Starting from rest at t = 0, a wheel undergoes a constant angular acceleration. When t = 2.0 s, the angular velocity of the wheel is 5.0 rad/s. The acceleration continues until t = 20 s, when it abruptly ceases. Through what angle does the wheel rotate in the interval t = 0 to t = 40 s?

Point $A$ is given a constant acceleration $a$ to the right starting from rest with $x$ essentially zero. Determine the angular velocity $\omega$ of link $A B$ in terms of $x$ and $a$.

A wheel has a constant angular acceleration of 5.0 rad/s². Starting from rest, it turns through 300 rad. (a) What is its final angular velocity? (b) How much time elapses while it turns through the 300 radians?

A flywheel has a constant angular deceleration of

$$2.0 rad/s^2.$$

(a) Find the angle through which the flywheel turns as it comes to rest from an angular speed of 220 rad/s. (b) Find the time required for the flywheel to come to rest.

The shaft of a pump starts from rest and has an angular acceleration of $3.00 \mathrm{rad} / \mathrm{s}^2$ for $18.0 \mathrm{~s}$. At the end of this interval, what is (a) the shaft's angular speed and (b) the angle through which the shaft has turned?

$$\left. \begin{array} { l } { \text { Suppose that } H \text { is a subgroup of } S _ { n } \text { of odd order. Prove that } H \text { is a } } \\ { \text { subgroup of } A _ { n } } \end{array} \right.$$

The Ruby-throated Hummingbird (Archillochus colubris) can flap its wings with an angular speed of $346 \mathrm{rad} / \mathrm{s}$. What is the period of its flapping motion?

A rotating object starts from rest at t=0 s and has a constant angular acceleration. At a time of t=7.0 s the object has an angular velocity of $\omega=16 \mathrm{rad} / \mathrm{s}$. What is its angular velocity at a time of t=14 s?

A transverse wave on a string is described by $y ( x , t ) = ( 0.35 \mathrm { mm } ) \sin \left\{ \left( \frac { \pi } { 3.0 } \frac { \mathrm { rad } } { \mathrm { m } } \right) \left[ x - \left( 66 \frac { \mathrm { m } } { \mathrm { s } } \right) \right] t \right\}$. Find the (a) amplitude, (b) wavelength, and (c) frequency of this wave. (d) In what direction does the wave travel? (e) What is the maximum transverse speed of a point on the string?

Find the projection of

$$\mathbf { v }\text { onto } \mathbf { u }.$$

$$\mathbf { u } = \left[ \begin{array} { l } { 0.5 } \\ { 1.5 } \end{array} \right] , \mathbf { v } = \left[ \begin{array} { l } { 2.1 } \\ { 1.2 } \end{array} \right]$$

A figure skater is spinning with an angular velocity of + 15 rad/s. She then comes to a stop over a brief period of time. During this time, her angular displacement is +5 .1 rad. Determine (a) her average angular acceleration and (b) the time during which she comes to rest.

A spot of paint on a bicycle tire moves in a circular path of radius 0.33 m. When the spot has traveled a linear distance of 1.95 m, through what angle has the tire rotated? Give your answer in radians.

A $110 \mathrm{~kg}$ ice hockey player skates at $3.00 \mathrm{~m} / \mathrm{s}$ toward a railing at the edge of the ice and then stops himself by grasping the railing with outstretched arms. During this stopping process, the player's torso moves $30.0 \mathrm{~cm}$ toward the railing. What is the change in the kinetic energy of the center of mass during the stop?

A motorcyclist is traveling along a road and accelerates for 4.50 s to pass another cyclist. The angular acceleration of each wheel is

$$+ 6.70 \mathrm { rad } / \mathrm { s } ^ { 2 }$$

and, just after passing, the angular velocity of each is

$$+ 74.5 \mathrm { rad } / \mathrm { s }$$

where the plus signs indicate counterclockwise directions. What is the angular displacement of each wheel during this time?