Flywheels are large, massive wheels used to store energy. They can be spun up sl

Question

Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.8 m diameter and a mass of 220 kg . Its maximum angular velocity is 1600 rpm .

Part A

A motor spins up the flywheel with a constant torque of 57 N?m . How long does it take the flywheel to reach top speed?

Express your answer to two significant figures and include the appropriate units.

Part B

How much energy is stored in the flywheel?

Express your answer to two significant figures and include the appropriate units.

Part C

The flywheel is disconnected from the motor and connected to a machine to which it will deliver energy. Half the energy stored in the flywheel is delivered in 2.1 s . What is the average power delivered to the machine?

Express your answer to two significant figures and include the appropriate units.

Part D

How much torque does the flywheel exert on the machine?

Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

here,

mass of flywheel , m1 = 220 kg

diameter , d = 1.8 m

radius , r = d/2= 0.9 m

angular velocity , w = 1600 rpm

w = 167.5 rad/s

moment of inertia , I = 0.5 * m1 * r^2 = 89.1 kg.m^2

a)

constant torque , T = 57 N.m

angular accelration , alpha = T/I = 0.64 rad/s^2

let the time taken be t

1600 = 0 + alpha * t

1600 = 0 + 0.64 * t

t = 2501.1 s

b)

the energy stored in flywheel , E = 0.5 * I * w^2

E = 0.5 * 89.1 * 1600^2 J

E = 1.14 * 10^8 J

c)

time taken , t1 = 2.1 s

the average power delivered , P = E/ (2t1)

P = (1.14 * 10^8)/( 2 * 2.1) W

P = 2.72 * 10^7 W

d)

let the final angular speed be w'

as E = 0.5 * E'

0.5 * I * w'^2 = 0.5 * 0.5 * I * w^2

w' = 0.5 * 1600^2

w' = 1136 rad/s

let the angular accelration be alpha'

w' = w + alpha * t

1600 = 1136 + alpha * 2.1

alpha = 221 rad/s^2

the torque exerted by flywheel on machine , T' = I * alpha

T' = 89.1 * 221 N.m = 1.97 * 10^4 N.m

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