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The initial angular speed is ${\mathrm{蝇}}_{0\mathrm{z}}=500\mathrm{rev}/\min$.

The final angular speed is ${\mathrm{蝇}}_{\mathrm{z}}=200\mathrm{rev}/\min$.

The time is t = 4 s.

Angular velocity at time t of a rigid body with constant angular acceleration is given by,

${\mathit{蝇}}_{\mathbf{z}}\mathbf{=}{\mathit{蝇}}_{\mathbf{0}\mathbf{z}}\mathbf{卤}{\mathit{伪}}_{\mathbf{2}}\mathit{t}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Here, ${\mathit{伪}}_{\mathbf{z}}$is the angular acceleration, ${\mathit{蝇}}_{\mathbf{0}\mathbf{z}}$is the angular velocity of body at time 0, t is the time.

Angular position at time t of a rigid body with constant angular acceleration is given by,

role="math" localid="1667980841517" $\mathit{胃}\mathbf{=}{\mathit{胃}}_{\mathbf{0}}\mathbf{+}{\mathit{蝇}}_{\mathbf{0}\mathbf{z}}\mathit{t}\mathbf{卤}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{伪}}_{\mathbf{z}}{\mathit{t}}^{\mathbf{2}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$

Here,${\mathit{胃}}_{\mathbf{0}}$ is Angular position of body at time 0.

Convert the initial angular speed in .

$$\begin{gathered} {\mathrm{蝇}}_{0\mathrm{z}}=\left(500\frac{\mathrm{rev}}{\min }\right)\left(\frac{1\min }{60\sec }\right) \\ =8.33\mathrm{rev}/\mathrm{s} \end{gathered}$$

Convert the final angular speed in .

$$\begin{gathered} {\mathrm{蝇}}_{\mathrm{z}}=\left(200\frac{\mathrm{rev}}{\min }\right)\left(\frac{1\min }{60\sec }\right) \\ =3.33\mathrm{rev}/\mathrm{s} \end{gathered}$$

From the initial and final angular speed, it can be concluded that the motion of the body is deceleration.

Substitute ${\mathrm{蝇}}_{0\mathrm{z}}=8.33\mathrm{rev}/\mathrm{s},\mathrm{t}=4\mathrm{s}$and${\mathrm{蝇}}_{\mathrm{z}}=3.33\mathrm{rev}/\mathrm{s}$ in equation${蝇}_z={蝇}_z-{伪}_{z}t$ .

$$\begin{gathered} 3.33\mathrm{rev}/\mathrm{s}=8.33\mathrm{rev}/\mathrm{s}-{\mathrm{伪}}_{\mathrm{z}}\left(4\mathrm{s}\right) \\ {\mathrm{伪}}_{\mathrm{z}}=\frac{8.33\mathrm{rev}/\mathrm{s}-3.33\mathrm{rev}/\mathrm{s}}{4\mathrm{s}} \\ =\frac{5\mathrm{rev}/\mathrm{s}}{4\mathrm{s}} \\ =1.25\mathrm{rev}/{\mathrm{s}}^2 \end{gathered}$$

Therefore, the angular acceleration is$1.25\mathrm{rev}/{\mathrm{s}}^2$ .

Substitute ${\mathrm{胃}}_0=0,{\mathrm{蝇}}_{0\mathrm{z}}=8.33\mathrm{rev}/\mathrm{s},\mathrm{t}=4\mathrm{s},$and${\mathrm{伪}}_{\mathrm{z}}=1.25\mathrm{rev}/{\mathrm{s}}^2$ in equation .

$$\begin{gathered} \mathrm{胃}={\mathrm{胃}}_0+{\mathrm{蝇}}_{0\mathrm{z}}\mathrm{t}-\frac{1}{2}{\mathrm{伪迟}}^2 \\ \mathrm{胃}=0+\left(8.33\mathrm{rev}/\mathrm{s}\right)\left(4\mathrm{s}\right)-\frac{1}{2}\left(1.25\mathrm{rev}/{\mathrm{s}}^2\right){\left(4\mathrm{s}\right)}^2 \\ =23.32\mathrm{rev} \end{gathered}$$

Therefore, the number of revolutions are 23.32 rev.

When the body stops then ${蝇}_z=0$.

Substitute role="math" localid="1667981776572" ${\mathrm{蝇}}_{0_{\mathrm{z}}}=8.33\mathrm{rev}/\mathrm{s},{\mathrm{伪}}_{\mathrm{z}}=1.25\mathrm{rev}/{\mathrm{s}}^2$,and${\mathrm{蝇}}_{\mathrm{z}}=0\mathrm{rev}/\mathrm{s}$ in equation .

${\mathrm{蝇}}_{\mathrm{z}}={\mathrm{蝇}}_{0_{\mathrm{z}}}-{\mathrm{伪}}_{\mathrm{z}}\mathrm{t}$

$$\begin{gathered} 0\mathrm{rev}/\mathrm{s}=8.33\mathrm{rev}/\mathrm{s}-\left(1.25\mathrm{rev}/{\mathrm{s}}^2\right)\mathrm{t} \\ \mathrm{t}=\frac{8.33\mathrm{rev}/\mathrm{s}}{1.25\mathrm{rev}/{\mathrm{s}}^2} \\ =6.66\mathrm{rev}/{\mathrm{s}}^2 \end{gathered}$$

Find the time it takes to stop after t = 4 s .

$$\begin{gathered} {\mathrm{t}}_{\mathrm{rest}}=6.66\mathrm{s}-4\mathrm{s} \\ =2.66\mathrm{s} \end{gathered}$$

Therefore, the required time is 2.66 s.