A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 40.0 kg and diameter 75.0 cm. The power is off for 30.0 s, and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 200 complete revolutions. (a) At what rate is the flywheel spinning when the power comes back on?

Verified step by step guidance

1. Convert the initial angular velocity from rpm to rad/s. Use the conversion factor where 1 rpm equals \(\frac{2\pi}{60}\) rad/s.
2. Calculate the initial angular velocity, \(\omega_i\), using the formula \(\omega_i = 500 \times \frac{2\pi}{60}\).
3. Determine the total angular displacement during the power outage by converting the number of revolutions to radians. Use the fact that one revolution equals \(2\pi\) radians, so \(\theta = 200 \times 2\pi\) radians.
4. Use the formula for angular displacement \(\theta = \omega_i \times t + \frac{1}{2} \times \alpha \times t^2\) to solve for the angular deceleration, \(\alpha\). Rearrange to find \(\alpha = \frac{2(\theta - \omega_i \times t)}{t^2}\).
5. Calculate the final angular velocity, \(\omega_f\), using the formula \(\omega_f = \omega_i + \alpha \times t\). This will give the rate at which the flywheel is spinning when the power comes back on.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second or revolutions per minute (rpm). In this scenario, the initial angular velocity of the flywheel is given as 500 rpm, which can be converted to radians per second for calculations involving angular displacement and time.

Friction and Deceleration

Friction is a force that opposes the motion of an object, causing it to decelerate. In the case of the flywheel, friction in the axle bearings acts to slow down its rotation when the power is off. Understanding the effects of friction is crucial for determining how much the flywheel's speed decreases over the 30 seconds it is not powered.

Angular Displacement

Angular displacement refers to the angle through which an object has rotated about a specific axis, measured in radians. In this problem, the flywheel makes 200 complete revolutions during the power outage, which can be converted into radians to calculate the change in angular velocity and determine the final speed when power is restored.