Question

Two points $A$ and $B$ on a disc have velocities $v_{1}$ and $v_{2}$, respectively, at some moment. Their directions make angles $60^{\circ}$ and $30^{\circ}$, respectively, with the line of separation as shown in figure. The angular velocity of disc is:

• A. $\frac{\sqrt{3} v_{1}}{d}$
• B. $\frac{v_{2}}{\sqrt{3} d}$
• C. $\frac{v_{2}-v_{1}}{d}$
• D. $\frac{v_{2}}{d}$

Answer 1

Answer: (D)

For rigid body, separation between two points remains same.
$v_{1} \cos 60^{\circ}=v_{2} \cos 30^{\circ}$
$\frac{v_{1}}{2}=\frac{\sqrt{3} v_{2}}{2} \Rightarrow v_{1}=\sqrt{3} v_{2}$

$\omega_{ disc }\left|\frac{v_{2} \sin 30^{\circ}-v_{1} \sin 60^{\circ}}{d}\right|=\left|\frac{\frac{v_{2}}{2}-\frac{\sqrt{3} v_{1}}{2}}{d}\right|$
$=\left|\frac{1 v_{2}-\sqrt{3} \times \sqrt{3} v_{2}}{2 d}\right|=\frac{2 v_{2}}{2 d}=\frac{v_{2}}{d} \Rightarrow \omega_{ disc }=\frac{v_{2}}{d}$