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.

Answer 1

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