Question

g A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 39.0kg and diameter 78.0cm. The power is off for 34.0s, and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 170 complete revolutions.At what rate is the flywheel spinning when the power comes back on?

Answer 1

Answer:

$10.54\ \text{rad/s}$

Explanation:

$\omega_i$ = Initial angular velocity = 500 rpm = $500\times \dfrac{2\pi}{60}\ \text{rad/s}$

$\omega_f$ = Final angular velocity

t = Time = 34 s

$\theta$ = Angular displacement = 170 revs = $170\times 2\pi\ \text{rad}$

$\alpha$ = Angulr acceleration

From the kinematic equations of angular motion we have

$\theta=\omega_it+\dfrac{1}{2}\alpha t^2\\\Rightarrow \alpha=\dfrac{\theta-\omega_it}{\dfrac{1}{2}t^2}\\\Rightarrow \alpha=\dfrac{170\times 2\pi-500\times \dfrac{2\pi}{60}\times 34}{\dfrac{1}{2}\times 34^2}\\\Rightarrow \alpha=-1.23\ \text{rad/s}^2$

$\omega_f=\omega_i+\alpha t\\\Rightarrow \omega_f=500\times \dfrac{2\pi}{60}+(-1.23)\times 34\\\Rightarrow \omega_f=10.54\ \text{rad/s}$

The rate at which the wheel is spinning when the power comes back on is $10.54\ \text{rad/s}$.