Question

A ball of mass `m` suspended by a weightless spring can perform vertical oscillations with damping coefficient `beta`. The natural oscillation frequency is equal to `omega_(0)`. Due to the external vertical force varying as `F=F_(0) cos omegat ` the ball performs steady `-` state harmonic oscillations. Find `:`
`(a)` the mean power `( :P : )` , develocped by the force `F`. averaged over one oscillations perod,
`(b)` the frequency `omega` of the force `F` at which `( : P : )` is maximum, what is `( : P : )_(max)` equal to ?

Answer 1

Here as usual ` tan varphi=( 2 beta omega)/( omega_(0)^(2)- omega^(2))` where `varphi` is the phase lag of the displacement
`x= a cos ( omegat - varphi), a =(F_(0))/(m) (1)/( sqrt((omega_(0)^(2)- omega^(2))^(2)+ 4 bets ^(2) omega^(2)))`
`(a) ` Mean power developed by the force over one oscillation period
`=(pi F_(0) a sin varphi)/(T)=(1)/(2) F _(0) a sin varphi`
`=(F_(0))/( m) ( beta omega^(2))/( (omega_(0)^(2)- omega^(2))^(2)+ 4 beta^(2) omega^(2))=( F_(0)^(2)beta)/( m) (1)/( ((omega_(0)^(2))/( omega)-omega)^(2)+4 beta^(2))`
`(b)` Mean power `lt Pgt` is maximum when `omega= omega_(0)` `(` for the denominator is then minimum
Also
`lt P gt _(max)=(F_(0)^(2))/( 4 m beta)`