Rotational Inertia - AP Physics C Mechanics

1. Calculate torque for a rotational inertia of 2 kg m² and angular acceleration 4 rad/s².

   $\tau = 8 \text{ Nm}$. Using $\tau = I\alpha$ with given values.

2. Which object has higher rotational inertia: hoop or solid disk of same mass and radius?

   Hoop. Hoop has all mass at maximum radius from center.

3. What is the rotational inertia of a point mass $m$ at distance $r$ from axis?

   $I = mr^2$. Fundamental definition for point mass rotational inertia.

4. State the formula for rotational inertia of a hollow sphere.

   $I = \frac{2}{3} MR^2$. All mass concentrated at surface of sphere.

5. State the formula for rotational inertia of a thin rod about its end.

   $I = \frac{1}{3} ML^2$. Rod pivots about one end, maximizing distance from axis.

6. What is the parallel axis theorem formula?

   $I = I_c + Md^2$. Relates inertia about parallel axes to center-of-mass inertia.

7. What is the effect of rotational inertia on rolling motion?

   Higher inertia slows acceleration. Greater inertia requires more torque for same acceleration.

8. What happens to inertia when the axis of rotation is changed?

   It may increase or decrease. Inertia depends on distance from specific rotation axis.

9. Identify the variable representing radius in rotational inertia formulas.

   $R$. Radius determines distance from rotation axis.

10. What does $d$ represent in the parallel axis theorem?

    Distance from the center of mass to the new axis. Perpendicular distance between the two parallel axes.

11. Identify the variable representing mass in rotational inertia formulas.

    $M$. Mass appears in all rotational inertia formulas.

12. Find rotational inertia of a 4 kg hoop, radius 0.7 m, about its center.

    $I = 1.96 \text{ kg m}^2$. Using hoop formula $I = MR^2$ with given values.

13. Find the rotational inertia of a 5 kg solid sphere with radius 0.5 m.

    $I = 0.5 \text{ kg m}^2$. Using solid sphere formula with given values.

14. Identify the dependency of rotational inertia on shape.

    Depends on the mass distribution. Different shapes have different inertia formulas.

15. What is rotational inertia's role in angular motion?

    It resists changes in angular velocity. Rotational analog of mass in linear motion.

16. What is the formula for rotational inertia of a solid cylinder?

    $I = \frac{1}{2} MR^2$. For solid cylinder rotating about central axis.

17. State the formula for rotational inertia of a hoop about its center.

    $I = MR^2$. All mass concentrated at maximum distance from center.

18. How does increasing mass affect rotational inertia?

    Directly increases. Rotational inertia is linearly proportional to mass.

19. What is rotational inertia of a uniform rod about its center?

    $I = \frac{1}{12} ML^2$. Rod rotates about perpendicular axis through center.

20. Which factor does not affect rotational inertia?

    Color of the object. Rotational inertia depends only on mass distribution and geometry.

21. Find rotational inertia of a 6 kg rod, length 2 m, about its center.

    $I = 2 \text{ kg m}^2$. Using rod formula $I = \frac{1}{12}ML^2$ about center.

22. What is the effect of increasing radius on rotational inertia?

    Rotational inertia increases. Inertia depends on $R^2$ in most formulas.

23. What happens to rotational inertia if mass is doubled?

    Rotational inertia doubles. Rotational inertia is directly proportional to mass.

24. Find the rotational inertia of a $5 \, \text{kg}$ solid sphere with radius $0.5 \, \text{m}$.

    $I = 0.5 \text{ kg m}^2$. Using solid sphere formula with given values.

25. Calculate the rotational inertia of a 3 kg disk, radius 0.4 m, about its center.

    $I = 0.24 \text{ kg m}^2$. Using solid disk formula $I = \frac{1}{2}MR^2$.

26. What is the formula for rotational inertia of a thin spherical shell?

    $I = \frac{2}{3} MR^2$. All mass located at the spherical surface.

27. What is rotational inertia of a disk with mass 4 kg and radius 0.5 m?

    $I = 0.5 \text{ kg m}^2$. Using solid disk formula $I = \frac{1}{2}MR^2$.

28. What is the effect of a larger radius on the ease of spinning an object?

    Harder to spin. Larger radius increases rotational inertia significantly.

29. What is the unit of rotational inertia in the SI system?

    Kilogram meter squared ($\text{kg m}^2$). Derived from $[M][L]^2$ dimensional analysis.

30. Calculate rotational inertia of a 8 kg hollow sphere, radius 0.3 m, about its center.

    $I = 0.48 \text{ kg m}^2$. Using hollow sphere formula with given values.

31. How does mass distribution affect rotational inertia?

    Farther mass increases inertia. Mass farther from axis contributes more to inertia.

32. Find rotational inertia for a 3 kg hoop, radius 1 m, about its diameter.

    $I = 3 \text{ kg m}^2$. Hoop about diameter uses $I = \frac{1}{2}MR^2$.

33. Calculate the rotational inertia for a 2 kg point mass 3 m from the axis.

    $I = 18 \text{ kg m}^2$. Using $I = mr^2$ with $m=2$ kg, $r=3$ m.

34. Identify the axis of rotation in the rotational inertia formula.

    The fixed line about which the object rotates. The reference line for measuring distances in inertia calculations.

35. What is the formula for rotational inertia of a solid sphere?

    $I = \frac{2}{5} MR^2$. Mass distributed throughout sphere's volume.

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