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)`.

Explanation

`x=alpha cos (omegat+theta)`
`v=(dx)/(dt) =-a omega sin(omegat+theta)`
when `t=0, x=x_(0) ` and `(dx)/(dt) =-v_(0)`
`:. X_(0)=a cos theta.....(i)`
and `-v_(0)=-a omega sin theta` or ` a sin theta=(v_(0))/(omega)....(ii)`
Squaring and adding (i) and (ii) , we get
`a^(2)(cos^(2) theta+sin^(2) theta)=x_(0)^(2)+(v_(0)^(2))/(omega^(2))`
or `a=sqrt((x_(0)^(2)+(v_(0)^(2))/(omega^(2)))`