Q: A gymnast swings through 2.5 revolutions in 1.90 s on a high bar. What is her average angular velocity in radians per second?

$$w = \frac{2.5 \times 2\pi}{1.90} = 8.27\ \text{rad/s}$$

Q: A disk rotates from an initial angular position of $$+7\ \text{rad} to a $$final angular position of $$-3\ \text{rad}. $$What is the angular displacement?

$$\Delta \theta = -3 - 7 = -10\ \text{rad}$$

Q: A turntable rotates with a constant angular acceleration of $$2.25\ \text{rad/s}^2. $$After $$4.00\ \text{s}$$, it has rotated through an angle of $$30.0\ \text{rad}. $$What was the angular velocity at the beginning of this interval?

$$\theta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$ \Rightarrow \$$omega_0$$ = \frac{\theta - \frac{1}{2} \alpha t^2$}{t} = \frac{30.0 - 0.5 \times 2.25 \times 16}{4.00} = 3.19\ \text{rad/s}$$

Q: An airplane propeller is rotating at $$1900\ \text{rpm}. $$What is its angular velocity in radians per second?

$$\omega = 1900 \times \frac{2\pi}{60} = 199\ \text{rad/s}$$

Q: A skater is spinning with an angular velocity of $$+15\ \text{rad/s}$$ and comes to a stop after rotating through $$+5.1\ \text{rad}. $$What is her average angular acceleration?

$$\$$omega^2$$ = \$$omega_0$$^2 + 2\alpha \Delta \theta \Rightarrow \alpha = \frac{\$$omega^2$$ - \$$omega_0$$^2}{2\Delta \theta} = \frac{0 - 225}{2 \times 5.1} = -22.1\ \text{rad/s}^2$$

Q: A helicopter blade has an angular speed of $$6.50\ \text{rev/s}$$ and an angular acceleration of $$1.30\ \text{rev/s}^2. At a $$point $$r = 5.70\ \text{m}$$ from the axis, what is the tangential acceleration?

$$a_t = r \alpha = 5.70 \times (1.30 \times 2\pi) = 46.5\ \text{m/s}^2$$

Q: What is the length of an arc subtended by an angle of $$2.00^{\circ} on a $$circle of radius $$4.23 \times 10^7$\ \text{m}$$?

$$s = r \theta = 4.23 \times 10^7$ \times \frac{2.00 \times \pi}{180} = 1.48 \times 10^6$\ \text{m}$$

Q: A blender’s blades rotate with an angular velocity of $$+1740\ \text{rad/s}$$ after an angular displacement of $$+44.0\ \text{rad}$$, starting from rest. What is the angular acceleration?

$$\$$omega^2$$ = \$$omega_0$$^2 + 2\alpha \Delta \theta \Rightarrow \alpha = \frac{\$$omega^2$$ - \$$omega_0$$^2}{2\Delta \theta} = \frac{(1740)^2$ - 0}{2 \times 44.0} = 34,409\ \text{rad/s}^2$$

Question 1

A gymnast swings through 2.5 revolutions in 1.90 s on a high bar. What is her average angular velocity in radians per second?

Accepted Answer

$$w = \frac{2.5 	imes 2\pi}{1.90} = 8.27\ 	ext{rad/s}$$

Question 2

A disk rotates from an initial angular position of $$+7\ 	ext{rad} to a $$final angular position of $$-3\ 	ext{rad}. $$What is the angular displacement?

Accepted Answer

$$\Delta 	heta = -3 - 7 = -10\ 	ext{rad}$$

Question 3

A turntable rotates with a constant angular acceleration of $$2.25\ 	ext{rad/s}^2. $$After $$4.00\ 	ext{s}$$, it has rotated through an angle of $$30.0\ 	ext{rad}. $$What was the angular velocity at the beginning of this interval?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$ \Rightarrow \$$omega_0$$ = \frac{	heta - \frac{1}{2} \alpha t^2$}{t} = \frac{30.0 - 0.5 	imes 2.25 	imes 16}{4.00} = 3.19\ 	ext{rad/s}$$

Question 4

An airplane propeller is rotating at $$1900\ 	ext{rpm}. $$What is its angular velocity in radians per second?

Accepted Answer

$$\omega = 1900 	imes \frac{2\pi}{60} = 199\ 	ext{rad/s}$$

Question 5

A skater is spinning with an angular velocity of $$+15\ 	ext{rad/s}$$ and comes to a stop after rotating through $$+5.1\ 	ext{rad}. $$What is her average angular acceleration?

Accepted Answer

$$\$$omega^2$$ = \$$omega_0$$^2 + 2\alpha \Delta 	heta \Rightarrow \alpha = \frac{\$$omega^2$$ - \$$omega_0$$^2}{2\Delta 	heta} = \frac{0 - 225}{2 	imes 5.1} = -22.1\ 	ext{rad/s}^2$$

Question 6

A helicopter blade has an angular speed of $$6.50\ 	ext{rev/s}$$ and an angular acceleration of $$1.30\ 	ext{rev/s}^2. At a $$point $$r = 5.70\ 	ext{m}$$ from the axis, what is the tangential acceleration?

Accepted Answer

$$a_t = r \alpha = 5.70 	imes (1.30 	imes 2\pi) = 46.5\ 	ext{m/s}^2$$

Question 7

What is the length of an arc subtended by an angle of $$2.00^{\circ} on a $$circle of radius $$4.23 	imes 10^7$\ 	ext{m}$$?

Accepted Answer

$$s = r 	heta = 4.23 	imes 10^7$ 	imes \frac{2.00 	imes \pi}{180} = 1.48 	imes 10^6$\ 	ext{m}$$

Question 8

A blender’s blades rotate with an angular velocity of $$+1740\ 	ext{rad/s}$$ after an angular displacement of $$+44.0\ 	ext{rad}$$, starting from rest. What is the angular acceleration?

Accepted Answer

$$\$$omega^2$$ = \$$omega_0$$^2 + 2\alpha \Delta 	heta \Rightarrow \alpha = \frac{\$$omega^2$$ - \$$omega_0$$^2}{2\Delta 	heta} = \frac{(1740)^2$ - 0}{2 	imes 44.0} = 34,409\ 	ext{rad/s}^2$$

Question 9

Which of the following best explains why it is more difficult to balance on a stopped bike than on a moving bike?

Accepted Answer

A moving bike has angular momentum, which stabilizes its motion due to gyroscopic effects.

Question 10

The Moon is gradually slowing Earth's rotation. What is the primary physical mechanism responsible for this effect?

Accepted Answer

Tidal friction between Earth's oceans and the Moon's gravitational pull transfers angular momentum, slowing Earth's rotation.

Question 11

A wheel rotates from $$\$$theta_1$$ = -3\ 	ext{rad} to$$ $$\$$theta_2$$ = +5\ 	ext{rad}. $$What is the sign and value of the angular displacement?

Accepted Answer

$$\Delta 	heta = +5 - (-3) = +8\ 	ext{rad}$$, positive

Question 12

A point on a rotating disk is $$0.25\ 	ext{m}$$ from the axis and the disk rotates at $$12\ 	ext{rad/s}. $$What is the tangential speed of the point?

Accepted Answer

$$v_t = r \omega = 0.25 	imes 12 = 3.0\ 	ext{m/s}$$

Question 13

If a merry-go-round makes one full revolution, what is the angular displacement in radians?

Accepted Answer

$$\Delta 	heta = 2\pi\ 	ext{rad}$$

Question 14

A wheel starts from rest and accelerates at $$4.0\ 	ext{rad/s}^2$$ for $$3.0\ 	ext{s}. $$What is its final angular velocity?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t = 0 + 4.0 	imes 3.0 = 12.0\ 	ext{rad/s}$$

Question 15

A car tire rotates at $$2.0\ 	ext{rev/s}. $$What is the angular velocity in radians per second?

Accepted Answer

$$\omega = 2.0 	imes 2\pi = 12.57\ 	ext{rad/s}$$

Question 16

A wheel rotates through $$1.50\ 	ext{m} of $$arc on a circle of radius $$2.50\ 	ext{m}. $$What is the angle subtended in radians?

Accepted Answer

$$	heta = \frac{s}{r} = \frac{1.50}{2.50} = 0.60\ 	ext{rad}$$

Question 17

Which direction is considered positive for angular displacement according to the right-hand rule?

Accepted Answer

Counterclockwise

Question 18

A synchronous satellite orbits Earth at a radius of $$4.23 	imes 10^7$\ 	ext{m}$$ with an angular separation of $$2.00^{\circ}. $$What is the arc length between the satellites?

Accepted Answer

$$s = r 	heta = 4.23 	imes 10^7$ 	imes \frac{2.00 	imes \pi}{180} = 1.48 	imes 10^6$\ 	ext{m}$$

Rotational Kinematics: Displacement, Velocity, and Units

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