Question

Two particles A and B are moving on circular tracks of radii 3 m and 4 m at a speed of 4 m/s and 3 m/s respectively. Angular velocity (in rad/s) of A with respect to B at the instant shown in the figure is (Write the answer until two digits after the decimal point.)

Answer 1

$\tan \theta =\frac{4}{3}\Rightarrow \theta ={53}^{\circ }$
The relative velocity of A wrt B is as shown in the figure.
The component of A's relative horizontal velocity perpendicular to the line joining AB is $3\sin \theta$.
The component of A's relative vertical velocity perpendicular to the line joining AB is $4\cos \theta$.
So, the component of A's relative velocity perpendicular to the line joining AB is $3\sin \theta +4\cos \theta$.
Therefore, the angular velocity of A as seen by B is
$\frac{3\sin \theta +4\cos \theta }{AB}=\frac{3\sin \theta +4\cos \theta }{5}$
$=\frac{3sin\theta +4cos\theta }{5}$
$=\frac{3\times \frac{4}{5}+4\times \frac{3}{5}}{5}=\frac{12+12}{25}=\frac{24}{25}$
$\omega =0.96$