Question

The maximum tension that an inextensible ring of radius $1\text{m}$ and mass density $0.1{\text{kgm}}^{-1}$ can bear is $40\text{N}$. The maximum angular velocity with which it can be rotated in a circular path is

• A. $20\text{rad/s}$
• B. $18\text{rad/s}$
• C. $16\text{rad/s}$
• D. $15\text{rad/s}$

Answer 1

The correct option is A $20\text{rad/s}$

To find tension in the ring, let us take a small arc which subtends $2d\theta$ angle at the centre.


The net force responsible for providing centripetal acceleration for small arc is,

$F_c=2T\sin d\theta$

$\Rightarrow 2T\sin d\theta =\left(dm\right){\omega }^{2}R........\left(1\right)$

$[$Here $dm=\lambda R2d\theta =\lambda \times$ (linear length of arc) $]$

For small angles, $\sin d\theta \approx d\theta$

Therefore, from equation $\left(1\right)$,

$2Td\theta =2\lambda R^2{\omega }^{2}d\theta$

$\Rightarrow T=\lambda R^2{\omega }^2$

$\Rightarrow {\omega }_{max}=\sqrt{\frac{T_{max}}{\lambda }}\times \frac{1}{R}$

$(\because T_{max}=40\text{N},\lambda =0.1{\text{kgm}}^{-1})$

${\omega }_{max}=\sqrt{\frac{40}{0.1}}\times \frac{1}{1}=20\text{rad/s}$

Hence, option $\left(a\right)$ is correct.

Why this question? Concept: For a very small arc subtending angle $2d\theta$ at centre, tension can be assumed constant for the small element $dm$.