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Angular Velocity

Two particles...

Question

Two particles $A$ and $B$ are moving on two different concentric circles of radii $r_{A}$ and $r_{B}$ ( $r_{B} > r_{A}$ ) with different velocities $V_{A}$ and $V_{B}$ respectively. Then angular velocity of $B$ as observed by $A$ is given by

A

\frac{V_{B} - V_{A}}{r_{B} - r_{A}}

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B

\frac{V_{A}}{r_{A}}

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C

\frac{V_{B}}{r_{B}}

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D

\frac{V_{B} + V_{A}}{r_{B} + r_{A}}

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Solution

The correct option is A $\frac{V_{B} - V_{A}}{r_{B} - r_{A}}$

Velocity of B w.r.t A perpendicular to the line joining them is $V_{B A_{⊥}}$
Here $V_{B A_{⊥}} = V_{B} - V_{A}$ and
$B A = r_{B} - r_{A}$
$\Rightarrow ω_{B A} = \frac{V_{B A_{⊥}}}{B A} = \frac{V_{B} - V_{A}}{r_{B} - r_{A}}$

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