Question

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. [Hint: Start with the equation x = a cos (ωt+θ) and note that the initial velocity is negative.]

Answer 1

standard equation of SHM,
x = Acos(wt + ∅)
Where x is displacement
A is amplitude
w is angular frequency
∅ is phase angle

x = Acos(wt + ∅)
differentiate wrt time t
dx/dt = -Awsin(wt + ∅)
V = - Awsin(wt + ∅)
v = instantaneous velocity of the particle at t
Now, at t = 0, x = xo
xo = Acos∅ ------(1)
Also at t = 0, v = vo
vo = -wAsin∅
vo/w = -Asin∅------(2)

From eqns (1) and (2)
(vo/w)² + xo² = A²
A = √{xo² + (vo/w²)}