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A mass attached to a spring is free to o...

A mass attached to a spring is free to oscillate, with angular velocity `omega`, in a horizontal plane without friction or damping. It is pulled to a distance `x_(0)` and pushed towards the centre with a velocity `v_(0)` at time `t=0`. Determine the amplitude of the resulting oscillations in terms of the parameters `ometa, x_(0)and v_(0)`.

Text Solution

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When `t=0, x=x_(0)` and `(dx)/(dt)=-v_(0)` `:.` From `x=Acos(omegat+theta), x_(0)=Acostheta` …(i)
velocity `(dx)/(dt)=-Aomegasin(omegat+theta)`
`-v_(0)=-Aomegasintheta or A sin theta=v_(0)//omega` …(ii)
Squaring and adding (i) and (ii), we get
`A^(2)(sin^(2)theta+cos^(2)theta)=(v_(0)^(2)//omega^(2))+x_(0)^(2)`
`A=[v_(0)^(2)//omega^(2)+x_(0)^(2)]^(1//2)`
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