Question

Particles $A$ and $B$ move with constant and equal speeds in a circle as shown in Fig. Find the angular velocity of the particle A with respect to B, if the angular velocity of particle $A$ w.r.t $O$ is $\omega$.

Answer 1

Angular velocity of $A$ with respect O is

${\omega }_{AO}=\frac{\left(v_{A}O\right)\perp }{r_{AB}}=\frac{2v}{2r}=\omega$

Now ,${\omega }_{AB}=\frac{v_{AB}\perp }{r_{AB}}{\sqrt{v}}_{AB}=2v$

Since $v_{AB}isperpendicularto$r_{AB}$

$\Rightarrow \left(v_{A}B\right)\perp =v_{AB}=2v;r_{AB}=2r$

$\Rightarrow {\omega }_{AB}=\frac{\left(v_{AB}\right)\perp }{r_{AB}}=\frac{2v}{2r}=\omega$