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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 omegatthe ball performssteady - 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 ? |
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Answer» SOLUTION :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)` |
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