Question

A stone of mass $1\,kg$ 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 $ms^{- 1}$ ) at the highest point of the circle? (Taking $g=10\,ms^{- 2}$ )

Answer 1

$mg+T_{A}=\frac{m v_{A}^{2}}{L}$
$T_{B}-mg=\frac{m v_{B}^{2}}{L}$
$\frac{T_{B}}{T_{A}}=\frac{\left(\frac{m v_{B}^{2}}{L} + m g\right)}{\left(\frac{m v_{A}^{2}}{L} - m g\right)}=4$
$4v_{A}^{2}-v_{B}^{2}=5gL...\left(\right. i \left.\right)$
By energy conservation,
$\frac{1}{2}mv_{B}^{2}=\frac{1}{2}mv_{A}^{2}+mgh_{A}$
$v_{B}^{2}=v_{A}^{2}+4gL...\left(i i\right)$
From equation (i) and (ii)
$v_{A}^{2}=3gL=3\times 10\times \frac{10}{3}$
$v_{A}=10\,ms^{- 1}$