Question

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?
(correct answer + 1, wrong answer - 0.25)

• A. ${\omega }_1-{\omega }_2$
• B. $\sqrt{{\omega }_1^2+{\omega }_2^2}$
• C. $\sqrt{{\omega }_1^2-{\omega }_2^2}$
• D. ${\omega }_1+{\omega }_2$

Answer 1

The correct option is B $\sqrt{{\omega }_1^2+{\omega }_2^2}$

Let us 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 $\overrightarrow{{\omega }_1}={\omega }_1\hat{k}$ and $\overrightarrow{{\omega }_2}={\omega }_2\hat{i}$
$\overrightarrow{{\omega }_{21}}=\overrightarrow{{\omega }_2}-\overrightarrow{{\omega }_1}={\omega }_2\hat{i}-{\omega }_1\hat{k}$
$\therefore \left|{\overrightarrow{\omega }}_{21}\right|=\sqrt{{\omega }_1^2+{\omega }_2^2}$