Question

Why is the following situation impossible? In a large city with an air-pollution problem, a bus has no combustion engine. It runs over its citywide route on energy drawn from a large, rapidly rotating flywheel under the floor of the bus. The flywheel is spun up to its maximum rotation rate of 3000 rev/minby an electric motor at the bus terminal. Every time the bus speeds up, the flywheel slows down slightly. The bus is equipped with regenerative braking so that the flywheel can speed up when the bus slows down. The flywheel is a uniform solid cylinder with mass 1200 kgand radius 0.500 m.The bus body does work against air resistance and rolling resistance at the average rate of 25.0 hpas it travels its route with an average speed of 35.0 km/h.

Short answer

The situation is impossible because the bus has to run over its citywide route in a large city and the range is so small for that.

Step-by-step solution

Given values

$\begin{array}{c}\omega (\mathrm{angular}\mathrm{speed}\mathrm{of}\mathrm{flywheel})=3000\left(\frac{\mathrm{rev}}{\min }\right)\left(\frac{2\pi \mathrm{rad}}{1\mathrm{rev}}\right)\left(\frac{1\min }{60\mathrm{s}}\right)\\ =314\mathrm{rad}/\mathrm{s}\end{array}$

$\begin{array}{c}m(\mathrm{mass}\mathrm{of}\mathrm{flywheel})=1200\mathrm{kg}\\ r(\mathrm{radius}\mathrm{of}\mathrm{flywheel})=0.5\mathrm{m}\\ P=25\left(\mathrm{hp}\right)\left(\frac{746\mathrm{W}}{1\mathrm{hp}}\right)\\ =18650\mathrm{W}\end{array}$

$\begin{array}{c}v(\mathrm{average}\mathrm{speed}\mathrm{of}\mathrm{bus})=35\left(\frac{\mathrm{km}}{\mathrm{h}}\right)\left(\frac{1000\mathrm{m}}{1\mathrm{km}}\right)\left[\frac{1\mathrm{h}}{(60\times 60)\mathrm{s}}\right]\\ =9.72\mathrm{m}/\mathrm{s}\end{array}$

Solve for Δx

The moment of inertia of the wheel ( solid cylinder) is given as:

$I=\frac{1}{2}m{r}^2$

Substitute numerical values:

$\begin{array}{c}I=\frac{1}{2}\left(1200\right)(0.5{)}^2\\ =150 \mathrm{kg}·{\mathrm{m}}^2\end{array}$

The rotational kinetic energy of the bus is given by:

$\begin{array}{c}E=\frac{1}{2}\left(150\right)(314{)}^2 \\ =7.39\times {10}^6 \mathrm{J}\end{array}$

The time the flywheel rolls is:

$\begin{array}{c}\mathrm{\Delta }t=\frac{\mathrm{\Delta }x}{v}\\ =\frac{\mathrm{\Delta }x}{9.72}\end{array}$

The power output is given by:

$\begin{array}{c}P=\frac{E}{\mathrm{\Delta }t}\\ P=\frac{7.39\times {10}^6}{\frac{\mathrm{\Delta }x}{9.72}}\end{array}$

Solve for$∆x$.

$\begin{array}{c}\mathrm{\Delta }x=\frac{\left(7.39\times {10}^6\right)(9.72)}{18650}\\ =3895 \mathrm{m}\\ =3.86 \mathrm{km}\end{array}$

Therefore, the situation is impossible because the bus has to run over its citywide route in a large city and the range is so small for that.