Question

A stone of mass 1 kg tied to a light inextensible string of length 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 m/$s^2$ , the speed of the stone at the highest point of the circle is

• A. 20 m/sec
• B. 10$\sqrt{3}$ m/sec
• C. 5$\sqrt{2}$ m/sec
• D. 10 m/sec

Answer 1

The correct option is D 10 m/sec

Since the maximum tension $T_B$ in the string moving in the vertical circle is at the bottom and minimum tension $T_T$ is at the top.
$T_B$ = $\frac{m{v_B}^2}{L}$ = mg and $T_T$ = $\frac{m{v_T}^2}{L}$ - mg
$\therefore$ $\frac{T_B}{T_T}$ = $\frac{\frac{m{v_B}^2}{L}+mg}{\frac{m{v_T}^2}{L}-mg}$ or $\frac{{v_B}^2+gL}{{v_T}^2+gL}$ = $\frac{4}{1}$
or ${v_B}^2$ + gL = 4${v_T}^2$ - 4gL but ${v_B}^2$ = ${v_T}^2$ + 4gL
$\therefore$ ${v_T}^2$ + 4gL +gL $\Rightarrow$ 3 ${v_T}^2$ = 3gL
$\therefore$ ${v_T}^2$ = 3 $\times$ g $\times$ L = 3 $\times$ 10 $\times$ $\frac{10}{3}$ or $v_T$ = 10 m/sec