question_answer

Two solid bodies rotate about stationary mutually perpendicular intersecting axes with constant angular velocities \[{{\omega }_{1}}\] and \[{{\omega }_{2}}\]. What is the magnitude of angular velocity of one with respect to the other?

A) \[{{\omega }_{1}}-{{\omega }_{2}}\]

B) \[\sqrt{\omega _{1}^{2}+\omega _{2}^{2}}\]

C) \[\sqrt{\omega _{1}^{2}-\omega _{2}^{2}}\]

D) \[{{\omega }_{1}}+{{\omega }_{2}}\]

Correct Answer: B

Solution :

Let is consider any two particles of the body. Let a particle of the first body rotate in the x-y plane in a circle of radius \[{{r}_{1}}.\]

Similarly, let a particle of body 2 move in the y-z plane in a circle of radius \[{{r}_{2}}\]as shown in the figure.

Then \[{{\vec{\omega }}_{1}}={{\omega }_{1}}\hat{k}\] and \[{{\vec{\omega }}_{2}}={{\omega }_{2}}\hat{i}\]

\[{{\vec{\omega }}_{21}}={{\vec{\omega }}_{2}}-{{\vec{\omega }}_{1}}={{\omega }_{2}}\hat{i}-{{\omega }_{1}}\hat{k}\]

\[\therefore \,\,\,\left| \,{{{\vec{\omega }}}_{21}}\, \right|=\sqrt{\omega _{1}^{2}+\omega _{2}^{2}}\]