A particle is vibrating in SHM. If its velocities are v1 and v2 when the displacements from the mean position are y1 and y2 respectively…

Q: A particle is vibrating in SHM. If its velocities are v1 and v2 when the displacements from the mean position are y1 and y2 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}} $