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Particle A an...

Question

Particle A and B moving in co-planar circular paths centered at O.They are rotating in the same sense.Time periods of rotation of A and B around O are $T_{A} a n d T_{B}$ , respectively, with $T_{B} > T_{A}$ . Time required for B to make one rotation around O relative to a is :

A

T_{B} - T_{A}

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B

T_{B} + T_{A}

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C

\frac{T_{A} T_{B}}{T_{A} - T_{B}}

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D

\frac{T_{B} T_{A}}{T_{B} - T_{A}}

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Solution

The correct option is D $\frac{T_{A} T_{B}}{T_{A} - T_{B}}$
Time period of rotation of $A$ and $B$ are $T_{A}$ and $T_{B}$ respectively.

Let time required for $B$ to make one rotation relative to $A$ be $T_{B A}$

If the angular velocities of $A$ and $B$ are $ω_{A}$ and $ω_{B}$ respectively. then,

$ω_{A} = \frac{2 π}{T_{A}}$

and

$ω_{B} = \frac{2 π}{T_{B}}$

$ω_{B A} = ω_{B} - ω_{A}$

$⟹ \frac{2 π}{T_{B A}} = \frac{2 π}{T_{B}} - \frac{2 π}{T_{A}}$

$⟹ \frac{1}{T_{B A}} = \frac{1}{T_{B}} = \frac{1}{T_{A}}$

$⟹ \frac{1}{T_{B A}} = \frac{T_{A} - T_{B}}{T_{A} T_{B}}$

$⟹ T_{B A} = \frac{T_{A} T_{B}}{T_{A} - T_{B}}$

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