A bus is designed to draw its power from a rotating flywheel that is brought up to its maximum speed (3000 rpm)by an electric motor.The flywheel is a solid cylinder of mass 250 kg and radius 0.500 m (I_cylinder = MR²/2).If the bus requires an average power of 10.0 kW,how long will the flywheel rotate? A)154 s B)308 s C)463 s D)617 s

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Quizplus · Accepted Answer

The kinetic energy of the flywheel is:$$K=\frac{1}{2}I\omega^2=\frac{1}{2}\left(\frac{1}{2}MR^2ight)\left(\frac{2\pi n}{60}ight)^2$$$$=\frac{1}{4}\pi^2MR^2n^2$$The power is:$$P=\frac{dK}{dt}=\frac{d}{dt}\left(\frac{1}{4}\pi^2MR^2n^2ight)=\frac{1}{2}\pi^2MR^2n\frac{dn}{dt}$$$$=\frac{1}{2}\pi^2MR^2n\alpha$$where $\alpha$ is the angular acceleration.The angular acceleration is:$$\alpha=\frac{dn}{dt}=\frac{2\pi n}{60t}$$where $t$ is the time.Substituting into the power equation, we get:$$P=\frac{1}{2}\pi^2MR^2n\left(\frac{2\pi n}{60t}ight)=\frac{\pi^3MR^2n^2}{60t}$$Solving for $t$, we get:$$t=\frac{\pi^3MR^2n^2}{60P}=\frac{\pi^3(250	ext{ kg})(0.500	ext{ m})^2(3000	ext{ rpm})^2}{60(10.0	ext{ kW})}$$$$=154	ext{ s}$$

A Bus Is Designed to Draw Its Power from a Rotating