A stone of mass 1 kg tied to a light inextensible string of length L = $\frac{10}{3}\mathrm{m}$ is whirling in a circular path of radius L in a vertical plane. If the ratio of the maximum tension in the string to the minimum tension in the string is 4 and if g is taken to be 10 $\mathrm{m}/{\sec }^2$, the speed of the stone at the highest point of the circle is

Since the maximum tension ${\mathrm{T}}_{\mathrm{B}}$ in the string moving in the vertical circle is at the bottom and minimum tension ${\mathrm{T}}_{\mathrm{T}}$ is at the top.

$\therefore {\mathrm{T}}_{\mathrm{B}} = \frac{{{\mathrm{mv}}_{\mathrm{B}}}^2}{\mathrm{L}}+\mathrm{mg} \mathrm{and} {\mathrm{T}}_{\mathrm{T}} = \frac{{{\mathrm{mv}}_{\mathrm{T}}}^2}{\mathrm{L}}-\mathrm{mg}$

$\therefore \frac{{\mathrm{T}}_{\mathrm{B}}}{{\mathrm{T}}_{\mathrm{T}}} = \frac{{\displaystyle \frac{\mathrm{m}{{\mathrm{v}}_{\mathrm{B}}}^2}{\mathrm{L}}+\mathrm{mg}}}{{\displaystyle \frac{{{\mathrm{mv}}_{\mathrm{T}}}^2}{\mathrm{L}}-\mathrm{mg}}} = \frac{4}{1} \mathrm{or} \frac{{{\mathrm{v}}_{\mathrm{B}}}^2+\mathrm{gL}}{{{\mathrm{v}}_{\mathrm{T}}}^2-\mathrm{gL}} = \frac{4}{1}$

or ${{\mathrm{v}}_{\mathrm{B}}}^2+\mathrm{gL} = 4{{\mathrm{v}}_{\mathrm{T}}}^2 -4\mathrm{gL} \mathrm{but} {{\mathrm{v}}_{\mathrm{B}}}^2 = {{\mathrm{v}}^2}_{\mathrm{T}}+4\mathrm{gL}$

$\therefore {{\mathrm{v}}_{\mathrm{T}}}^2+4\mathrm{gL}+\mathrm{gL} = 4{{\mathrm{v}}_{\mathrm{T}}}^2-4\mathrm{gL} \Rightarrow 3{{\mathrm{v}}_{\mathrm{T}}}^2 = 9\mathrm{gL}$

$\therefore {{\mathrm{v}}_{\mathrm{T}}}^2 = 3\times \mathrm{g}\times \mathrm{L} = 3\times 10\times \frac{10}{3} \mathrm{or} {\mathrm{v}}_{\mathrm{T}}= 10 \mathrm{m}/\sec$