Question

A circular rope of weight $W$ and radius $r=\frac{R}{2}$ is resting on a smooth sphere of radius $R$. Find the tension in rope.

• A. $W/\left(\pi \sqrt{6}\right)$
• B. $W/\left(\pi \sqrt{12}\right)$
• C. $W/\left(5\pi \sqrt{12}\right)$
• D. $W/\left(\pi \sqrt{3}\right)$

Answer 1

The correct option is B $W/\left(\pi \sqrt{12}\right)$

Taking a small segment of the rope with an angle $d\theta$ in the circle of radius $r$. Angle between plane of rope and line joining center of sphere to edge of rope is ${60}^0$.

$Nsin\left({60}^0\right)=\frac{mgd\theta }{2\pi }$

$Ncos\left({60}^0\right)=2Tsin\left(\frac{d\theta }{2}\right)\approx Td\theta$

Dividing the two equations,

$T=\frac{mg}{2\pi \sqrt{3}}=\frac{W}{\pi \sqrt{12}}$