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 x_{0} and pushed towards the centre with a velocity v_{0} at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x_{0} and v_{0}. [Hint: Start with the equation x = a cos (ωt+θ) and note that the initial velocity is negative.]

Asked by Abhisek | 1 year ago | 195

The angular velocity of the spring = ω

x = a cos (ωt+θ)

At t = 0, x = x_{0}

Substituting these values in the above equation we get

x_{0} = A cos θ —–(1)

Velocity, v=\( \dfrac{dx}{dt}\)

= – Aω sin (ωt+θ)

At t = 0, v = – v_{0}

Substituting these values in the above equation we get

– v_{0 }= – Aω sin θ

Asin θ = v_{0/ω}———-(2)

Squaring and adding (1) and (2) we get

\( A^{2}(cos^{2}\theta+sin^{2}\theta )=x_{0}^{2}+\dfrac{v_{0}^{2}}{\omega ^{2}}\)

A=\( \sqrt{x_{0}^{2}+\dfrac{v_{0}^{2}}{\omega^{2}}}\)

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