## A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity v0 at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0 and v0. | Learn NCERT solution | Education portal Class 11 Physics | Study online Unit-14 Oscillations

### Q.25:- A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity v0 at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0and v0. [Hint: Start with the equation x = a cos (ωt+θ) and note that the initial velocity is negative.]

The displacement rquation for an oscillating mass is given by:
x = Acos(ωt + θ)
where,
A is the amplitude
x is the displacement
θ is the phase constant

Velcoity, v = dx/dt = –Aωsin(ωt + θ)
At t = 0, x = x0
x0 = Acos θ = x0    ….(i)
and, dx/dt = –v0Aωsinθ
Asinθ = v0/ω          …(ii)

Squaring and adding equations (i) and (ii), we get: