Ncert Exercises & Solutions

14.25 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=acos(ωt+θ) and note that the initial velocity is negative.]

The displacement equation is given by :

x = Acos (ωt + θ)

Velocity, v = \frac{dx}{dt} = - Aωsin (ωt + θ)

At t = 0, x = x_{0} \Rightarrow Acosθ = x_{0} . . . (i)

And, \frac{dx}{dt} = - v_{0} = Aωsin θ

Asin θ = \frac{v_{0}}{ω} . . . (ii)

Squaring and adding equations (i) and (ii), we get :

\begin{array}{l} A^{2} (\cos^{2} θ + \sin^{2} θ) = x_{0}^{2} + (\frac{v_{0}^{2}}{ω^{2}}) \\ ∴ A = \sqrt{x_{0}^{2} + {(\frac{v_{0}}{ω})}^{2}} \end{array}

Hence, the amplitude of the resulting oscillation is \sqrt{x_{0}^{2} + {(\frac{v_{0}}{ω})}^{2}} .

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