Question

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

• A. 3 : 1
• B. 5 : 1
• C. 7 : 1
• D. 9 : 1

Answer 1

The correct option is C 7 : 1

Let $v_1$ and $v_2$ be the velocities at the lowest and the highest points.
$\therefore \frac{v_1}{v_2}=\frac{\sqrt{3}}{1}$
Conserving energy between the lowest and highest points gives
${v_1}^2={v_2}^2+2gh$
$\Rightarrow 3{v_2}^2={v_2}^2+2gh$
$\Rightarrow {v_2}^2=gh$
$\Rightarrow {v_1}^2=3gh$
$h$ is distance between the lowest and the highest points which is $2r$.

$h=2r$

Tensions at the lowest and highest point
$T_1=\frac{m{v_1}^2}{r}+mg$

$T_2=\frac{m{v_2}^2}{r}-mg$

$\Rightarrow \frac{T_1}{T_2}=\frac{{v_1}^2/r+g}{{v_2}^2/r-g}$

$\Rightarrow \frac{T_1}{T_2}=\frac{6gr/r+g}{2gr/r-g}$

$\Rightarrow \frac{T_1}{T_2}=7/1$