Conservation of Angular Momentum - AP Physics C Mechanics

1. What is the relationship between torque and the rate of change of angular momentum?

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

2. Explain how decreasing the radius of rotation affects angular velocity if angular momentum is conserved.

   Angular velocity increases. Smaller radius means larger $\omega$ to conserve $L$.

3. Describe what happens to the angular momentum of a satellite orbiting Earth when no external forces act.

   It remains constant. No external forces means angular momentum is conserved.

4. State the formula for the moment of inertia of a solid sphere.

   $I = \frac{2}{5}mr^2$. Standard formula for uniform solid spheres rotating about center.

5. If a rotating system is isolated, what can be said about its angular momentum?

   It is conserved. Isolated systems have no external torques acting.

6. Name a common real-world example of conservation of angular momentum.

   A figure skater spinning. Classic demonstration of changing moment of inertia.

7. State the formula for angular momentum of a point mass.

   $L = mvr \sin \theta$. Where $m$ is mass, $v$ is velocity, $r$ is distance, and $\theta$ is angle.

8. What is the effect on a rotating system's angular velocity if its radius is reduced?

   The angular velocity increases. Smaller radius decreases $I$, increasing $\omega$.

9. What is the principle of conservation of angular momentum?

   Total angular momentum remains constant if no external torque acts. This is Newton's first law applied to rotation.

10. How does angular momentum relate to the rotational equivalent of linear momentum?

    It is the rotational equivalent. Angular momentum parallels linear momentum in rotation.

11. What principle explains why a diver rotates faster when tucking in their limbs?

    Conservation of angular momentum. Reducing $I$ increases $\omega$ to maintain constant $L$.

12. In $L = I\omega$, what does $\omega$ represent?

    Angular velocity. The rate of rotational motion in radians per second.

13. What is the unit of angular momentum in the SI system?

    Kilogram meter squared per second (kg·m²/s). Derived from $[M][L^2][T^{-1}]$ dimensional analysis.

14. What is the term for the rotational equivalent of mass in the context of angular momentum?

    Moment of inertia. The rotational analog of mass for angular motion.

15. How does conservation of angular momentum affect a spinning ice skater's speed when she pulls in her arms?

    Her speed increases. Reduced moment of inertia increases angular velocity.

16. What is the angular momentum of a 1 kg mass at 1 m radius moving at 1 m/s?

    $L = 1 \text{ kg·m}^2/\text{s}$. Using $L = mvr = 1 \times 1 \times 1 = 1$.

17. What is the expression for angular momentum of a rigid body with moment of inertia $I$ and angular velocity $\omega$?

    $L = I\omega$. Standard formula for rigid body angular momentum.

18. Find the new angular velocity when a disk's moment of inertia changes from $4 \, \text{kg·m}^2$ to $2 \, \text{kg·m}^2$.

    Doubles. Conservation requires $I_1\omega_1 = I_2\omega_2$, so $\omega$ doubles.

19. What happens to angular momentum if no external torque acts on a system?

    It remains constant. Direct consequence of conservation of angular momentum.

20. Calculate the angular momentum of a 3 kg mass moving at 5 m/s in a circle of radius 2 m.

    $L = 30 \text{ kg·m}^2/\text{s}$. Using $L = mvr = 3 \times 5 \times 2 = 30$.

21. State the relationship between linear and angular momentum for a rotating object.

    Angular momentum is the rotational equivalent of linear momentum. Both describe momentum but for different types of motion.

22. What is the angular momentum for a mass $m$ at distance $r$ moving with velocity $v$?

    $L = mvr \sin \theta$. Cross product of position and momentum vectors.

23. What happens to angular velocity when a rotating object's moment of inertia decreases?

    Angular velocity increases. Conservation requires $\omega$ to increase as $I$ decreases.

24. What is the angular momentum of an object with $m = 5$ kg, $r = 2$ m, and $v = 10$ m/s?

    $L = 100 \text{ kg·m}^2/\text{s}$. Using $L = mvr = 5 \times 10 \times 2 = 100$.

25. What is the relationship between angular momentum and rotational inertia?

    $L = I\omega$. Fundamental equation linking angular momentum and rotation.

26. Find the new angular velocity when a disk's moment of inertia changes from 4 kg·m² to 2 kg·m².

    Doubles. Conservation requires $I_1\omega_1 = I_2\omega_2$, so $\omega$ doubles.

27. What does $\tau$ represent in $\tau = \frac{dL}{dt}$?

    Torque. The rotational force causing angular acceleration.

28. Find the change in angular momentum given $\tau = 10$ Nm and $\Delta t = 2$ s.

    $\Delta L = 20 \text{ kg·m}^2/\text{s}$. Using $\Delta L = \tau \Delta t = 10 \times 2 = 20$.

29. What is the effect on the angular momentum of a system if external forces are applied?

    It can change. External forces can apply torques that alter $L$.

30. What is the relationship between torque and angular momentum?

    Torque is the rate of change of angular momentum. Torque is defined as $\frac{dL}{dt}$.

31. Determine the new angular velocity if a figure skater pulls her arms in and her moment of inertia decreases.

    Angular velocity increases. Conservation requires $I\omega$ to remain constant, so $\omega$ rises.

32. Which physical quantity must be conserved in a closed system with no external torques?

    Angular momentum. No external torques means no change in rotational momentum.

33. Identify the variable $L$ in the angular momentum formula $L = I\omega$.

    Angular momentum. The rotational momentum quantity in the equation.

34. Choose the correct formula for the angular momentum of a rotating solid disk.

    $L = I\omega$. Standard angular momentum formula for rotating objects.

35. State the formula for the moment of inertia of a solid cylinder.

    $I = \frac{1}{2}mr^2$. Standard formula for uniform solid cylinders about central axis.

36. What is the angular momentum of a rotating disk with $I = 2 \text{ kg·m}^2$ and $\omega = 3 \text{ rad/s}$?

    6 kg·m²/s. Using $L = I\omega = 2 \times 3 = 6$.

37. What is the effect of external torque on the angular momentum of a system?

    It changes the angular momentum. Torque is the rate of change of angular momentum.

38. What is the effect on angular momentum if a system is isolated and no torque is applied?

    Angular momentum remains constant. No external torques means angular momentum is conserved.

39. Calculate the angular momentum of a 2 kg mass moving in a circle of radius 3 m at 4 m/s.

    $L = 24 \text{ kg·m}^2/\text{s}$. Using $L = mvr = 2 \times 4 \times 3 = 24$.

40. What does $I$ represent in the equation $L = I\omega$?

    Moment of inertia. The rotational inertia of the object about the axis.

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