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 initial velocity is negative.]

Answer 1

Hii dear,

◆ Answer-
A = √(x₀^2+v₀^2/ω^2)

● Explaination-
Consider a particle of mass m moving in SHM.
The position of particle is given by-
x = Acos(ωt+θ)

Now, we can configure velocity,
v = dx/dt
v = -Aωsin(ωt+θ)

At t = 0,
x₀ = Acosθ ...(1)
v₀ = Aωsinθ
v₀/ω = Asinθ ...(2)

Squaring and adding (1) & (2),
x₀^2 + v₀^2/ω^2 = (Acosθ)^2 + (Asinθ)^2
x₀^2 + v₀^2/ω^2 = A^2
A = √(x₀^2+v₀^2/ω^2)


Hope that is useful...