Question

An external harmonic force `F` whose frequency can be varied, with amplitude maintained constant, acts in a vertical direction on a ball suspended by a weightless spring. The damping coefficient is `eta` times less than the natural oscillation frequency `omega_(0)` of the ball. How much, in per cent, does the mean power `( :P: )` developed differ from the maximum mean power `( :P: )_(max)`? Averaging is performed over one oscillation period.

Answer 1

Given ` beta = omega_(0)//eta` . Then from the previous problem
`lt Pgt =(F_(0)^(2)omega_(0))/( etam). (1)/( ((omega_(0)^(2))/(omega)-omega)^(2)+ 4 ( omega_(0)^(2))/( eta^(2)))`
At displacement resonance ` omega=sqrt(omega_(0)^(2)-2 beta^(2))`
`lt P gt_(res)=(F_(0)^(2)omega_(0))/( etam)(1)/((4 beta^(4))/( omega_(0)^(2)- 2 beta^(2))+(4 omega_(0)^(2))/( eta^(2)))=(F_(0)^(2)omega_(0))/( eta m)(1)/((4 omega_(0)^(4)//eta^(4))/( omega_(0)^(2)(1-(2)/( eta^(2))))+4( omega_(0)^(2))/( eta^(2)))`
`=(F_(0)^(2))/( 4 eta m omega_(0))(n^(2))/((1)/( n^(2)-2)+1)=(F_(0)^(2)eta)/( 4 m omega_(0))(n^(2)-2)/( n^(2)-1)`
while `lt P gt_(max)=(F_(0)^(2)eta)/(4 m omega_(0))`.
Thus `(lt P gt_(max)- lt P gt _(res))/(lt P gt_(max))=(100)/(eta^(2)-1)%`