91Ó°ÊÓ

The initial angular speed of theflywheel,${\mathrm{Ó\neg }}_{\mathrm{i}}=1.5\text{ }\frac{\text{rad}}{\text{s}}$

The final angular speed of the engine,$\mathrm{Ó\neg }=0\text{ }\frac{\text{rad}}{\text{s}}$

The angular displacement, $\mathrm{θ}=40\text{ r±ð±¹}=251.3\text{ }\text{rad}$

Use the kinematic equation for constant angular acceleration to calculate the time and angular acceleration. The time for 20 revolutions can be calculated by using the value of angular acceleration.

$\mathrm{Ó\neg }={\mathrm{Ó\neg }}_0+\mathrm{Î\pm ³Ù}$

${\mathrm{Ó\neg }}^2={\mathrm{Ó\neg }}_0^2+2\mathrm{Î\pm θ}$

$\mathrm{θ}=\frac{1}{2}(\mathrm{Ó\neg }+{\mathrm{Ó\neg }}_0)\mathrm{t}$

The kinematic equation for angular motion as

$\mathrm{θ}=\frac{1}{2}(\mathrm{Ó\neg }+{\mathrm{Ó\neg }}_0)\mathrm{t}$

By rearranging equation for time solve further as:

$\begin{array}{c}\mathrm{t}=\frac{2\mathrm{θ}}{\mathrm{Ó\neg }+{\mathrm{Ó\neg }}_0}\\ =\frac{2×251.3}{0+1.5}\\ =335\text{s}\\ ≈340\text{s}\end{array}$

The time for the flywheel to come to rest, $\mathrm{t}=340\text{s}$.

Consider the formula for the angular acceleration as:

$\begin{array}{c}\mathrm{Ó\neg }={\mathrm{Ó\neg }}_0+\mathrm{Î\pm ³Ù}\\ \mathrm{Î\pm }=\frac{\mathrm{Ó\neg }−{\mathrm{Ó\neg }}_0}{\mathrm{t}}\end{array}$

Substitute the values and solve as:

$\begin{array}{c}\mathrm{Î\pm }=\frac{0−1.5}{335}\\ \mathrm{Î\pm }=−4.5×{10}^{−3}\text{ }\frac{\text{rad}}{{\text{s}}^2}\end{array}$

Angular acceleration of the flywheel, $\mathrm{Î\pm }=−4.5×{10}^{−3}\text{ }\frac{\text{rad}}{{\text{s}}^2}$

Now, for $\mathrm{θ}=20\text{rev}=125.6637\text{rad}$, consider the formula:

$\begin{array}{c}\mathrm{θ}={\mathrm{Ó\neg }}_0\mathrm{t}+\frac{1}{2}{\mathrm{Î\pm ³Ù}}^2\\ 125.6637=1.5\mathrm{t}−\left(\frac{1}{2}×4.5×{10}^{−3}×{\mathrm{t}}^2\right)\\ 2.25×{10}^{−3}{\mathrm{t}}^2−1.5\mathrm{t}+125.6637=0\end{array}$

Solution to the equation is:

$\mathrm{t}=98.2576\text{s}\text{ o°ù}\text{ }568.409\text{sec}$

The practical value of t should be smaller, therefore,

$\begin{array}{c}\text{t}=98.2576\text{s}\\ ≈98\text{s}\end{array}$

The time required for the flywheel to complete first 20 revolutions is $98\text{ }\mathrm{s}$.