Question

A string is passing through a ring of mass $m$ as shown. The ring rotates in a horizontal plane of radius $r$ with a constant angular velocity. What should be the maximum speed of rotation so that the string does not get distorted?

• A. $rg$
• B. $\sqrt{rg}$
• C. $\frac{r}{g}$
• D. $\frac{g}{r}$

Answer 1

The correct option is B $\sqrt{rg}$

FBD of ring of mass $m$:


Since there is only one string, tension is same throughout.
Equating forces along horizontal direction :
$T\sin {30}^{\circ }+T\sin {60}^{\circ }=\frac{m{v}^2}{r}$....(1)
Equating forces along vertical direction :
$T\cos {30}^{\circ }+T\cos {60}^{\circ }=mg$ ....(2)

Dividing (1) by (2):
$\frac{T\sin {30}^{\circ }+T\sin {60}^{\circ }}{T\cos {30}^{\circ }+T\cos {60}^{\circ }}=$ $\frac{v^2}{rg}$
$\Rightarrow \frac{v^2}{rg}=1$
$\Rightarrow v=\sqrt{rg}$