How do I calculate?

Question

A flywheel in the shape of a solid cylinder with a radius of 0.499 m and mass of 10.1 kg can be brought to an angular speed of 5.88 rad/s in 0.531 s by a motor exerting a constant torque. After the motor is turned off, the flywheel makes 26.8 rev before coming to rest because of friction (which is assumed constant during rotation).

What percentate of the power generated by the motor is used to overcome friction?

Answer 1

For your solid cylinder I=m*r^2/2, m=10.1kg, r=0.499m,
w=w’*t1, w angular speed 5.88 rad/s, w’ angular acceleration, t1=0.531s
u=w’*t1^2/2 angular path to achieve w, u=w*t1/2.
Thus the total work of motor A=A1+A2, A1 useful work, A2 to overcome friction resistance,
A1= I*w^2/2, A2=T*u=T*w*t1/2, T friction torque or torque to overcome friction.
Now motor off.
y=y’*t2^2/2, y=26.8*2*pi rad, y’ friction deceleration, t2 time of deceleration
while w=y’*t2, thus t2=w/y’ & y=y’*(w/y’)^2/2, y=w^2/(2*y’) or y’=w^2/(2y)
on the other hand T=I*y’=I*w^2/(2y) or I=T*2*y/w^2
now p=A2/(A1+A2)=
=(T*w*t1/2)/((2y*T/w^2)*w^2/2+T*w*t1/2)=
=(w*t1/2)/(y+w*t1/2)=1/(1+2y/(w*t1))
that means: to find percentage p three values – y, w & t1 – are essential, others do not matter,
check it & delete everything unnecessary I’ve written here!