Q: A car tire rotates at 720 revolutions per minute (RPM). What is its angular velocity in radians per second?

$$\omega = 720 \times \frac{2\pi}{60} = 24\pi \text{ rad/s}$$

Q: A child sits 0.5 m from the center of a merry-go-round spinning at $$\omega = 2 \text{ rad/s}. $$What is the child's tangential velocity?

$$v = r\omega = 0.5 \times 2 = 1 \text{ m/s}$$

Q: A wrench is used to loosen a bolt. If a force of 40 N is applied perpendicular to the wrench at a distance of 0.25 m from the bolt, what is the torque produced?

$$\tau = rF = 0.25 \times 40 = 10 \text{ N}\cdot\text{m}$$

Q: A door is 1.2 m wide. If you push with a force of 15 N perpendicular to the edge, what is the torque about the hinges?

$$\tau = rF = 1.2 \times 15 = 18 \text{ N}\cdot\text{m}$$

Q: A wheel rotates through an angular displacement of $$\frac{\pi}{2}$$ radians. How many degrees has it rotated?

$$\frac{\pi}{2} \text{ radians} = 90^\circ$$

Question 1

A car tire rotates at 720 revolutions per minute (RPM). What is its angular velocity in radians per second?

Accepted Answer

$$\omega = 720 	imes \frac{2\pi}{60} = 24\pi 	ext{ rad/s}$$

Question 2

A child sits 0.5 m from the center of a merry-go-round spinning at $$\omega = 2 	ext{ rad/s}. $$What is the child's tangential velocity?

Accepted Answer

$$v = r\omega = 0.5 	imes 2 = 1 	ext{ m/s}$$

Question 3

Which of the following best explains why a hoop is harder to rotate than a solid disk of the same mass and radius?

Accepted Answer

The hoop's mass is distributed farther from the axis, increasing its rotational inertia ($$I = MR^2$) compared to the disk (I$$ = \frac{1}{2}$$MR^2).$$

Question 4

A wrench is used to loosen a bolt. If a force of 40 N is applied perpendicular to the wrench at a distance of 0.25 m from the bolt, what is the torque produced?

Accepted Answer

$$	au = rF = 0.25 	imes 40 = 10 	ext{ N}\cdot	ext{m}$$

Question 5

A ball is swung in a circle on a string. What force keeps the ball moving in a circular path?

Accepted Answer

Centripetal force, which acts toward the center of the circle.

Question 6

A door is 1.2 m wide. If you push with a force of 15 N perpendicular to the edge, what is the torque about the hinges?

Accepted Answer

$$	au = rF = 1.2 	imes 15 = 18 	ext{ N}\cdot	ext{m}$$

Question 7

A figure skater spins with arms extended, then pulls them in. What happens to her angular velocity, assuming no external torque?

Accepted Answer

Her angular velocity increases because her moment of inertia decreases.

Question 8

A wheel rotates through an angular displacement of $$\frac{\pi}{2}$$ radians. How many degrees has it rotated?

Accepted Answer

$$\frac{\pi}{2} 	ext{ radians} = 90^\circ$$

Question 9

A cyclist increases the radius of her path from 2 m to 4 m while maintaining the same angular velocity. What happens to her tangential velocity?

Accepted Answer

It doubles, since $$v = r\omega$$ and $$r$$ increases.

Question 10

Which scenario best illustrates the difference between centripetal and centrifugal force?

Accepted Answer

Feeling pushed outward in a turning car (centrifugal) vs. the friction keeping the car on the road (centripetal).

Question 11

A thin ring (mass 3 kg, radius 0.4 m) and a solid disk (mass 3 kg, radius 0.4 m) are spun about their centers. Which has greater rotational inertia and why?

Accepted Answer

The ring, because its mass is farther from the axis: $$I_{	ext{ring}} = MR^2$ vs. I$$_{	ext{disk}} = \frac{1}{2}$$MR^2.$$

Question 12

A spinning record has points A (0.05 m from center) and B (0.15 m from center). If the record spins at $$\omega = 5 	ext{ rad/s}$$, what is the ratio of their tangential velocities?

Accepted Answer

Point B's tangential velocity is three times that of point A: $$\frac{v_B}{v_A} = \frac{0.15}{0.05} = 3$$

Question 13

If a force is applied at an angle less than 90° to a lever arm, how does this affect the torque?

Accepted Answer

Torque is reduced because only the perpendicular component of the force contributes: $$	au = rF\sin	heta$$

Question 14

A mud droplet flies off a spinning tire. What best explains its motion after leaving the tire?

Accepted Answer

It moves tangentially to the circle due to inertia, not outward from the center.

Question 15

A rotating flywheel stores energy. Which factors most affect its rotational kinetic energy?

Accepted Answer

Its moment of inertia and the square of its angular velocity: $$KE = \frac{1}{2}I\$$omega^2$$

Question 16

A student pushes a door at its midpoint instead of the edge. How does the torque compare to pushing at the edge with the same force?

Accepted Answer

Torque is halved, since $$r is $$half as large: $$	au = rF$$

Question 17

Which of the following is NOT a real force in circular motion?

Accepted Answer

Centrifugal force

Question 18

A spinning top slows down due to friction. Which quantity decreases as it slows?

Accepted Answer

Angular velocity ($$\omega)$$

Question 19

If the circumference of a circle is 10 m and its diameter is 3.18 m, what is the value of $$\pi$$ based on these measurements?

Accepted Answer

$$\pi = \frac{10}{3.18} \approx 3.14$$

Rotational Motion: Study Guide and Key Concepts

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