Question

A stone of mass $1kg$ tied to a light inextensible string of length $L=\frac{10}{3}m$ is whirling in a circular path of the radius $L$ in the vertical plane. If the ratio of the maximum tension to the minimum tension in the string is $4$ . What is the speed of stone (in $m{s}^{-1}$ ) at the highest point of the circle? (Taking $g=10m{s}^{-2}$ )

Answer 1

Answer: 10

$mg+T_A=\frac{m{v}_A^2}{L}$
$T_B-mg=\frac{m{v}_B^2}{L}$
$\frac{T_B}{T_A}=\frac{\left(\frac{m{v}_B^2}{L}+mg\right)}{\left(\frac{m{v}_A^2}{L}-mg\right)}=4$
$4v_A^2-v_B^2=5gL...(i)$
By energy conservation,
$\frac{1}{2}m{v}_B^2=\frac{1}{2}m{v}_A^2+mg{h}_A$
$v_B^2=v_A^2+4gL...\left(ii\right)$
From equation (i) and (ii)
$v_A^2=3gL=3\times 10\times \frac{10}{3}$
$v_A=10m{s}^{-1}$