A flywheel with the initial angular velocity `omega_0` decelerates due to the forces whose moment relative to the axis is proportional to the square root of its angular velocity. Find the mean angular velocity of the flywheel averaged over the total decleration time.

Explanation

According to the question,
`I(domega)/(dt)=-ksqrtomega` or, `I=(domega)/(sqrtomega)=-kdt`
Integrating, `sqrtomega=-(kt)/(2I)+sqrt(omega_0)`
or, `omega=(k^2t^2)/(4I^2)-(sqrt(omega_0)kt)/(I)+omega_0`, (Nothing that at `t=0`, `omega=omega_0`.)
Let the flywheel stops at `t=t_0` then from Eq. (1), `t_0=(2Isqrt(omega_0))/(k)`
Hence sought average angular velocity
`lt omega gt = (underset(0)overset((2Isqrt(omega_0))/(k))int((k^2t^2)/(4I^2)-(sqrt(omega_0)kt)/(I)+omega_0)dt)/(underset(0)overset((2Isqrt(omega_0))/(k))intdt)=omega_0/3`