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Two balls of mass $M = 9 g$ and $m = 3 g$ are attached by threads $A D$ and $O B$ whose combined length is $l = 1 m$ to a vertical axis $O$ (figure) and are set in rotational motion in a horizontal plane about this axis with a constant angular velocity $ω$ . Determine the ratio of the lengths $A O$ and $O B$ for which the tensions in the threads will be the same. Disregard the weight of the threads.

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$O A = 75 c m, O B = 25 c m$

$O A = 25 c m, O B = 50 c m$

$O A = 25 c m, O B = 75 c m$

$O A = 50 c m, O B = 25 c m$

answer is A.

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Detailed Solution

If $l_{1}$ and $l_{2}$ are the lengths of the parts $O B$ and $A O$ respectively, then for same tension in both the threads we can write

$T = m ω^{2} l_{1} - - - (i) a n d T = M ω^{2} l_{2} - - - (i i) ∴ \frac{l_{1}}{l_{2}} = \frac{M}{m} o r \frac{l_{1}}{l_{2}} = \frac{9}{3} = 3 a l s o l_{1} + l_{2} = 1$

From equations, we get

$l_{1} = 0.75 m a n d l_{2} = 0.25 m$

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