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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 `omega, x_(0)and v_(0)`.

Text Solution

Verified by Experts

As `x = A cos (omegat + phi) " "…(i)`
At `t = 0, x = x_(0)`
`x_(0) = A cos theta`
`A cos theta = x_(0)" "…(a)`
differentiating (i) w.r.t. time
`(dx)/(dt) = - A omega sin (omegat + 0)`
`V_(0) = - A omega sin theta`
`A sin theta = (V_(0))/(omega) " "...(b)`
Squaring and adding (a) and (b) we get
`A^(2) cos^(2)theta + A^(2) sin^(2) theta = x_(0)^(2) + ((V_(0))/(omega))^(2)`
`A^(2) = x_(0)^(2) + ((V_(0))/(omega))^(2)`
`A = sqrt(x_(0)^(2) + ((V_(0))/(omega))^(2))`
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