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 x0 and pushed towards the centre with a velocityv_0 at time t=0 . Determine the amplitude of the resulting oscillations in terms of the parameters omega, x_0 and v_0 . x= A cos (omega t + theta)

A mass attached to a spring is free to oscillate, with angular velocit

1.	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 x0 and pushed towards the centre with a velocityv_0 at time t=0 . Determine the amplitude of the resulting oscillations in terms of the parameters omega, x_0 and v_0 . x= A cos (omega t + theta)
Answer» Solution :velocity, `(dx)/(dt) = - A omega sin (omega t + theta)` When `t= 0 , x = x_0 and (dx)/(dt)= - v_0 "" THEREFORE x_0 = A COS theta""……..(i)` `-v_0 =- A omega sin theta ` (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)`