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Two spheres of equal masses are attached to a string of length $2 m$ as shown. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant rate. If $T_{1}$ is the tension in the string between $P & Q$ and $T_{2}$ is the tension in the string between $P & O$ then $T_{1} / T_{2}$ equals_______. $P$ is connected to the mid point of string.

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$\frac{2}{3}$

$1$

$\frac{1}{2}$

$\frac{1}{3}$

answer is C.

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Detailed Solution

$T_{1} = 2 m ω^{2}, T_{2} = T_{1} + m ω^{2}$

$T_{2} = 2 m ω^{2} + m ω^{2} = 3 m ω^{2}$

$\frac{T_{1}}{T_{2}} = \frac{2 m ω^{2}}{3 m ω^{2}} = \frac{2}{3}$

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