Rotational Inertia - AP Physics 1

1. Calculate the rotational inertia of two point masses at opposite ends of a rod.

   Use $I = m_1 r_1^2 + m_2 r_2^2$. Apply point mass formula to each mass separately.

2. What is the relationship between torque and rotational inertia?

   $\tau = I \times \beta$. Rotational analog of Newton's second law ($F = ma$).

3. How does increasing mass affect rotational inertia?

   Increases rotational inertia. Rotational inertia is directly proportional to mass.

4. What is the formula for rotational inertia of a hollow sphere?

   $I = \frac{2}{3} m r^2$. Thin shell with all mass at surface radius.

5. State the parallel axis theorem formula.

   $I = I_{\text{cm}} + m d^2$. Relates inertia about any axis to center of mass axis.

6. What is the formula for rotational inertia of a thin rod about its end?

   $I = \frac{1}{3} m L^2$. Rod rotating about perpendicular axis at one end.

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

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

8. Determine the rotational inertia of a system of point masses.

   $I = \text{sum of } m_i r_i^2$. Add up $mr^2$ for each individual mass element.

9. What is the effect of doubling radius on rotational inertia?

   Quadruples rotational inertia. Radius appears squared in inertia formulas.

10. Which factor does rotational inertia depend on besides mass?

    Distribution of mass relative to axis. Mass farther from axis increases rotational inertia.

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

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

12. State the effect of rotational inertia on torque requirement.

    Higher inertia, more torque needed. From $\tau = I\alpha$: larger $I$ needs larger $\tau$.

13. What is the rotational inertia of a hoop about a tangent axis?

    $I = 2 m r^2$. Uses parallel axis theorem: $I = I_{center} + md^2$.

14. What is the effect of rotational inertia on angular velocity?

    Higher inertia, lower angular velocity. Conservation of angular momentum: $L = I\omega = constant$.

15. What is the effect of mass distribution on rotational inertia?

    Further mass increases inertia. Mass farther from rotation axis contributes more.

16. What is the rotational inertia of a disc about its diameter?

    $I = \frac{1}{4} m r^2$. Disc rotating about axis through its diameter.

17. Calculate the rotational inertia of two masses connected by a rod.

    Use $I = m_1 r_1^2 + m_2 r_2^2$. Treat each mass as point mass at its distance.

18. What is the rotational inertia of a solid sphere about a tangent axis?

    $I = \frac{7}{5} m r^2$. Uses parallel axis theorem with $d = r$.

19. Identify the formula for rotational inertia of a solid disc.

    $I = \frac{1}{2} m r^2$. Flat circular disc rotating about its center axis.

20. Identify the formula for rotational inertia of a disk about a tangent axis.

    $I = \frac{3}{2} m r^2$. Uses parallel axis theorem: $\frac{1}{2}mr^2 + mr^2$.

21. What is the rotational inertia of a ring about its diameter?

    $I = \frac{1}{2} m r^2$. Ring rotating about axis through its diameter.

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

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

23. What is the formula for rotational inertia of a hollow cylinder?

    $I = m r^2$. Cylindrical shell with all mass at outer radius.

24. Calculate the rotational inertia of a composite object.

    Sum of individual inertias. Add rotational inertias of each component part.

25. How does increasing the radius affect rotational inertia?

    Increases rotational inertia. Inertia depends on $r^2$, so larger radius increases it.

26. State the formula for rotational inertia of a sphere about its diameter.

    $I = \frac{2}{5} m r^2$. Same as solid sphere about any diameter through center.

27. What is the symbol for rotational inertia?

    Symbol: $I$. Standard physics notation for moment of inertia.

28. What does the rotational inertia of an object signify?

    Resistance to angular acceleration. Measures how hard it is to change angular motion.

29. State the formula for rotational inertia of a solid cylinder.

    $I = \frac{1}{2} m r^2$. Standard formula for uniform solid cylinder about its axis.

30. What is the rotational inertia of a thin spherical shell?

    $I = \frac{2}{3} m r^2$. Hollow spherical shell about any diameter through center.

31. What is the rotational inertia of a thin hoop about its diameter?

    $I = \frac{1}{2} m r^2$. Hoop rotating about perpendicular axis through diameter.

32. Identify the formula for rotational inertia of a thin rod about its center.

    $I = \frac{1}{12} m L^2$. For uniform rod rotating perpendicular to length at center.

33. Find the rotational inertia of a point mass at a distance $r$.

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

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