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A uniform horizontal disc fixed at its centre to an elastic vertical rod performs forced torsional oscillations dur to the moment of forces `N_(z)=N_(m)cos omegat`. The oscialltions obey the law `varphi=varphi_(m) cos ( omega t- alpha)`. `(a)` the work performed by friction forces acting on the disc during one oscillation period , `(b)` the quality factor of the given oscillator if th emoment of inertia of the disc relative to the axis is equal to `I`. |
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Answer» The equation of the disc is `ddot(varphi)+ 2 beta dot (varphi)+ omega_(0)^(2) varphi=(N_(m) cos omegat)/( I)` Then as before `varphi=varphi_(m) cos ( omegat- alpha)` where `varphi(m)=(N_(m))/(I[( omega_(0)^(2)-omega^(2))^(2)+ 4 beta^(2)omega^(2)] ^(1//2)), tan alpha=(2 beta omega)/( omega_(0)^(2)- omega^(2))` `(a)` Work performed by frictional forces `=-int N_(r) d varphi` where`N_(r)=-2 I beta dot(varphi)=- int_(0)^(T) 2 betaI dot(varphi^(2))dt=-2 pi beta omega I varphi_(m)^(2)` `=- pi I varphi_(m)^(2)[( omega_(0)^(2)- omega^(2))^(2)+4 beta^(2)omega^(2)]^(1//2)sin alpha=- pi N_(m) varphi_(m) sin alpha` `(b)` The quality factor `Q=(pi)/(lambda)=(pi)/(betaT)=(sqrt(omega_(0)^(2)-beta^(2)))/( 2 beta)=( omegasqrt(omega_(0)^(2)-beta^(2)))/( ( omega_(0)^(2)- omega^(2) ) tan alpha) =(1)/(tan alpha){( 4 omega^(2) omega_(0)^(2))/((omega_(0)^(2)-omega^(2))^(2))-(4 beta^(2)omega^(2))/((omega_(0)^(2)-omega^(2))^(2))}` `=(1)/(2 tan alpha) { ( 4 omega^(2) omega_(0)^(2) I^(2) varphi_(m)^(2))/( N_(m)^(2)cos^(2) alpha)- tan ^(2) alpha}`since` omega_(0)^(2)= omega^(2)+(N)/( Ivarphi_(m))cos alpha` `=(1)/(2 sin alpha){(4 omega^(2) omega_(0)^(2) I^(2) varphi_(m)^(2))/( N_(m)^(2)) - sin ^(2) alpha}^(1//2)` `=(1)/(2 sin alpha){ ( 4 omega^(2) I^(2) varphi_(m)^(2))/( N_(m)^(2))(omega^(2)+(N_(m) cos alpha)/( I varphi_(m)))+ 1- cos^(2) alpha}^(1//2)` `=(1)/( 2sin alpha){( 4 I^(2) varphi_(m)^(2))/( N_(m)^(2))omega^(4)+ ( 4Ivarphi_(m))/( N_(m))omega ^(2) cos alpha+ cos ^(2) alpha-1}^(1//2)=(1)/( 2sin alpha ){((2Ivarphi_(m)omega^(2))/( N_(m))+ cos alpha)^(2)-1}^(1//2)` |
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