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If two particles are moving on same circle or different circles in same plane with different angular velocities $ω_{1}$ and $ω_{2}$ and different time period $T_{1}$ and $T_{2}$ in same sense, then the time taken by particle 2 to complete one revolution w.r.t particle 1 is

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$T = \frac{T_{1} T_{2}}{T_{2} - T_{1}}$

$T = \frac{T_{1} + T_{2}}{2}$

$T = \frac{T_{1} T_{2}}{T_{1} + T_{2}}$

$T_{2} - T_{1}$

answer is A.

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Detailed Solution

$θ_{1} = θ_{2} + 2 π \Rightarrow ω_{1} t = ω_{2} t + 2 π$

$(ω_{1} - ω_{2}) t = 2 π \Rightarrow (\frac{2 π}{T_{1}} - \frac{2 π}{T_{2}}) t = 2 π$

$t = \frac{T_{1} T_{2}}{T_{2} - T_{1}}$

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