Connecting Linear and Rotational Motion - AP Physics 1

1. Calculate the torque for a force of $10 \text{ N}$ applied at $2 \text{ m}$ from the pivot.

   $\tau = 20 \text{ Nm}$. Apply $\tau = rF$: $\tau = 2 \times 10 = 20$ Nm.

2. What is the formula for the total acceleration of a point on a rotating body?

   $a_{total} = \text{√}(a_t^2 + a_c^2)$. Pythagorean sum of tangential and centripetal accelerations.

3. Calculate the angular momentum for a point mass with mass $m = 2 \text{ kg}$, radius $r = 3 \text{ m}$, and velocity $v = 4 \text{ m/s}$.

   $L = 24 \text{ kg m}^2/\text{s}$. Apply $L = mvr$: $L = 2 \times 4 \times 3 = 24$ kg⋅m²/s.

4. Find the linear velocity for an object rotating at $\theta = 4 \text{ rad/s}$ with radius $r = 0.5 \text{ m}$.

   $v = 2 \text{ m/s}$. Apply $v = r\omega$: $v = 0.5 \times 4 = 2$ m/s.

5. What is the formula for the centripetal acceleration in terms of linear velocity and radius?

   $a_c = \frac{v^2}{r}$. Standard formula for centripetal acceleration.

6. Identify the unit of torque.

   Newton meter (Nm). SI unit for torque measurement.

7. State the formula for torque in terms of force and radius.

   $\tau = r \times F$. Torque equals force times perpendicular distance from pivot.

8. State the equation for the work done by torque over an angular displacement.

   $W = \tau \times \theta$. Rotational work equals torque times angular displacement.

9. What is the formula for angular momentum in terms of mass, velocity, and radius?

   $L = mvr$. Angular momentum for linear motion in circular path.

10. What is the unit of angular velocity?

    Radians per second (rad/s). Standard SI unit for angular velocity measurement.

11. Calculate the tangential acceleration for $\beta = 5 \text{ rad/s}^2$ and $r = 0.3 \text{ m}$.

    $a_t = 1.5 \text{ m/s}^2$. Apply $a_t = r\alpha$: $a_t = 0.3 \times 5 = 1.5 \text{ m/s}^2$.

12. Calculate the centripetal acceleration for $v = 5 \text{ m/s}$ and $r = 2 \text{ m}$.

    $a_c = 12.5 \text{ m/s}^2$. Apply $a_c = v^2/r$: $a_c = 5^2/2 = 12.5 \text{ m/s}^2$.

13. What is the relationship between linear displacement and angular displacement?

    $s = r \times \theta$. Arc length equals radius times angle in radians.

14. Calculate the moment of inertia for a hoop with mass $m = 5 \text{ kg}$ and radius $r = 1 \text{ m}$.

    $I = 5 \text{ kg m}^2$. For a hoop, $I = mr^2$: $I = 5 \times 1^2 = 5$ kg⋅m².

15. What is the linear distance traveled by a point on the rim of a wheel with radius $r$ after one revolution?

    $d = 2\text{π}r$. Circumference of circle with radius $r$.

16. Convert an angular velocity of $3 \text{ rad/s}$ to linear velocity if $r = 2 \text{ m}$.

    $v = 6 \text{ m/s}$. Apply $v = r\omega$: $v = 2 \times 3 = 6$ m/s.

17. Find the angular acceleration if the tangential acceleration is $8 \text{ m/s}^2$ and the radius is $4 \text{ m}$.

    $\beta = 2 \text{ rad/s}^2$. Apply $\alpha = a_t/r$: $\alpha = 8/4 = 2 \text{ rad/s}^2$.

18. Calculate the angular velocity for a wheel with $v = 10 \text{ m/s}$ and $r = 2 \text{ m}$.

    $\theta = 5 \text{ rad/s}$. Apply $\omega = v/r$: $\omega = 10/2 = 5 \text{ rad/s}$

19. Calculate the rotational kinetic energy of a disc with $I = 2 \text{ kg m}^2$ and $\theta = 3 \text{ rad/s}$.

    $KE_{rot} = 9 \text{ J}$. Apply $KE = \frac{1}{2}I\omega^2$: $KE = \frac{1}{2} \times 2 \times 3^2 = 9$ J.

20. What is the relationship between work done and torque in rotational motion?

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

21. What is the moment of inertia for a solid cylinder about its central axis?

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

22. What is the equation for the moment of inertia of a point mass?

    $I = m \times r^2$. For a point mass at distance $r$ from the axis.

23. Identify the unit for moment of inertia.

    Kilogram meter squared (kg m$^2$). SI unit for moment of inertia.

24. Find the angular momentum for $I = 8 \text{ kg m}^2$ and $\theta = 3 \text{ rad/s}$.

    $L = 24 \text{ kg m}^2/\text{s}$. Apply $L = I\omega$: $L = 8 \times 3 = 24$ kg⋅m²/s.

25. Convert a linear velocity of $5 \text{ m/s}$ to angular velocity for $r = 0.5 \text{ m}$.

    $\theta = 10 \text{ rad/s}$. Apply $\omega = v/r$: $\omega = 5/0.5 = 10$ rad/s.

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