Figure 10-33 gives angular speed versus time for a thin rod that rotates around one end. The scale on the v axis is set by $\mathit{\omega }\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathit{r}\mathit{a}\mathit{d}\mathbf{/}\mathit{s}$ (a) What is the magnitude of the rod’s angular acceleration? (b) At t = 4.0s , the rod has a rotational kinetic energy of 1.60J. What is its kinetic energy at t = 0 ?

A flywheel with a diameter of $\mathbf{1}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{\text{m}}\mathbf{}$ is rotating at an angular speed of $\mathbf{200}\mathbf{ }\raisebox{1ex}{$\mathbf{\text{rev}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{\text{min}}$}\right.$.

(a) What is the angular speed of the flywheel in radians per second?

(b) What is the linear speed of a point on the rim of the flywheel?

(c) What constant angular acceleration (in revolutions per minute-squared) will increase the wheel’s angular speed to $\mathbf{1000}\mathbf{ }\raisebox{1ex}{$\mathbf{\text{rev}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{\text{min}}$}\right.$ in $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{s}}$ ?

(d) How many revolutions does the wheel make during that $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{s}}$ ?

A vinyl record on a turntable rotates at $\text{33}\frac{\text{1}}{\text{3}}\text{rev/min}$. (a) What is its angular speed in radians per second? What is the linear speed of a point on the record (b) $\text{15 \hspace{0.17em}cm}$ and (c) $\text{7.4 \hspace{0.17em}cm}$ from the turntable axis?

A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length $\mathbf{55}\mathbf{.0}\mathbf{}\mathbf{\text{\hspace{0.17em}m}}$. At the instant it makes an angle of $\mathbf{\text{35.0°}}$with the vertical as it falls, what are

(a) the radial acceleration of the top, and

(b) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque.)

(c) At what angle $\mathbf{\text{θ}}$ is the tangential acceleration equal to g?

A pulley, with a rotational inertia of $\mathbf{1}\mathbf{.0}\mathbf{}\mathbf{\times }\mathbf{}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}{\mathbf{\text{\hspace{0.17em}kg.m}}}^{\mathbf{2}}$ about its axle and a radius of $\mathbf{10}\mathbf{\text{\hspace{0.17em}cm}}$,is acted on by a force applied tangentially at its rim. The force magnitude varies in time as $\mathbf{F}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.50}\mathbf{t}\mathbf{}\mathbf{+}\mathbf{}\mathbf{0}\mathbf{.30}{\mathbf{t}}^{\mathbf{2}}\mathbf{}\mathbf{}$ ,with F in newtons and t in seconds. The pulley is initially at rest. At $\mathbf{t}\mathbf{=}\mathbf{}\mathbf{3.0}\mathbf{\text{\hspace{0.17em}s}}\mathbf{}$ what are its (a) angular acceleration and (b) angular speed?

A drum rotates around its central axis at an angular velocity of $\mathbf{12}\mathbf{.}\mathbf{60}\mathbf{}\mathbf{rad}\mathbf{/}\mathbf{s}$. If the drum then slows at a constant rate of $\mathbf{4.20}\mathbf{}\mathbf{rad}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$.

(a) how much time does it take and

(b) through what angle does it rotate in coming to rest?