down an incline.

    $v = \sqrt{\frac{2gh}{1 + \frac{I}{mr^2}}}$$. Energy conservation with rotational inertia term.

23. Determine the ratio of rotational to translational kinetic energy for a solid sphere.

    $\frac{KE_{rot}}{KE_{trans}} = \frac{2}{5}$. For solid sphere, $I = \frac{2}{5}mr^2$ gives this ratio.

24. Identify the formula for the frictional force in rolling without slipping.

    $f = \frac{I a}{r^2}$. Friction provides torque for rolling without slipping.

25. Find the total kinetic energy of a rolling object.

    $KE_{total} = KE_{trans} + KE_{rot}$. Sum of translational and rotational kinetic energies.

26. What is the center-of-mass acceleration down an incline for rolling without slipping, using $I_{\mathrm{cm}}$?

    $a=\frac{g\sin\theta}{1+\frac{I_{\mathrm{cm}}}{MR^2}}$. Derived from $F=Ma$ and $\tau=I\alpha$ with rolling constraint.

27. What is $a$ down an incline for a solid sphere rolling without slipping?

    $a=\frac{5}{7}g\sin\theta$. Substitute $I_{\mathrm{cm}}=\frac{2}{5}MR^2$ into general formula.

28. Determine the linear acceleration of a rolling object down an incline.

    $a = \frac{g \sin \theta}{1 + \frac{I}{mr^2}}$. Derived from applying Newton's laws to rolling motion.

29. What is the total kinetic energy of a rigid body rolling without slipping on a surface?

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

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

    $a_{\mathrm{cm}}=\alpha R$. Linear acceleration equals angular acceleration times radius for no slip.

31. What is the speed of the contact point of a wheel rolling without slipping, in the ground frame?

    $v_{\mathrm{contact}}=0$. No-slip condition requires contact point to be instantaneously at rest.

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