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 = acos (ωt+θ) and note that the initial velocity is negative.]
Solution
The displacement equation 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,
= -v0 = Aωsinθ
Asinθ = v0/ω ...(ii)
Squaring and adding equations (i) and (ii), we get
So, A is the required amplitude of oscillation.
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,
= -v0 = Aωsinθ Asinθ = v0/ω ...(ii)
Squaring and adding equations (i) and (ii), we get
So, A is the required amplitude of oscillation.
