An elastic string of constant k is attached to a block of mass m as shown. The block is given an extension of $\frac{2mg}{k}$from the equilibrium position and released. The time period of oscillations of the block is

Here string becomes slack when acceleration of body becomes more than g above equilibrium position. Hence during that period, SHM disappears:

upto P₃ , SHM appears and between P₃ & P₄ , SHM disappears. $=\frac{2T_1}{3}+\left(t_{P_3{P}_4}+t_{P_4{P}_3}\right)$

$\therefore T=t_{O{P}_2+}{t}_{P_{2}O}+t_{O{P}_3}+t_{P_3{P}_4}+t_{P_4{P}_3}+t_{P_{3}O}$

$\text{ where }:T_1=2\pi \sqrt{\frac{m}{k}}\text{ velocity at }{P}_3\text{ is : }$

$v=\omega \sqrt{A^2-x^2}=\omega \sqrt{4b^2-b^2}=b\omega \sqrt{3}$

$t_{P_3{P}_4}+t_{P_4{P}_3}=\frac{2v}{g}$