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Q.

A stone of mass 1kg tied to a light inextensible string of length L =10 / 3m is whirling in a circular path of radius L in a vertical plane. If the ratio of the maximum tension in the string to the minimum tension is 4 and if g is taken to be 10 ms^-2 , the speed of the stone at the highest point of the circle is

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a

10 ms^-1

b

$5 \sqrt{3} {ms}^{- 1}$

c

$20 {ms}^{- 1}$

d

$10 \sqrt{3} {ms}^{- 1}$

answer is A.

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Detailed Solution

$m = 1 kg \Rightarrow L = \frac{10}{3} m \frac{T_{max}}{T_{min}} = 4 And v_{L} = \sqrt{V_{H}^{2} + 4 gL}$
Tension at highest point, $T_{min} = \frac{{mv}_{H}^{2}}{L} - mg$
Tension at lowest point, $T_{max} = \frac{{mv}_{L}^{2}}{L} + mg = \frac{m (v_{H}^{2} + 4 gL)}{L} + mg$
Now Eq.(i) can be written as

$m [\frac{V_{H}^{2} + 4 gL}{L} + g] = 4 \times m (\frac{v_{H}^{2}}{L} - g)$

$\Rightarrow v_{H} = \sqrt{3 gL} = 10 {ms}^{- 1}$

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