Question
A particle is doing simple harmonic motion. If the velocity of the particle at distances $x_1$ and $x_2$ from the mean position are $v_1$ and $v_2$ respectively, then prove that its time period will be
$ T =2 \pi \sqrt{\frac{x_2^2-x_1^2}{ v _1^2- v _2^2}}. $

Answer

$ v=\omega \sqrt{A^2-y^2} $
$y=x_1, v=v_1$ and $y=x_2, v=v_2$
$ \begin{aligned} v_1 & =\omega \sqrt{A^2-x_1^2} \\ v_1^2 & =\omega^2\left(A^2-x_1^2\right) \\ v_2 & =\omega \sqrt{A^2-x_2^2} \\ v_2^2 & =\omega^2\left(A^2-x_2^2\right) \end{aligned} $
On subtracting equation (2) from (1)
$ \begin{aligned} v_1^2-v_2^2 & =\omega^2\left(x_2^2-x_1^2\right) \\ \omega & =\sqrt{\frac{v_1^2-v_2^2}{x_2^2-x_1^2}} \\ \text { Period } T & =\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{x_2^2-x_1^2}{v_1^2-v_2^2}} \end{aligned} $
Hence Proved.

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