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

`omega=omega_(0)-ktheta`
`(d theta)/(dt)=omega_(0)-ktheta`
`int_(0)^(theta)(d theta)/(omega_(0)-ktheta)=int_(0)^(t)t dt`
`(|In(omega_(0)-ktheta)|_(0)^(theta))/(-k)=t`
`In(omega_(0)-ktheta)-In(omega_(0))=-kt`
`In((omega_(0)-ktheta)/(omega_(0)))=-kt`
`(omega_(0)-ktheta)/(omega_(0))=e^(-kt)`
`omega_(0)-ktheta=omega_(0)e^(-kt)`
`theta=(omega_(0))/(k)(1-e^(-kt))`
`omega=(d theta)/(dt)=(omega_(0))/(k)[0-e^(-kt)(-k)]`
`omega=omega_(0)e^(-kt)`