Rotational Kinematics - AP Physics C Mechanics

1. Calculate the moment of inertia for a point mass $m$ at distance $r$.

   $I = m r^2$. Simplest case where all mass is concentrated at distance $r$.

2. What is the linear speed at the edge of a rotating disk?

   $v = r \omega$. Tangential velocity at any radius on a rotating object.

3. What is the relationship between frequency and angular velocity?

   $\omega = 2\pi f$. Angular velocity equals $2\pi$ times frequency in Hz.

4. What is the formula for net torque?

   $\tau_{net} = I \alpha$. Newton's second law for rotational motion with net torque.

5. Calculate the angular displacement given $\omega = 10$ rad/s, $t = 3$ s.

   $\theta = 30$ rad. Using $\theta = \omega t$ with constant angular velocity.

6. Identify the rotational analog of mass.

   Moment of inertia. Mass determines linear inertia; moment of inertia determines rotational inertia.

7. What is the formula for angular momentum?

   $L = I \omega$. Product of moment of inertia and angular velocity.

8. State the formula for angular displacement in radians.

   $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$. Kinematic equation analogous to $s = ut + \frac{1}{2}at^2$ for linear motion.

9. Identify the formula for tangential acceleration.

   $a_t = r \alpha$. Component of acceleration tangent to the circular path.

10. Define the term 'angular frequency'.

    The rate of rotation per unit time. How fast an object rotates, measured in radians per second.

11. What is the angular equivalent of Newton's second law?

    $\tau = I \alpha$. Rotational analog of $F = ma$ relating torque to angular acceleration.

12. Find the angular velocity if a wheel makes 20 revolutions in 10 seconds.

    $\omega = 4\pi$ rad/s. 20 rev in 10 s = 2 rev/s = $2 \times 2\pi = 4\pi$ rad/s.

13. What is the angular displacement for one complete rotation?

    $2\pi$ radians. One complete revolution equals $2\pi$ radians.

14. Calculate final $\omega$ for $\omega_0 = 2$ rad/s, $\alpha = 4$ rad/s², $t = 3$ s.

    $\omega = 14$ rad/s. Using $\omega = \omega_0 + \alpha t$ with given values.

15. What is the formula for angular displacement in terms of revolutions?

    $\theta = 2\pi N$. Each revolution equals $2\pi$ radians, so $N$ revolutions equals $2\pi N$.

16. What is the formula for angular frequency?

    $\omega = 2\pi f$. Same as angular velocity formula in terms of frequency.

17. Express the moment of inertia for a solid cylinder.

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

18. Convert $360$ degrees to radians.

    $2\pi$ radians. Full circle equals $2\pi$ radians by definition.

19. Define angular acceleration.

    Change of angular velocity per unit time. Rate of change of angular velocity, analogous to linear acceleration.

20. Define the term 'torque'.

    A measure of the force that can cause an object to rotate. Rotational analog of force that produces angular acceleration.

21. What is the formula for centripetal acceleration in terms of $\omega$?

    $a_c = r \omega^2$. Acceleration directed toward center of circular motion.

22. Calculate tangential speed given $r = 0.5$ m, $\omega = 2$ rad/s.

    $v = 1$ m/s. Direct application of $v = r\omega$ formula.

23. Find the angular acceleration given $\tau = 10$ Nm, $I = 2$ kg·m².

    $\alpha = 5$ rad/s². Using $\alpha = \frac{\tau}{I}$ with given values.

24. State the formula relating torque to force.

    $\tau = r F \sin \theta$. Cross product of position vector and force vector.

25. What is the relationship between linear and angular velocity?

    $v = r \omega$. Linear speed equals radius times angular velocity for circular motion.

26. Calculate $\theta$ if $\omega_0 = 0$, $\alpha = 2$ rad/s², $t = 4$ s.

    $\theta = 16$ rad. Using $\theta = \frac{1}{2}\alpha t^2$ when initial velocity is zero.

27. Define rotational kinetic energy.

    Energy due to the rotation of an object. Kinetic energy associated with spinning motion.

28. Define the term 'moment of inertia'.

    Resistance to change in rotational motion. Rotational inertia measuring resistance to angular acceleration.

29. State the angular displacement formula with constant $\omega$.

    $\theta = \omega t$. Simplified formula when angular velocity is constant.

30. What is the unit of angular displacement?

    Radians. Base SI unit for angular position measurement.

31. What is the formula for rotational kinetic energy?

    $KE_{rot} = \frac{1}{2} I \omega^2$. Kinetic energy formula for rotating objects with moment of inertia $I$.

32. What is the unit of moment of inertia?

    Kilogram meter squared (kg·m²). SI unit combining mass and distance squared.

33. What is the unit of torque?

    Newton meter (Nm). SI unit for rotational force measurement.

34. What is the formula for rotational work?

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

35. What is the unit of angular velocity?

    Radians per second (rad/s). Standard SI unit for rate of angular displacement change.

36. What is the rotational analog of linear momentum?

    Angular momentum. Conserved quantity in rotational motion, like linear momentum.

37. Calculate the angular velocity from $\theta = 3t^2$ at $t = 2$ s.

    $\omega = 12$ rad/s. Angular velocity is $\frac{d\theta}{dt} = 6t$, so at $t = 2$ s: $\omega = 12$ rad/s.

38. Find $\omega$ for a disk with $\theta = \pi$ rad in 0.5 s.

    $\omega = 2\pi$ rad/s. Using $\omega = \frac{\theta}{t}$ with constant angular velocity.

39. State the formula for final angular velocity.

    $\omega = \omega_0 + \alpha t$. Linear relationship between initial velocity, acceleration, and time.

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