At t = 0 a grinding wheel has an angular velocity of 24.0 rad/s. It has a constant angular acceleration of $30.0 \text { rad} / \mathrm { s } ^ { 2 }$ until a circuit breaker trips at t = 2.00 s. From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. Through what total angle did the wheel turn between t = 0 and the time it stopped?

A rotating wheel has a constant angular acceleration. It has an angular velocity of 5.0 rad/s at time t=0 s, and 3.0 s later has an angular velocity of 9.0 rad/s. What is the angular displacement of the wheel during the 3.0-s interval? (a) 15 rad (b) 21 rad (c) 27 rad (d) There is not enough information given to determine the angular displacement.

A bicycle wheel has an initial angular velocity of 1.50 rad/s. (a) If its angular acceleration is constant and equal to $0.200 \mathrm { rad } / \mathrm { s } ^ { 2 }$, what is its angular velocity at t = 2.50 s? (b) Through what angle has the wheel turned between t = 0 and t = 2.50 s?

A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to $\theta(t)=\gamma t+\beta t^3$, where $\gamma=0.400 \mathrm{~rad} / \mathrm{s}$ and $\beta=$ $0.0120 \mathrm{~rad} / \mathrm{s}^3$.
(b) What is the initial value of the angular velocity?

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

A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 40.0 kg and diameter 75.0 cm. The power is off for 30.0 s, and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 200 complete revolutions. At what rate is the flywheel spinning when the power comes back on?

A wheel rotates from rest with constant angular acceleration. If it rotates through 8.00 revolutions in the first 2.50 s, how many more revolutions will it rotate through in the next 5.00 s?

A bicycle wheel has an initial angular velocity of $1.50$ rad$/$s.
(b) Through what angle has the wheel turned between $t=0$ and $t=2.50$ s?

At t = 0 the current to a dc electric motor is reversed, resulting in an angular displacement of the motor shaft given by $\theta ( t ) = ( 250 t - \left( 20.0 \mathrm { rad } / \mathrm { s } ^ { 2 } \right) \mathrm { t } ^ { 2 } - \left( 1.50 \mathrm { rad } / \mathrm { s } ^ { 3 } \right) \mathrm { t } ^ { 3 }$. (a) At what time is the angular velocity of the motor shaft zero? (b) Calculate the angular acceleration at the instant that the motor shaft has zero angular velocity. (c) How many revolutions does the motor shaft turn through between the time when the current is reversed and the instant when the angular velocity is zero? (d) How fast was the motor shaft rotating at t = 0, when the current was reversed? (e) Calculate the average angular velocity for the time period from t = 0 to the time calculated in part (a).

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. (a) Find the angular acceleration in $\mathrm { rev } / \mathrm { s } ^ { 2 }$ and the number of revolutions made by the motor in the 4.00-s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to $\theta(t)=\gamma t+\beta t^3$, where $\gamma=0.400 \mathrm{~rad} / \mathrm{s}$ and $\beta=$ $0.0120 \mathrm{~rad} / \mathrm{s}^3$.
($c$) Calculate the instantaneous value of the angular velocity $\omega_z$ at $t=5.00 \mathrm{~s}$ and the average angular velocity $\omega_{\mathrm{av}-z}$ for the time interval $t=0$ to $t=5.00 \mathrm{~s}$. Show that $\omega_{\mathrm{av}-z}$ is $\it{not}$ equal to the average of the instantaneous angular velocities at $t=0$ and $t=5.00 \mathrm{~s}$, and explain.

Consider three identical metal spheres, A, B, and C. Sphere A carries a charge of +5q. Sphere B carries a charge of - q. Sphere C carries no net charge. Spheres A and B are touched together and then separated. Sphere C is then touched to sphere A and separated from it. Last, sphere C is touched to sphere B and separated from it. (a) How much charge ends up on sphere C? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched ?

A wheel initially rotating at 60 rpm experiences the angular acceleration . What is the wheel's angular velocity, in rpm, at $t=3.0 \mathrm{~s} ?$

A small grinding wheel has a moment of inertia of $4.0 \times 10 ^ { - 5 } \mathrm { kg } \cdot \mathrm { m } ^ { 2 }$. What net torque must be applied to the wheel for its angular accelaration to be $150 \mathrm { rad } / \mathrm { s } ^ { 2 }$?