A mass attached to a spring is free to oscillate, with an 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 tmet `t=0`. Determine the amplitude of the resulting oscillations in terms of the parameter `omega, x_(0), v_(0)` (Hint : Start with the equation `x=a cos (omega t+theta)` and note that the initial velocity is negative.)

Explanation

Given `x=a cos (omegat+theta)`
velocity `v=(dx)/(dt)=-(a omega)sin(omegat+theta)`
`"i.e. "x_(0)=a cos theta`
`"i.e. "v_(0)=-(a omega)sin(0+theta)" ……(1)"`
`v_(0)=-omega sin(theta)`
or `a sin theta=-(v_(0))/(omega)" .........(2)"`
Solving (1) and (2) and adding we get `a^(2)=x_(0)^(2)+(v_(0)^(2))/(omega^(2))`
or `a=sqrt(x_(0)^(2)+((v_(0)^(2))/(omega^(2))))`