A body is moving in a vertical circle of radius $r$ by a string. If the ratio of maximum to minimum speed is $\sqrt{3} :1$, the ratio of maximum to minimum tensions in the string is

$\frac{V_{max}}{V_{min}}=\frac{\sqrt{3}}{1};\frac{T_{max}}{T_{min}}=\frac{\frac{mV{max}_{max}^2+mg}{r}}{\frac{mV{V}_{min}^2}{r}-mg}$

$\begin{array}{l}{v}_{max}:v_{min}=\sqrt{3}:1\\ v_{max}=\sqrt{3}{v}_{min}\\ v^2=v^2-2gl(1-\cos \theta )\\ v_{min}^2={\left(\sqrt{3}{v}_{min}\right)}^2-2gl(1-\cos \theta )\\ v_{\mathrm{m}}^2=\sqrt{3}{v}_m^2-2gl(1-\cos \theta )\\ \theta ={180}^{\circ }\end{array}$

$\begin{array}{l}2{\mathrm{v}}_{\mathrm{m}}^2=2\mathrm{g}l\left(2\right)\\ {\mathrm{v}}_{\mathrm{m}}^2=2\mathrm{g}l\\ \frac{{\mathrm{v}}_{\mathrm{m}}^2}{2\mathrm{g}}=l\\ {\mathrm{T}}_{max}=\frac{m{v}^2}{\mathrm{L}}+\mathrm{mgcos}\theta \\ =\frac{\mathrm{m}{\left(\sqrt{3}{\mathrm{v}}_{\mathrm{m}}\right)}^2}{\frac{v_{\mathrm{m}}^2}{2\mathrm{g}}}+\mathrm{mg}\\ {\mathrm{T}}_{max}=7\mathrm{mg}\\ {\mathrm{T}}_{min}=\frac{{\mathrm{mvv}}_{\mathrm{m}}^2}{\frac{v_{\mathrm{m}}^2}{2g}}-\mathrm{mg}\\ {\mathrm{T}}_{min}=2\mathrm{gm}-\mathrm{mg}\\ {\mathrm{T}}_{max}:{\mathrm{T}}_{min}=7:1\end{array}$