91影视

The initial angular speed is ${蝇}_{0_z}=500鈥�rpm$.

The angular displacement is $胃-{胃}_0=200鈥�rev$.

The time is t=30 s or t =0.5 m in..

Derive the required expression to find the rate of spinning of flywheel as follows.

$$\begin{gathered} \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{胃}\mathbf{鈭�}{\mathbf{胃}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left({蝇}_z+{蝇}_{0_z}\right)\mathbf{t} \\ \mathbf{2}\left(胃鈭�{胃}_0\right)\mathbf{=}\left({蝇}_z+{蝇}_{0_z}\right)\mathbf{t} \\ \frac{\mathbf{2}\left(胃鈭�{胃}_0\right)}{\mathbf{t}}\mathbf{=}{\mathbf{蝇}}_{\mathbf{z}}\mathbf{+}{\mathbf{蝇}}_{{\mathbf{0}}_{\mathbf{z}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{蝇}}_{\mathbf{z}}\mathbf{=}\frac{\mathbf{2}\left(胃鈭�{胃}_0\right)}{\mathbf{t}}\mathbf{鈭�}{\mathbf{蝇}}_{{\mathbf{0}}_{\mathbf{z}}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right) \end{gathered}$$

Substitute ${蝇}_{0_z}=500鈥�rpm$, $胃-{胃}_0=200鈥�rev$, and t =0.5 min in equation (1).

$$\begin{gathered} {\mathrm{蝇}}_{\mathrm{z}}=\frac{2\left(200\mathrm{rav}\right)}{0.5\min }-500\mathrm{rpm} \\ =800\mathrm{rpm}-500\mathrm{rpm} \\ =300\mathrm{rpm} \end{gathered}$$

Therefore, the rate of spinning when the power comes back is 300 rpm.

From the above results, it can be observed that the speed is decreasing which means the flywheel is decelerated.

Use the following expression to find the angular acceleration.

$$\begin{gathered} {蝇}_z={蝇}_{0_z}-{伪}_{z}t \\ {伪}_z=\frac{{蝇}_{0_z}-{蝇}_z}{t}......\left(2\right) \end{gathered}$$

Substitute ${蝇}_{0_z}=500鈥�rpm$, t=30 s , and${蝇}_z=300鈥�rpm$in equation ${蝇}_z^2={{蝇}_{o_z}}^2+2{伪}_z胃$.

$$\begin{gathered} {伪}_z=\frac{500\mathrm{rpm}鈭�300\mathrm{rpm}}{30\mathrm{s}} \\ =\frac{200\mathrm{rpm}}{30\mathrm{s}} \\ =\frac{1}{30\mathrm{s}}\left(200\frac{\mathrm{rev}}{\min }脳\frac{1\min }{60\sec }\right) \\ =0.11\mathrm{rev}/{\mathrm{s}}^2 \end{gathered}$$

At the end the flywheel is stopped, so take ${蝇}_z=0$.

Substitute ${蝇}_{0_z}=500鈥�rpm$,${伪}_z=0.11鈥�rev/s^2$ , and${蝇}_z=0鈥�rpm$in equation (2) to find time.

$$\begin{gathered} t=\frac{\left(500\frac{\mathrm{rev}}{\min }脳\frac{1\min }{60\sec }\right)鈭�0}{0.11\mathrm{rev}/{\mathrm{s}}^2} \\ =\frac{8.33\mathrm{rev}/\mathrm{s}}{0.11\mathrm{rev}/{\mathrm{s}}^2} \\ =75.7\sec \end{gathered}$$

Therefore, the required time is 75.7 sec.

Use the following expression to find the angular displacement.

$螖胃={蝇}_{0_z}t-1/2{伪}_z{t}^2鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�......\left(3\right)$

Substitute ${蝇}_{0_z}=8.33鈥�rev/s$,${伪}_z=0.11鈥�rev/s^2$ , and t=75.7 sec in equation (3).

$$\begin{gathered} \mathrm{螖}胃=(8.33\mathrm{rev}/\mathrm{s})\left(75.7\sec \right)鈭�\frac{1}{2}\left(0.11\mathrm{rev}/{\mathrm{s}}^2\right)(75.7\sec {)}^2 \\ =630\mathrm{rev}鈭�315\mathrm{rev} \\ =315\mathrm{rev} \end{gathered}$$

Therefore, the angular displacement is 315 rev.