Rotational Kinematics - AP Physics 1

1. Determine rotational kinetic energy of a $2$ kg·m² disk rotating at $4$ rad/s.

   $K = 16$ J. Use $K = \frac{1}{2}I\omega^2$ formula.

2. State the relationship between torque and angular momentum change.

   $\tau = \frac{dL}{dt}$. Torque equals the time rate of change of angular momentum.

3. What is the new angular velocity if a $2$ kg·m² object increases its moment of inertia to $4$ kg·m²?

   $\text{ω}_{\text{new}} = \frac{1}{2}\text{ω}_{\text{initial}}$. Conservation of angular momentum: $I_1\omega_1 = I_2\omega_2$.

4. What is the unit of angular velocity in the International System of Units (SI)?

   Radians per second (rad/s). Standard SI unit for rotational motion rate.

5. Which quantity measures an object's resistance to changes in rotational motion?

   Moment of Inertia. Inertia determines how hard it is to change rotation.

6. Find the centripetal acceleration of a point on a wheel $0.2$ m from the center, rotating at $10$ rad/s.

   $a_c = 20$ m/s². Use $a_c = r\omega^2$ formula.

7. What is the relationship between frequency ($f$) and period ($T$) of rotation?

   $f = \frac{1}{T}$. Frequency is the reciprocal of period.

8. What is the formula for the torque ($\tau$) acting on a rotating object?

   $\tau = rF\text{sin}(\theta)$. Torque depends on force, distance, and angle.

9. What is the term for the rate of change of angular velocity?

   Angular Acceleration. The rate of change of angular velocity.

10. What is the linear displacement of a point $1$ m from the axis after a $2$ rad rotation?

    $s = 2$ m. Arc length equals radius times angle in radians.

11. Identify the formula for angular velocity in terms of angular displacement and time.

    $\text{Angular velocity} = \frac{\theta}{t}$. Angular velocity equals angular displacement divided by time.

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

    Moment of Inertia. Rotational inertia measures resistance to angular acceleration.

13. What is the formula for rotational work done by a torque through an angle?

    $W = \tau\theta$. Work done equals torque times angular displacement.

14. State the formula for the period of a rotating object in terms of angular velocity.

    $T = \frac{2\text{π}}{\text{ω}}$. Period is $2\pi$ divided by angular velocity.

15. State the formula for the centripetal force required for circular motion.

    $F_c = \frac{mv^2}{r}$. Force needed to maintain circular motion.

16. What is the relationship between centripetal acceleration and angular velocity?

    $a_c = r \omega^2$. Alternative form using angular velocity and radius.

17. State the relationship between linear displacement ($s$) and angular displacement ($\theta$).

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

18. Identify the conservation law applicable to a closed system with no external torques.

    Conservation of Angular Momentum. Angular momentum remains constant without external torques.

19. Which principle relates torque to angular acceleration?

    Newton's Second Law for Rotation. States that net torque equals $I\alpha$.

20. Identify the term for the rate of change of angular displacement.

    Angular Velocity. The rate of change of angular position.

21. Identify the angular displacement if a wheel turns $4$ revolutions.

    $\theta = 8\text{π}$ rad. One revolution equals $2\pi$ radians.

22. State the formula for angular displacement in terms of initial and final angle.

    $\theta = \theta_f - \theta_i$. Angular displacement is the change in angular position.

23. Identify the formula for centripetal acceleration in terms of velocity and radius.

    $a_c = \frac{v^2}{r}$. Acceleration directed toward the center of circular motion.

24. Calculate the period of rotation for an object rotating at $5$ rad/s.

    $T = \frac{2\text{π}}{5}$ s. Period equals $2\pi$ divided by angular velocity.

25. What is the formula for angular momentum ($L$) of a rotating object?

    $L = I\text{ω}$. Angular momentum equals moment of inertia times angular velocity.

26. Identify the tangential speed of a point $2$ m from the center of a wheel rotating at $3$ rad/s.

    $v = 6$ m/s. Use $v = r\omega$ relationship.

27. Calculate the angular momentum of a $3$ kg·m² object rotating at $10$ rad/s.

    $L = 30$ kg·m²/s. Angular momentum equals $I$ times $\omega$.

28. What is the formula for angular acceleration in terms of change in angular velocity and time?

    $\text{Angular acceleration} = \frac{\text{Δω}}{t}$. Rate of change of angular velocity over time.

29. Find the angular velocity if a wheel rotates $180$ degrees in $2$ seconds.

    $\text{ω} = 1.57$ rad/s. Convert $180°$ to $\pi$ rad, then divide by time.

30. Determine the torque if a $5$ N force is applied $0.5$ m from the pivot.

    $\tau = 2.5$ N·m. Torque equals force times perpendicular distance.

31. State the relationship between linear velocity ($v$) and angular velocity ($\text{ω}$).

    $v = r\text{ω}$. Linear velocity equals radius times angular velocity.

32. State the equation of rotational kinetic energy.

    $K = \frac{1}{2}I\text{ω}^2$. Rotational analog of linear kinetic energy.

33. Identify the formula for moment of inertia for a point mass.

    $I = mr^2$. Moment of inertia for a point mass at distance $r$.

34. State the unit of angular acceleration in the International System of Units (SI).

    Radians per second squared (rad/s²). Standard SI unit for rotational acceleration.

35. Find the angular acceleration for a $10$ kg disk with $1$ m radius and $5$ N·m torque.

    $\text{α} = 1.25$ rad/s². Use $\tau = I\alpha$ where $I = \frac{1}{2}mr^2$ for disk.

36. Identify the formula relating tangential speed and angular speed.

    $v_{\text{tangential}} = r\text{ω}$. Tangential speed at distance $r$ from rotation axis.

37. Calculate the moment of inertia of a $2$ kg mass at $3$ m from the axis.

    $I = 18$ kg·m². Use $I = mr^2$ with given mass and distance.

38. Determine the frequency of a rotating object with a period of $0.5$ seconds.

    $f = 2$ Hz. Frequency equals one divided by period.

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