Q: If the pitcher’s arm starts from rest and accelerates uniformly to $$\omega = 40\ \mathrm{rad/s}$$ over $$500$$ degrees of rotation, what is the magnitude of the angular acceleration $$\alpha$$?

$$\alpha = \dfrac{\$$omega^2$$}{2\Delta\theta} = \dfrac{(40)^2$}{2 \times \dfrac{500 \times 2\pi}{360}} \approx 9.7\ \mathrm{$$rad/s^2$$}$$

Q: A gardener winds a hose onto a reel with a radius $$r = 0.10\ \mathrm{m}$$, starting from rest. The reel undergoes an angular acceleration of $$\alpha = 15\ \mathrm{$$rad/s^2$$}$$ for $$3.00\ \mathrm{s}. $$What is the length of hose wound onto the reel?

$$s = r \Delta \theta = 0.10 \times \dfrac{1}{2} \times 15 \times (3.00)^2$ = 6.75\ \mathrm{m}$$

Q: A reel rotates through $$67.5\ \mathrm{rad} as a $$hose is wound onto it. If the radius of the reel is $$0.10\ \mathrm{m}$$, what is the total length of hose wound?

$$s = r \Delta \theta = 0.10 \times 67.5 = 6.75\ \mathrm{m}$$

Q: A hose reel starts from rest and is given a constant angular acceleration $$\alpha$$ for a time $$t. $$What is the total angular displacement $$\Delta \theta$$?

$$\Delta \theta = \dfrac{1}{2} \alpha t^2$

Question 1

A fast-pitch softball pitcher uses a circular 'windmill' motion to pitch a ball at a speed of $$v = 32\ \mathrm{m/s}$$ with an arm length of $$L = 0.80\ \mathrm{m}. $$What is the angular velocity $$\omega of $$her arm at the moment the ball is released?

Accepted Answer

$$\omega = \dfrac{v}{L} = \dfrac{32}{0.80} = 40\ \mathrm{rad/s}$$

Question 2

If the pitcher’s arm starts from rest and accelerates uniformly to $$\omega = 40\ \mathrm{rad/s}$$ over $$500$$ degrees of rotation, what is the magnitude of the angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \dfrac{\$$omega^2$$}{2\Delta	heta} = \dfrac{(40)^2$}{2 	imes \dfrac{500 	imes 2\pi}{360}} \approx 9.7\ \mathrm{$$rad/s^2$$}$$

Question 3

How long does the pitcher’s arm take to reach the release speed, given $$\omega = 40\ \mathrm{rad/s}$$ and $$\alpha = 9.7\ \mathrm{$$rad/s^2$$}$$?

Accepted Answer

$$\Delta t = \dfrac{\omega}{\alpha} = \dfrac{40}{9.7} \approx 0.437\ \mathrm{s}$$

Question 4

A merry-go-round is spinning clockwise as viewed from above. Using the right-hand rule, what is the direction of the angular velocity vector?

Accepted Answer

Into the page

Question 5

For the same merry-go-round, what is the direction of the angular momentum vector?

Accepted Answer

Into the page

Question 6

If a torque vector points into the page for a clockwise-spinning merry-go-round, what effect does this have on its rotation?

Accepted Answer

It speeds up the clockwise rotation

Question 7

A gardener winds a hose onto a reel with a radius $$r = 0.10\ \mathrm{m}$$, starting from rest. The reel undergoes an angular acceleration of $$\alpha = 15\ \mathrm{$$rad/s^2$$}$$ for $$3.00\ \mathrm{s}. $$What is the length of hose wound onto the reel?

Accepted Answer

$$s = r \Delta 	heta = 0.10 	imes \dfrac{1}{2} 	imes 15 	imes (3.00)^2$ = 6.75\ \mathrm{m}$$

Question 8

Which equation correctly relates torque $$	au$$, moment of inertia $$I$$, and angular acceleration $$\alpha$$ for rotational motion?

Accepted Answer

$$	au = I \alpha$$

Question 9

A door of width $$L is $$acted on by a force $$F at an $$angle $$	heta to $$the door. What is the magnitude of the torque about the hinge?

Accepted Answer

$$	au = F L \sin 	heta$$

Question 10

A force $$F is $$applied perpendicular to a door at its edge, a distance $$L$$ from the hinge. What is the direction of the resulting torque vector?

Accepted Answer

Out of the page

Question 11

If a torque vector points out of the page for a clockwise-spinning merry-go-round, what effect does this have on its rotation?

Accepted Answer

It slows down the clockwise rotation

Question 12

A reel rotates through $$67.5\ \mathrm{rad} as a $$hose is wound onto it. If the radius of the reel is $$0.10\ \mathrm{m}$$, what is the total length of hose wound?

Accepted Answer

$$s = r \Delta 	heta = 0.10 	imes 67.5 = 6.75\ \mathrm{m}$$

Question 13

A door of width $$L is $$acted on by a force $$F at an $$angle $$	heta to $$the door. If $$	heta = 0$$^{\circ}$$, what is the torque about the hinge?

Accepted Answer

$$	au = 0$$

Question 14

Which of the following best describes the right-hand rule for determining the direction of the angular velocity vector for a rotating object?

Accepted Answer

Curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the angular velocity vector.

Question 15

A force $$F is $$applied at a distance $$L$$ from the hinge of a door, but the force is directed along the door (parallel to its length). What is the torque produced?

Accepted Answer

$$	au = 0$$

Question 16

A merry-go-round is spinning clockwise as viewed from above. If you want to slow it down, in which direction should you apply a torque?

Accepted Answer

Out of the page

Question 17

A hose reel starts from rest and is given a constant angular acceleration $$\alpha$$ for a time $$t. $$What is the total angular displacement $$\Delta 	heta$$?

Accepted Answer

$$\Delta 	heta = \dfrac{1}{2} \alpha t^2$

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