The flywheel of an engine is rotating at $\mathbf{25}\mathbf{.}\mathbf{0}\raisebox{1ex}{$\mathbf{r}\mathbf{a}\mathbf{d}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$ . When the engine is turned off, the flywheel slows at a constant rate and stops in $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{}\mathit{s}$ . Calculate

(a) The angular acceleration of the flywheel,

(b) The angle through which the flywheel rotates in stopping, and

(c) The number of revolutions made by the flywheel in stopping

In Fig. $\mathbf{10}\mathbf{-}\mathbf{41}$ , block $\mathbf{1}$ has mass ${\mathit{m}}_{\mathbf{1}}\mathbf{=}\mathbf{460}\mathbf{\text{\hspace{0.17em}g}}$ , block $\mathbf{\text{2}}$ has mass ${\mathbf{m}}_{\mathbf{2}}\mathbf{=}\mathbf{500}\mathbf{\text{\hspace{0.17em}g}}$ , and the pulley, which is mounted on a horizontal axle with negligible friction, has radius $\mathbf{R}\mathbf{=}\mathbf{5}\mathbf{.00}\mathbf{\text{\hspace{0.17em}cm}}$ . When released from rest, block $\mathbf{\text{2}}$ falls $\mathbf{75}\mathbf{.0}\mathbf{\text{\hspace{0.17em}cm}}$ in $\mathbf{5.00}\mathbf{\text{\hspace{0.17em}s}}$ without the cord slipping on the pulley. (a) What is the magnitude of the acceleration of the blocks? What are (b) tension ${\mathbf{\text{T}}}_{\mathbf{2}}$ and (c) tension ${\mathbf{\text{T}}}_{\mathbf{1}}$ ? (d) What is the magnitude of the pulley’s angular acceleration? (e) What is its rotational inertia?

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 thin spherical shell has a radius of 1.90 m. An applied torque of 960 N.m gives the shell an angular acceleration of 6.20 rad/s² about an axis through the center of the shell. What are (a) the rotational inertia of the shell about that axis and (b) the mass of the shell?

Figure$\mathbf{10}\mathbf{-}\mathbf{48}\mathbf{}$ shows a rigid assembly of a thin hoop (of mass m and radius$\mathbf{\text{R}}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{150}\mathbf{}\mathbf{}\mathbf{\text{m}}$ ) and a thin radial rod (of mass m and length$\mathbf{\text{L}}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\text{R}}$ ). The assembly is upright, but if we give it a slight nudge, it will rotate around a horizontal axis in the plane of the rod and hoop, through the lower end of the rod. Assuming that the energy given to the assembly in such a nudge is negligible, what would be the assembly’s angular speed about the rotation axis when it passes through the upside-down (inverted) orientation?

A thin rod of length $\mathbf{0}\mathbf{.75}\mathbf{\text{\hspace{0.17em}m}}$ and mass $\mathbf{\text{0.42 kg}}$ is suspended freely from one end. It is pulled to one side and then allowed to swing like a pendulum, passing through its lowest position with angular speed $\mathbf{4.0}\mathbf{\text{\hspace{0.17em}rad}}\mathbf{/}\mathbf{\text{s}}$ . Neglecting friction and air resistance, find (a) the rod’s kinetic energy at its lowest position and (b) how far above that position the center of mass rises.