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.]

Asked by Abhisek | 1 year ago |  196

##### Solution :-

The angular velocity of the spring = ω

x = a cos (ωt+θ)

At t = 0, x = x0

Substituting these values in the above equation we get

x0 = A cos θ —–(1)

Velocity, v=$$\dfrac{dx}{dt}$$

= – Aω sin (ωt+θ)

At t = 0, v = – v0

Substituting these values in the above equation we get

– v= – Aω sin θ

Asin θ =  v0/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=$$\sqrt{x_{0}^{2}+\dfrac{v_{0}^{2}}{\omega^{2}}}$$

Answered by Pragya Singh | 1 year ago

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