# Q. 14.25 A mass attached to a spring is free to oscillate, with angular velocity $\omega$, in a horizontal plane without friction or damping. It is pulled to 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 $\omega$, $x_{0}$ and $v_{0}$ . [Hint : Start with the equation $\inline x=a\; cos(\omega t+\theta )$ and note that the initial velocity is negative.]

At the maximum extension of spring, the entire energy of the system would be stored as the potential energy of the spring.

Let the amplitude be A

$\\\frac{1}{2}kA^{2}=\frac{1}{2}mv_{0}^{2}+\frac{1}{2}kx_{0}^{2}\\ A=\sqrt{x_{0}^{2}+\frac{m}{k}v_{0}^{2}}$

The angular frequency of a spring-mass system is always equal to $\sqrt{\frac{k}{m}}$

Therefore

$A=\sqrt{x_{0}^{2}+\frac{v_{0}^{2}}{\omega ^{2}}}$

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