A particle suspended by a light inextensible thread of length $\mathrm{l}$ is projected horizontally from its lowest position with velocity $\sqrt{7\mathrm{gl}/2}$. The string will slack after swinging through an angle equal to

$\mathrm{h}=\mathrm{I}+\mathrm{Isin\theta }=\mathrm{I}(1+\mathrm{sin\theta })$

${\mathrm{v}}^2 ={\mathrm{u}}^2-2\mathrm{gh}$

=u^{2}-2 g l(1+\sin \theta)

String will slack, where component of weight towards centre is just equal to centripetal force or, $\mathrm{mgsin\theta }=\frac{{\mathrm{mv}}^2}{\mathrm{l}}=\frac{\mathrm{m}}{\mathrm{l}}\left[{\mathrm{u}}^2-2\mathrm{gl}(1+\mathrm{sin\theta })\right]$

Substituting ${\mathrm{u}}^2=\frac{7\mathrm{gl}}{2}$, we get

$\mathrm{sin\theta }=\frac{1}{2} \text{ or } \mathrm{\theta }=30°$

$\therefore$ The desired angle is $90°+30°$ or $120°$.