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 38.0 kg and diameter 77.0 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.

Part A

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

Part 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?

Part C

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 = 38.0 kg
Radius = ½ * 77.0 cm = 38.5 cm = 0.385 m
Moment of inertial = ½ * 38 * 0.385^2 = 2.816


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
150 revolutions = 150 * 2 * = 300 * radians

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

300 * = 16.667 * * 35 + ½ * * 35^2
Solve for the angular acceleration
= (300 * – 16.667 * * 35) ÷ (½ * 35^2)
= -1.453 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 * + -1.453 * 35 = 1.506 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 * + -1.453 * t
Time = 36.04 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 1.453 rad/s^2 for 36.04 seconds.
= i * t + ½ * * t^2
= 16.667 * * 36.04 + ½ * -1.453 * 36.04^2
= 943.45 radians

Number of revolutions = number of radians ÷ (2 * ) = 150.15

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