Question

A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 37.0 kg and diameter 73.0 cm. The power is off for 35.0 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 150 complete revolutions.

A) At what rate is the flywheel spinning when the power comes back on?

B) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on?

C) How many revolutions would the wheel have made during this time?

Answer 1

Initial angular velocity = 500 * π/30 = 16.667 * π rad/s

150 revolutions = 150 * 2 * π = 300 * π radians

Initially the velocity of the flywheel was 16.667 * π rad/s
In 35 seconds, the flywheel rotated an angle of 300 * π radians.
θ = ωi * t + ½ * α * t^2
θ = 300 * π radians
ωi = 16.667 * π rad/s
t = 35 seconds

300 * π = 16.667 * π * 35 + ½ * α * 35^2
Solve for the angular acceleration
α = (300 * π – 16.667 * π * 35) ÷ (½ * 35^2)
α = ( 942- 1831.7 ) / 612.5

= - 1.45 rad/s^2

a)ωf = ωi + α * t

= 16.667 * π + -1.45 * 35

= 52.33- 50.75

= 1.58438 rad / s

b)Use the value of the angular acceleration to determine the time to reduce the angular velocity from 16.667 * π rad/s to 0 rad/s.

ωf = ωi + α * t
0 = 16.667 * π + -1.45* t  

t = 36.1 sec

C)

The flywheel’s initial angular velocity = 16.667 * π rad/s. It decelerated at the rate of 1.45 rad/s^2 for 36.1seconds.

θ = ωi * t + ½ * α * t^2
θ = 16.667 * π * 36.1 + ½ * -1.45 * 36.1^2
= 1888.88 - 944.83

= 944.05 rad

Number of revolutions = number of radians ÷ (2 * π)

= 150.33 rev