Q: A particle is vibrating in SHM. If its velocities are v_{1} and v_{2} when the displacements from the mean position are y_{1} and y_{2} respectively, then find its amplitude and time period.

Sol: $\large v_1 = \omega \sqrt{a^2 – y_1^2}$

$\large v_2 = \omega \sqrt{a^2 – y_2^2}$

$\large \frac{v_1^2}{v_2^2} = \frac{a^2 – y_1^2}{a^2 – y_2^2} $

$\large a = \sqrt{\frac{y_2^2 – y_1^2}{v_1^2 – v_2^2}} $

$\large v_1^2 – v_2^2 = \omega^2 (y_2^2 – y_1^2) $

$\large T = 2\pi \sqrt{\frac{y_2^2 – y_1^2}{v_1^2 – v_2^2}} $