A small sphere is given vertical velocity of magnitude$v_0=5m{s}^{-1}$ and it swings in a vertical plane about the end of a massless string. The angle θ with the vertical at which string will break, knowing that it can withstand a maximum tension equal to twice the weight of the sphere, is

$$\begin{gathered} \mathrm{from} \mathrm{equation} \mathrm{of} \mathrm{motion}, {\mathrm{v}}^2 = {\mathrm{v}}_0^2+2\mathrm{gl} \cos \mathrm{\theta } \\ \mathrm{for} \mathrm{vertical} \mathrm{circular} \mathrm{motion} \mathrm{tension} \mathrm{in} \mathrm{string} \mathrm{is} \mathrm{T}= \mathrm{mg} \cos \mathrm{\theta } + \frac{{\mathrm{mv}}^2}{\mathrm{l}} \\ \mathrm{m} \mathrm{is} \mathrm{mass} \mathrm{attached} \mathrm{to} \mathrm{string}, \mathrm{g} \mathrm{is} \mathrm{acceleration} \mathrm{due} \mathrm{to} \mathrm{gravity}, \\ \mathrm{\theta } \mathrm{is} \mathrm{angle} \mathrm{made} \mathrm{by} \mathrm{string} \mathrm{with} \mathrm{vertical}, \mathrm{v} \mathrm{is} \mathrm{velocity} \mathrm{of} \mathrm{mass}, \mathrm{l} \mathrm{is} \mathrm{length} \mathrm{of} \mathrm{string} \\ \mathrm{Given} \mathrm{string} \mathrm{can} \mathrm{withstand} \mathrm{tension} \mathrm{T} = 2\mathrm{mg} , \mathrm{substitute} \mathrm{in} \mathrm{above} \mathrm{equation} \\ 2\mathrm{mg} = \mathrm{mg} \cos \mathrm{\theta }+\frac{\mathrm{m}}{\mathrm{l}}({\mathrm{v}}_0^2+2\mathrm{gl} \mathrm{cos\theta }) \\ \mathrm{on} \mathrm{simplification} \\ \Rightarrow \cos \mathrm{\theta } =\frac{1}{4} \\ \mathrm{angle} \mathrm{made} \mathrm{by} \mathrm{the} \mathrm{string} \mathrm{with} \mathrm{vertical} \mathrm{\theta }={\cos }^{-1}\left[\frac{1}{4}\right] \end{gathered}$$