The maximum and minimum tensions in a string of length 0.1m with a stone 5g tied to it and whirled in a vertical circle are in the ratio 13:1. The velocity of the body at highest point is

Tension at bottom most point is given by ${\mathrm{T}}_{\mathrm{b}}=\mathrm{mg}+\frac{\mathrm{m}{{\mathrm{v}}_{\mathrm{b}}}^2}{\mathrm{l}}$.

Tension at top most point is given by ${\mathrm{T}}_{\mathrm{t}}=\frac{\mathrm{m}{{\mathrm{v}}_{\mathrm{t}}}^2}{\mathrm{l}}-\mathrm{mg}$

And the relation between the velocities at top most point and bottom most point is given by ${{\mathrm{v}}_{\mathrm{b}}}^2-4\mathrm{gl}={{\mathrm{v}}_{\mathrm{t}}}^2$

using this relation, we get $\frac{\mathrm{mg}+\frac{{\mathrm{mv}}_{\mathrm{t}}^2+4\mathrm{gl}}{\mathrm{I}}}{\frac{{\mathrm{mv}}_{\mathrm{t}}^2}{\mathrm{I}}-\mathrm{mg}}=\frac{13}{1}$

$\mathrm{gl}+\mathrm{m}{{\mathrm{v}}_{\mathrm{t}}}^2+4\mathrm{gl}=13(\mathrm{m}{{\mathrm{v}}_{\mathrm{t}}}^2-\mathrm{gl})\Rightarrow 12\mathrm{m}{{\mathrm{v}}_{\mathrm{t}}}^2=18\mathrm{gl}$

$\Rightarrow {\mathrm{v}}_{\mathrm{t}}=\sqrt{\frac{18\mathrm{gl}}{12\mathrm{m}}}=\sqrt{\frac{18(9.8)0.1}{12(0.05)}}=0.7\sqrt{3}{\mathrm{ms}}^{-1}$