Newton's Second Law in Rotational Form - AP Physics C Mechanics

1. What does $\alpha$ represent in the formula $\tau = I\alpha$?

   Angular acceleration. Rate of change of angular velocity.

2. What is the rotational analog of Newton's First Law?

   A body at rest stays at rest; a rotating body stays rotating unless acted upon. Rotational inertia maintains rotational state.

3. Identify the relationship between net torque and angular acceleration.

   $\tau_{\text{net}} = I\alpha$. Direct statement of Newton's second law rotation.

4. State the principle of superposition for torques.

   Net torque is the vector sum of individual torques. Torques add vectorially like forces.

5. What is the rotational equivalent of linear momentum?

   Angular momentum. Rotational equivalent of $p = mv$.

6. Calculate angular acceleration for $\tau = 6 \text{ N·m}$, $I = 2 \text{ kg·m}^2$.

   $\alpha = 3 \text{ rad/s}^2$. Apply $\alpha = \tau/I = 6/2$.

7. Calculate $I$ with $r = 1 \text{ m}$, $m = 4 \text{ kg}$ for a point mass.

   $I = 4 \text{ kg·m}^2$. Use $I = mr^2$ formula: $4 \times 1^2$.

8. What is the effect of $\sin \theta$ when $\theta = 0$ in torque calculation?

   Torque is zero because $\sin 0 = 0$. Force parallel to lever arm produces no rotation.

9. State the formula for torque ($\tau$) in rotational dynamics.

   $\tau = rF \sin \theta$. Torque equals lever arm times force times sine of angle.

10. What happens to $\alpha$ if net external torque is zero?

    $\alpha = 0$ (no angular acceleration). Newton's first law: no net torque means constant motion.

11. What is the unit of angular acceleration in the SI system?

    radians per second squared (rad/s²). Units of angular velocity per time.

12. What is the formula for the moment of inertia of a hoop?

    $I = MR^2$. All mass concentrated at radius for thin ring.

13. Determine the torque direction if $F$ is applied counterclockwise.

    Counterclockwise torque is positive. Convention: counterclockwise rotation is positive direction.

14. What effect does increasing $r$ have on torque for a given force?

    Increases torque. Longer lever arm increases rotational effect.

15. Determine the direction of torque with $F$ applied clockwise.

    Clockwise torque is negative. Convention: clockwise rotation is negative direction.

16. Which variable represents angular acceleration?

    $\alpha$. Greek letter alpha represents angular acceleration.

17. What is the moment of inertia of a solid cylinder rotating about its axis?

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

18. What is the angular acceleration if $I = 3 \text{ kg·m}^2$ and $\tau = 9 \text{ N·m}$?

    $\alpha = 3 \text{ rad/s}^2$. Apply $\alpha = \tau/I = 9/3$.

19. Find $\alpha$ given $\tau = 20 \text{ N·m}$ and $I = 4 \text{ kg·m}^2$.

    $\alpha = 5 \text{ rad/s}^2$. Apply $\tau = I\alpha$: $\alpha = 20/4$.

20. Identify the rotational quantity analogous to linear velocity.

    Angular velocity ($\omega$). Rate of change of angular position.

21. What is the effect of increasing $I$ on angular acceleration for a given torque?

    Decreases angular acceleration. Higher inertia reduces acceleration for same torque.

22. Calculate the net torque: $\tau_1 = 5 \text{ N·m}$, $\tau_2 = -3 \text{ N·m}$.

    $\tau_{\text{net}} = 2 \text{ N·m}$. Add torques algebraically: $5 + (-3) = 2$.

23. Find the torque with $r = 0.5 \text{ m}$, $F = 10 \text{ N}$ at $\theta = 90^\circ$.

    $\tau = 5 \text{ N·m}$. Use $\tau = rF\sin\theta$ with $\sin 90^\circ = 1$.

24. State Newton's Second Law in rotational form.

    $\tau = I\alpha$. Rotational analog of $F = ma$.

25. State the relationship between torque and angular momentum.

    Torque is the rate of change of angular momentum. Fundamental relationship: $\tau = \frac{dL}{dt}$.

26. Find $\tau$ given $r = 0.3 \text{ m}$, $F = 15 \text{ N}$, $\theta = 60^\circ$.

    $\tau = 3.9 \text{ N·m}$. Calculate: $0.3 \times 15 \times \sin 60^\circ$.

27. What is the moment of inertia for a solid sphere?

    $I = \frac{2}{5}MR^2$. Moment of inertia for uniform solid sphere.

28. Which variable represents torque in rotational dynamics equations?

    $\tau$. Greek letter tau denotes torque.

29. What is the unit of torque in the SI system?

    Newton-meter (N·m). Torque has units of force times distance.

30. What is the role of the angle $\theta$ in the torque formula $\tau = rF \sin \theta$?

    $\theta$ is the angle between force and lever arm. Sine function gives effective force component.

31. Calculate torque if $r = 2 \text{ m}$, $F = 8 \text{ N}$, $\theta = 45^\circ$.

    $\tau = 11.3 \text{ N·m}$. Use $\tau = 2 \times 8 \times \sin 45^\circ$.

32. What is the rotational analog of force in linear dynamics?

    Torque ($\tau$). Rotational equivalent of linear force.

33. Identify the rotational analog of mass in linear motion.

    Moment of inertia ($I$). Rotational resistance to acceleration.

34. What does $I$ represent in the formula $\tau = I\alpha$?

    Moment of inertia. Rotational inertia, resistance to angular acceleration.

35. Identify the reference point for calculating torque in lever systems.

    The pivot or axis of rotation. Fixed point about which rotation occurs.

36. Explain the significance of the perpendicular distance in torque calculation.

    Determines effective lever arm length. Only perpendicular force component creates torque.

37. What is the formula for moment of inertia for a point mass?

    $I = mr^2$. Mass times radius squared for point masses.

Study these flashcards