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A mass attached to a spring is free to oscillate with angular velocity ω, in a horizontal plane without friction or damping.

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 x=a\; cos(\omega t+\theta ) and note that the initial velocity is negative.]

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S Sayak

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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