A high-speed flywheel in a motor is spinning at 500 rpm when a power failure sud

Question

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 77.0 cm . The power is off for 31.0 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 180 complete revolutions.

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

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?

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

Explanation / Answer

The flywheel is solid cylindrical disc. Moment of inertial = ½ * mass * radius^2
Mass = 40 kg
Radius = ½ * 77.0 cm = 38.5 cm = 0.385 meter
Moment of inertial = ½ * 40 * 0.385^2 = 2.9645


Convert rpm to radians/second
The distance of 1 revolution = 1 circumference = 2 * * r
The number of radians/s in 1 revolution = 2 *
1 minute = 60 seconds
1 revolution per minute = 2 * radians / 60 seconds = /30 rad/s

Initial angular velocity = 500 * /30 = 16.667 * rad/s
180 revolutions = 180 * 2 * = 360 * radians

Initially the velocity of the flywheel was 16.667 * rad/s
In 31 seconds, the flywheel rotated an angle of 360 * radians.
= i * t + ½ * * t^2
= 340 * radians
i = 16.667 * rad/s
t = 31 seconds

360 * = 16.667 * * 29 + ½ * * 29^2
Solve for the angular acceleration
= (360 * – 16.667 * * 31) ÷ (½ * 31^2)
= -1.02 rad/s^2


A) At what rate is the flywheel spinning when the power comes back on?
w= ? rad/s
f = i + * t = 16.667 * + -1.02 * 29 = 22.78 rad/s
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?
t= ? s

Use the value of the angular acceleration to determine the time to reduce the angular velocity from 16.667 * rad/s to 0 rad/s.
f = i + * t
0 = 16.667 * + -1.071 * t
Time = 51.33 seconds

C) How many revolutions would the wheel have made during this time?
N= ? rev
The flywheel’s initial angular velocity = 16.667 * rad/s. It decelerated at the rate of 1.02 rad/s^2 for 48.89 seconds.
= i * t + ½ * * t^2
= 16.667 * * 51.33 + ½ * -1.02 * 51.33^2
2559.9 - 1280
= 1343.95 radians

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