A solid body rotates about a stationary axis so that its angular velocity depends on the rotation angle `varphi` as `omega=omega_0-avarphi`, where `omega_0` and a are positive constants. At the moment `t=0` the angle `varphi=0`. Find the time dependence of
(a) the rotation angle,
(b) the angular velocity.

Explanation

We have `omega=omega_0-avarphi=(dvarphi)/(dt)`
Integration this Eq. within its limit for `(varphi)t`
`underset(0)overset(varphi)int(dvarphi)/(omega_0-kvarphi)=underset0oversett int dt` or, `1n(omega_0-k varphi)/(omega_0)=-kt`
Hence `varphi=(omega_0)/(k)(1-e^(-kt))` (1)
(b) From the Eq., `omega=omega_0-kvarphi` and Eq. (1) or by differentiating Eq. (1)
`omega=omega_0e^(-kt)`