A body performes torsional oscillations according to the law `varphi=varphi_(0)e^(-betat)cos omegat`. Find `:` `(a)` the angular velocity `dot(varphi)` and the angular acceleration `ddot(varphi)` of the body ar the moment `t=0`,
`(b)` the moment of time at which the angular velocity becomes maximum.

Explanation

Given `varphi=varphi_(0)e^(-betat)cos omegat`
we have ` dot(varphi)=-beta varphi-omegavarphi_(0)e^(- betat)sin omegat`
` ddot(varphi)=-beta dot(varphi)+betaomegavarphi_(0)e^(-betat)sin omegat -omega^(2) varphi_(0)-e^(-betat)cos omegat `
`=beta^(2)varphi+2 beta omega varphi_(0) e^(-betat) sin omegat - omega^(2) varphi`
So
(a) `(varphi)_(0)=-betavarphi_(0), (ddot(varphi))_(0)=(beta^(2)-omega^(2))varphi_(0)`
(b) `dot(varphi)=-varphi_(0)e^(-betat)( beta cos omega t + omega sin omega t )` becomes maximum `(` or minimum `)` when
`ddot(varphi)=varphi_(0)(beta^(2)-omega^(2))e^(-betat)cos omegat+2 beta omegavarphi_(0) e^(-betat)sin omega t =0`
or `tan omegat =(omega^(2)-beta^(2))/( 2 beta omega)`
and ` t_(n)=(1)/(omega)[(tan ^(-1)((omega^(2)-beta^(2))/(2 beta omega))+n pi )], n=0,1,2,,.....`