Connecting Linear and Rotational Motion - AP Physics C Mechanics

1. What is the formula for linear displacement in terms of angular displacement?

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

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

   $I = mr^2$. Point mass inertia is mass times radius squared.

3. Find the moment of inertia for a hoop of mass $M$ and radius $R$.

   $I = MR^2$. For a hoop, all mass is at radius $R$ from center.

4. Calculate the torque with $F = 10 \text{ N}$, $r = 0.5 \text{ m}$, and angle $90^\text{o}$.

   $\tau = 5 \text{ Nm}$. Use $\tau = rF\sin(90°) = 0.5 \times 10 \times 1 = 5$ Nm.

5. State the formula for the angular displacement in terms of linear displacement.

   $\theta = \frac{s}{r}$. Angular displacement equals arc length divided by radius.

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

   Newton meter (Nm). Torque has units of force times distance.

7. Identify the unit of angular velocity in the SI system.

   radians per second (rad/s). Angular velocity measures rotation rate in radians per time.

8. State the relationship between rotational and translational kinetic energy.

   $KE_{total} = KE_{trans} + KE_{rot}$. Total kinetic energy is sum of translational and rotational parts.

9. Calculate the frequency if angular velocity is $4\text{π} \text{ rad/s}$.

   $f = 2 \text{ Hz}$. Use $f = \frac{\omega}{2\pi} = \frac{4\pi}{2\pi} = 2$ Hz.

10. Identify the rotational counterpart of force in linear motion.

    Torque. Torque causes rotational motion like force causes linear motion.

11. Find the angular displacement if $s = 8 \text{ m}$ and $r = 2 \text{ m}$.

    $\theta = 4 \text{ rad}$. Use $\theta = \frac{s}{r} = \frac{8}{2} = 4$ rad.

12. What is the unit of angular displacement in the SI system?

    radians (rad). Angular displacement measures rotation angle in radians.

13. State the formula for the conservation of angular momentum.

    $L_i = L_f$. Angular momentum is conserved when no external torque acts.

14. Identify the rotational counterpart of mass in linear motion.

    Moment of Inertia. Moment of inertia resists rotational motion like mass resists linear motion.

15. Calculate the period if angular frequency is $6\text{π} \text{ rad/s}$.

    $T = \frac{1}{3} \text{ s}$. Use $T = \frac{2\pi}{\omega} = \frac{2\pi}{6\pi} = \frac{1}{3}$ s.

16. What is the formula for torque?

    $\tau = rF\text{sin}(\theta)$. Torque equals perpendicular force times lever arm distance.

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