Question 1

How long does it take for a rotating object to speed up from  15.0 rad/s to  33.3 rad/s if it has a uniform angular acceleration of $3.45 \mathrm { rad } / \mathrm { s } ^ { 2 } ?$

Accepted Answer

The answer of How long does it take for a...

Question 2

A string is wound tightly around a fixed pulley having a radius of 5.0 cm. As the string is pulled, the pulley rotates without any slipping of the string. What is the angular speed of the pulley when
The string is moving at 5.0 m/s?

Accepted Answer

The answer of A string is wound tightly around a...

Question 3

A wheel accelerates with a constant angular acceleration of $4.5 \mathrm { rad } / \mathrm { s } ^ { 2 }$ from an initial angular speed of  1.0 rad/s . 
(a) Through what angle does the wheel turn in the first  2.0 s, and 
(b) what is its anqular speed at that time?

Accepted Answer

The answer of A wheel accelerates with a constant angular...

Question 4

A rolling wheel of diameter of 68 cm slows down uniformly from 8.4 m/s to rest over a distance of 115 m. What is the magnitude of its angular acceleration if there was no slipping?

Accepted Answer

The angular acceleration can be found using the relation between linear acceleration and angular acceleration, $a = \alpha r$, where $a$ is the linear acceleration, $\alpha$ is the angular acceleration, and $r$ is the radius of the wheel. First, we find the linear acceleration using the equation $v^2 = u^2 + 2as$, where $v$ is the final velocity (0 m/s), $u$ is the initial velocity (8.4 m/s), $a$ is the acceleration, and $s$ is the distance (115 m). Solving for $a$, we get $a = -u^2 / (2s) = -(8.4)^2 / (2 \times 115) = -0.306 \, \text{m/s}^2$. The negative sign indicates deceleration. The radius of the wheel is half the diameter, so $r = 68 / 2 = 34 \, \text{cm} = 0.34 \, \text{m}$. Thus, $\alpha = a / r = -0.306 / 0.34 \approx -0.90 \, \text{rad/s}^2$. The magnitude of angular acceleration is $0.90 \, \text{rad/s}^2$, ignoring the negative sign which indicates direction.

Question 5

A scooter has wheels with a diameter of 120 mm. What is the angular speed of the wheels when the scooter is moving forward at 6.00 m/s?

Accepted Answer

The angular speed, $\omega$, can be calculated using the formula $\omega = v/r$, where $v$ is the linear speed and $r$ is the radius of the wheel. Given the diameter is 120 mm, the radius $r = 60$ mm or $0.06$ m. With $v = 6.00$ m/s, $\omega = 6.00 / 0.06 = 100$ rad/s. To convert to rpm, use the conversion factor $1 \text{ rad/s} = 9.55 \text{ rpm}$, thus $100 \text{ rad/s} = 955$ rpm.

Question 6

An electrical motor spins at a constant 2695.0 rpm. If the rotor radius is 7.165 cm, what is the linear acceleration of the edge of the rotor?

Accepted Answer

The answer of An electrical motor spins at a constant...

Question 7

A bowling ball of mass 7.5 kg and diameter 18 cm rolls without slipping down a 10-m
bowling lane with a constant speed 4.3 m/s.
(a) Through what angle does the bowling ball turn as it travels the length of the lane?
(b) What is the angular speed of the bowling ball?
(c) Calculate the maximum radial acceleration that a point on the surface of the bowling
ball could have.
(d) Calculate the tangential acceleration of a point on the surface of the bowling ball.

Accepted Answer

The answer of A bowling ball of mass 7.5 kg...

Question 8

A wheel accelerates from rest to  59 rad/s at a uniform rate of $58 \mathrm { rad } / \mathrm { s } ^ { 2 }$ Through what angle (in radians) did the wheel turn while accelerating?

Accepted Answer

Explanation:The wheel's angular acceleration is 58 rad/s^2, and it accelerates from rest to 59 rad/s. We can use the equation:$$\omega^2 = \omega_0^2 + 2\alpha	heta$$where:$$\omega$$ is the final angular velocity (59 rad/s)$$\omega_0$$ is the initial angular velocity (0 rad/s)$$\alpha$$ is the angular acceleration (58 rad/s^2)$$	heta$$ is the angle through which the wheel turns (in radians)Plugging in the values, we get:$$59^2 = 0^2 + 2(58)	heta$$$$	heta = \frac{59^2}{2(58)} = 30 	ext{ rad}$$

Question 9

A wheel starts from rest and has a uniform angular acceleration of $4.0 \mathrm { rad } / \mathrm { s } ^ { 2 }$
After the wheel completes its first revolution, how long does it take for it to make its second complete revolution?

Accepted Answer

The answer of A wheel starts from rest and has...

Question 10

A machinist turns on the power on to a grinding wheel at time  t=0 s. The wheel accelerates uniformly from rest for  10 s and reaches the operating angular speed of  96 rad/s. The wheel is  run  at that angular velocity for  40 s and then power is shut off. The wheel slows down uniformly at $1.5 \mathrm { rad } / \mathrm { s } ^ { 2 }$ until the wheel stops. For how long a time after the power is shut off does it take the wheel to stop?

Accepted Answer

The time taken to stop the wheel after the power is shut off can be calculated using the formula $ t = \frac{\omega}{\alpha} $, where $ \omega = 96 \, \text{rad/s} $ is the angular velocity and $ \alpha = 1.5 \, \text{rad/s}^2 $ is the angular deceleration. Thus, $ t = \frac{96}{1.5} = 64 \, \text{s} $.

Question 11

A machinist turns on the power to a grinding wheel at time  t=0 s. The wheel accelerates
uniformly from rest for  10 s and reaches the operating angular speed of  38 rad/s . The wheel is run
at that angular speed for  30 s and then power is shut off. The wheel slows down uniformly at
$2.1 \mathrm { rad } / \mathrm { s } ^ { 2 }$
between  t=0 s and  t=10 s ?

Accepted Answer

The answer of A machinist turns on the power to...

Question 12

A bicycle has wheels that are 60 cm in diameter. What is the angular speed of these wheels when it is moving at 4.0 m/s?

Accepted Answer

The angular speed (ω) can be calculated using the formula ω = v/r, where v is the linear speed and r is the radius of the wheel. The radius is half the diameter, so r = 60 cm / 2 = 30 cm = 0.3 m. Therefore, ω = 4.0 m/s / 0.3 m = 13.33 rad/s, which is approximately 13 rad/s.

Question 13

In the figure, point P is on the rim of a wheel of radius  2.0 m . At time  t=0 , the wheel is at rest, and  P  is on the  x -axis. The wheel undergoes a uniform counterclockwise angular acceleration of $0.010 \mathrm { rad } / \mathrm { s } 2$ about the center  O.
(a) At time  t=0 , what is the tangential acceleration of P?
(b) What is the linear speed of  P  when it reaches the  y -axis?
(c) What is the magnitude of the net linear acceleration of  P  when it reaches the  y -axis?
(d) How long after starting does it take for  P to return to its original position on the  x -axis?

Accepted Answer

The answer of In the figure, point P is on...

Question 14

A cylinder of radius 8.0 cm rolls 20 cm in 5.0 s without slipping. Through how many
degrees does the cylinder turn during this time?

Accepted Answer

The answer of A cylinder of radius 8.0 cm rolls...

Question 15

A machinist turns on the power on to a grinding wheel at time  t=0 s. The wheel accelerates uniformly from rest for  10 s and reaches the operating angular speed of  58 rad/s. The wheel is run at that angular velocity for  30 s , and then power is shut off. The wheel slows down uniformly at $1.4 \mathrm { rad } / \mathrm { s } ^ { 2 }$ until the wheel stops. What is the total number of revolutions made by the wheel in this situation?

Accepted Answer

The answer of A machinist turns on the power on...

Question 16

A child is riding a merry-go-round that has an instantaneous angular speed of 1.25 rad/s and an angular acceleration o $0.745 \mathrm { rad } / \mathrm { s } ^ { 2 }$ The child is standing 4.65 m from the center of the
Merry-go-round. What is the magnitude of the linear acceleration of the child?

Accepted Answer

The answer of A child is riding a merry-go-round that...

Question 17

When an old LP turntable was revolving at $33 \frac { 1 } { 3 } \mathrm { rpm }$ it was shut off and uniformly
slowed down and stopped in  5.5  seconds.
(a) What was the magnitude of its angular acceleration $\left( \text { in } \mathrm { rad } / \mathrm { s } ^ { 2 } \right)$
as it slowed down?
(b) Through how many revolutions did it turn while stopping?

Accepted Answer

The answer of When an old LP turntable was revolving...

Question 18

A bicycle wheel has an initial angular speed of  7.2 rad/s. After turning through one-half of a revolution, the angular speed is reduced to  2.2 rad/s. If the angular acceleration of the wheel was constant during the motion, how long will it take the wheel to make the one-half revolution?

Accepted Answer

The answer of A bicycle wheel has an initial angular...

Question 19

A wheel of diameter 0.70 m rolls on the floor without slipping. A point at the top of the
wheel moves with a speed 2.0 m/s relative to the floor.
(a) At what speed is the central axis of the wheel moving relative to the floor?
(b) What is the angular speed of the wheel?

Accepted Answer

The answer of A wheel of diameter 0.70 m rolls...

Question 20

A child is riding a merry-go-round that is turning at 7.18 rpm. If the child is standing 4.65 m from the center of the merry-go-round, how fast is the child moving?

Accepted Answer

The speed of the child can be calculated using the formula for linear velocity, $v = r\omega$, where $v$ is the linear velocity, $r$ is the radius (distance from the center), and $\omega$ is the angular velocity in radians per second. First, convert the angular velocity from revolutions per minute (rpm) to radians per second: $7.18 \, \text{rpm} = 7.18 \times \frac{2\pi}{60} \, \text{rad/s}$. Then, calculate the linear velocity: $v = 4.65 \times 7.18 \times \frac{2\pi}{60} \approx 3.50 \, \text{m/s}$.

Quiz 8: Rotational Motion

How long does it take for a rotating object to speed up from 15.0 rad/s to 33.3 rad/s if it has a uniform angular acceleration of 3.45rad/s2?3.45 \mathrm { rad } / \mathrm { s } ^ { 2 } ?3.45rad/s2?

A string is wound tightly around a fixed pulley having a radius of 5.0 cm. As the string is pulled, the pulley rotates without any slipping of the string. What is the angular speed of the pulley when The string is moving at 5.0 m/s?

A wheel accelerates with a constant angular acceleration of 4.5rad/s24.5 \mathrm { rad } / \mathrm { s } ^ { 2 }4.5rad/s2 from an initial angular speed of 1.0 rad/s . (a) Through what angle does the wheel turn in the first 2.0 s, and (b) what is its anqular speed at that time?

A rolling wheel of diameter of 68 cm slows down uniformly from 8.4 m/s to rest over a distance of 115 m. What is the magnitude of its angular acceleration if there was no slipping?

A scooter has wheels with a diameter of 120 mm. What is the angular speed of the wheels when the scooter is moving forward at 6.00 m/s?

An electrical motor spins at a constant 2695.0 rpm. If the rotor radius is 7.165 cm, what is the linear acceleration of the edge of the rotor?

A wheel accelerates from rest to 59 rad/s at a uniform rate of 58rad/s258 \mathrm { rad } / \mathrm { s } ^ { 2 }58rad/s2 Through what angle (in radians) did the wheel turn while accelerating?

A wheel starts from rest and has a uniform angular acceleration of 4.0rad/s24.0 \mathrm { rad } / \mathrm { s } ^ { 2 }4.0rad/s2 After the wheel completes its first revolution, how long does it take for it to make its second complete revolution?

A bicycle has wheels that are 60 cm in diameter. What is the angular speed of these wheels when it is moving at 4.0 m/s?

A cylinder of radius 8.0 cm rolls 20 cm in 5.0 s without slipping. Through how many degrees does the cylinder turn during this time?

A bicycle wheel has an initial angular speed of 7.2 rad/s. After turning through one-half of a revolution, the angular speed is reduced to 2.2 rad/s. If the angular acceleration of the wheel was constant during the motion, how long will it take the wheel to make the one-half revolution?

A wheel of diameter 0.70 m rolls on the floor without slipping. A point at the top of the wheel moves with a speed 2.0 m/s relative to the floor. (a) At what speed is the central axis of the wheel moving relative to the floor? (b) What is the angular speed of the wheel?

A child is riding a merry-go-round that is turning at 7.18 rpm. If the child is standing 4.65 m from the center of the merry-go-round, how fast is the child moving?

How long does it take for a rotating object to speed up from 15.0 rad/s to 33.3 rad/s if it has a uniform angular acceleration of $3.45 \mathrm { rad } / \mathrm { s } ^ { 2 } ?$

A wheel accelerates with a constant angular acceleration of $4.5 \mathrm { rad } / \mathrm { s } ^ { 2 }$ from an initial angular speed of 1.0 rad/s . (a) Through what angle does the wheel turn in the first 2.0 s, and (b) what is its anqular speed at that time?

A wheel accelerates from rest to 59 rad/s at a uniform rate of $58 \mathrm { rad } / \mathrm { s } ^ { 2 }$ Through what angle (in radians) did the wheel turn while accelerating?

A wheel starts from rest and has a uniform angular acceleration of $4.0 \mathrm { rad } / \mathrm { s } ^ { 2 }$ After the wheel completes its first revolution, how long does it take for it to make its second complete revolution?