Rotational Kinetic Energy - AP Physics C Mechanics

1. How is the moment of inertia $I$ calculated for a solid sphere rotating about its center?

   $I = \frac{2}{5} m r^2$. Standard formula for uniform solid sphere about center.

2. What is the moment of inertia of a rod of length $L$ and mass $m$ about its center?

   $I = \frac{1}{12} m L^2$. Standard formula for thin rod about center.

3. Calculate the rotational kinetic energy for $I = 0.5 \text{ kg m}^2$ and $\theta' = 4 \text{ rad/s}$.

   $K = 4 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.5)(16) = 4$ J.

4. What is the rotational kinetic energy of a solid disk with $I = 1 \text{ kg m}^2$ and $\theta' = 2 \text{ rad/s}$?

   $K = 2 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(1)(4) = 2$ J.

5. What is the rotational analog of mass in linear motion?

   Moment of inertia. Both represent inertial resistance to acceleration.

6. Determine the moment of inertia for a point mass $m$ at radius $r$.

   $I = m r^2$. Definition of moment of inertia for point mass.

7. What effect does increasing mass have on the moment of inertia?

   Moment of inertia increases. Moment of inertia proportional to mass.

8. Find the rotational kinetic energy of a hoop with moment of inertia $I = 0.4 \text{ kg m}^2$ and angular velocity $3 \text{ rad/s}$.

   $K = 1.8 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.4)(9) = 1.8$ J.

9. What is the rotational kinetic energy of a disk with $I = 0.2 \text{ kg m}^2$ and $\theta' = 5 \text{ rad/s}$?

   $K = 2.5 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.2)(25) = 2.5$ J.

10. What is the moment of inertia for a thin cylindrical shell about its axis?

    $I = m r^2$. All mass concentrated at radius $r$ from axis.

11. Which formula relates torque $\tau$ and angular acceleration $\theta''$?

    $\tau = I \theta''$. Newton's second law for rotational motion.

12. What symbol represents angular velocity in rotational motion?

    $\theta'$. Standard notation for angular velocity in mechanics.

13. Calculate the rotational kinetic energy for $I = 3 \text{ kg m}^2$ and $\theta' = 4 \text{ rad/s}$.

    $K = 24 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(3)(16) = 24$ J.

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

    $I = \frac{1}{2} m r^2$. Mass distributed uniformly throughout solid cylinder.

15. What is the change in kinetic energy when angular velocity is halved?

    It decreases by a factor of four. KE proportional to $\theta'^2$, so halving gives 1/4.

16. Determine the kinetic energy for $I = 0.1 \text{ kg m}^2$ and $\theta' = 10 \text{ rad/s}$.

    $K = 5 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(0.1)(100) = 5$ J.

17. What is the moment of inertia for a thin spherical shell about its diameter?

    $I = \frac{2}{3} m r^2$. Standard formula for hollow sphere about diameter.

18. What is the moment of inertia of a rod of length $L$ and mass $m$ about an end?

    $I = \frac{1}{3} m L^2$. Standard formula for thin rod about end.

19. Calculate the kinetic energy of a rotating wheel with $I = 2 \text{ kg m}^2$ and $\theta' = 3 \text{ rad/s}$.

    $K = 9 \text{ J}$. Using $K = \frac{1}{2}I\theta'^2 = \frac{1}{2}(2)(9) = 9$ J.

20. What is the relationship between torque and rotational kinetic energy?

    Torque causes changes in rotational kinetic energy. Work done by torque equals change in rotational KE.

21. Identify the direction of angular momentum for a rotating body.

    Perpendicular to the plane of rotation. Angular momentum vector follows right-hand rule.

22. What is the unit of angular velocity in the SI system?

    Radians per second (rad/s). Standard SI unit for angular velocity.

23. How does doubling the angular velocity affect rotational kinetic energy?

    It quadruples the rotational kinetic energy. KE depends on $\theta'^2$, so doubling $\theta'$ quadruples KE.

24. What does the symbol $I$ denote in the context of rotational motion?

    Moment of inertia. Rotational analog of mass, resistance to angular acceleration.

25. What is the formula for rotational kinetic energy?

    $K = \frac{1}{2} I \theta'^2$. Analogous to $\frac{1}{2}mv^2$ but with $I$ and $\theta'^2$.

26. Identify the unit of rotational kinetic energy in the SI system.

    Joules (J). Energy has units of joules in SI system.

27. What happens to rotational kinetic energy if the moment of inertia doubles?

    It doubles if angular velocity is constant. KE is proportional to moment of inertia.

28. Which physical quantity is conserved in an isolated rotational system?

    Angular momentum. No external torques means angular momentum constant.

29. An object has a moment of inertia $I$ and $\theta' = 0$. What is $K$?

    $K = 0 \text{ J}$. Zero angular velocity means zero rotational kinetic energy.

30. State the relationship between linear velocity $v$ and angular velocity $\theta'$.

    $v = r \theta'$. Linear velocity equals radius times angular velocity.

31. Identify the rotational equivalent of force in linear motion.

    Torque. Both cause acceleration in their respective motions.

32. How is energy conserved in a system with both translational and rotational motion?

    Total kinetic energy is conserved. Sum of translational and rotational energies conserved.

33. What is the effect of increasing radius on the moment of inertia?

    Moment of inertia increases. Larger radius increases moment of inertia.

34. What is the formula for kinetic energy of a rigid body in pure rotation?

    $K = \frac{1}{2} I \theta'^2$. Standard rotational kinetic energy formula.

35. What is the formula for the total kinetic energy of a rolling object?

    $K_{\text{total}} = K_{\text{trans}} + K_{\text{rot}}$. Rolling objects have both translational and rotational energy.

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