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

The displacement equation for an oscillation mass is given by:
`x=Acos(omegat+theta)` Where, A is the amplitude
xis the displacement
0is the phase constant
`v= (dx)/(dt)=-Aomegasin(omegat+theta)`
Velocity ,
At t=o, `x=x_(o)`
`x_(o)=Acostheta = x_(0)`........(i)
And, `(dx)/(dt)=-v_(0)=Aomegasintheta`
`Asin theta = (v_(0))/(omega)`......(ii)
Squaring and adding equations (i)and (ii),we get:
`A^(2)(cos^(2)theta+sin^(2)theta)=x_(0)^(2)+((v_(0)^(2))/(omega^(2)))`
`A=sqrt(x_(0)^(2)+((v_(0))/(omega^(2)))`
Hence, the amplitude of the resulting oscillation is `sqrt(x_(0)^(2)+((v_(0))/(omega^(2)))`