Question

A hoop rolls on a horizontal ground without Slipping with linear speed $v$. Speed of a particle $P$ on the circumference of the hoop at angle $\theta$ is

• A. $2v\sin \left(\theta /2\right)$
• B. $v\sin \theta$
• C. $2v\cos \left(\theta /2\right)$
• D. $v\cos \theta$

Answer 1

The correct option is A $2v\sin \left(\theta /2\right)$


Since it is rolling without slipping, then net velocity will be resultant of linear speed of centre of mass $\left(v\right)$ and velocity because of pure rotation about centre $\left(rw=v\right)$.
Angle between both the velocities will be $\left(180°-\theta °\right)$ from geometry.
$$v_p=\sqrt{v^2+v^2+2vv\cos (180-\theta )}$$ $$\Rightarrow v_p=\sqrt{2v^2[1-cos\theta ]}$$ $$=2v\sin \left(\theta /2\right)$$