Question

A stone of mass 1 kg tied to a light in extensible string of length $L=\frac{10}{3}m$ is whirling in a circular path of radius L in a vertical plane. 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 $10\frac{m}{s^2}.$ The speed of stone at the highest point of the circle is

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

Answer 1

The correct option is D $10\frac{m}{s}$

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.
$\therefore T_B=\frac{m{v}_B^2}{L}+mg$
and $T_T=\frac{m{v}_T^2}{L}-mg$
$\therefore \frac{T_B}{T_T}=\frac{\frac{m{v}_B^2}{L}+mg}{\frac{m{v}_T^2}{L}-mg}=\frac{4}{1}$
Or $\frac{V_B^2+gL}{V_L^2-gL}=\frac{4}{1}$
Or $V_B^2+gL=4v_T^2-4gL$
but from energy conservation $V_B^2=V_T^2+4gL$
$\therefore V_T^2+4gL+gL=4V_T^2-4gL$
$\Rightarrow 3V_T^2=9gL$
$\therefore V_T^2=3\times g\times L=3\times 10\times \frac{10}{3}$
Or $V_T=10\frac{m}{s}$