question_answer

Two points A and B on a disc have velocities \[{{v}_{1}}\]& \[{{v}_{2}}\] at some moment. Their directing make angles \[60{}^\circ \] and \[30{}^\circ \] respectively with the lnie 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) \[\frac{{{v}_{2}}}{d}\]

Correct Answer: D

Solution :

[D]

For rigid body body separation between two point remains same.

\[{{v}_{1}}\cos 60{}^\circ ={{v}_{2}}\cos 30{}^\circ \]

\[\frac{{{v}_{1}}}{2}=\frac{\sqrt{3}{{v}_{2}}}{2}\Rightarrow {{v}_{1}}=\sqrt{3}\,\,{{v}_{2}}\]

\[{{\omega }_{disc}}=\left| \frac{{{v}_{2}}\sin 30{}^\circ -{{v}_{1}}\sin 60{}^\circ }{d} \right|=\left| \frac{\frac{{{v}_{2}}}{2}-\frac{\sqrt{3}{{v}_{1}}}{2}}{d} \right|\]

\[=\left| \frac{{{v}_{2}}-\sqrt{3}\times \sqrt{3}{{v}_{2}}}{2d} \right|=\frac{2{{v}_{1}}}{2d}=\frac{{{v}_{2}}}{d}\]

\[{{\omega }_{disc}}=\frac{{{v}_{2}}}{d}\]