A solid sphere of mass `3 kg` and radius `2m` is free to rotate about an axis passing through its centre. Find a constant tangential force `F` required to rotate the sphere with a velocity of `10 rad//s` in `2 sec`. Starting from rest. Also, find the number of rotations made by the sphere in that time interval.

Explanation

Here, `m = 3kg, r = 2m omega_(0) = 0`
`omega = 10 rad//s, t = 2s`
`alpha = (omega - omega_(0))/(t) = (10 - 0)/(2) = 5 rad//s^(2)`
`theta = omega_(0)t + (1)/(2)alpha t^(2) = 0 + (1)/(2)xx5 xx 2^(2) = 10 rad`
Now, `tau = F xx r = I alpha - (2)/(5)m r^(2) alpha`
`F = (2)/(5)m r alpha = (2)/(5) xx 3 xx2 xx 5 = 12 N`
Number of revolutions `= (theta)/(2pi) = (10)/(2pi) = (5)/(pi)`