A stone of mass $2 \mathrm{~kg}$ tied to a light inextensible string of length $\frac{5}{3} \mathrm{~m}$ is…

A stone of mass $2 \mathrm{~kg}$ tied to a light inextensible string of length $\frac{5}{3} \mathrm{~m}$ is whirling in a circular path in a vertical plane. If the ratio of the maximum tension to the minimum tension in the string is 4 , then the speed of the stone at the highest point of the circle is $\left(g=10 \mathrm{~ms}^{-2}\right)$
  1. $20 \mathrm{~ms}^{-1}$
  2. $10 \sqrt{3} \mathrm{~ms}^{-1}$
  3. $\sqrt{50} \mathrm{~ms}^{-1}$
  4. $10 \mathrm{~ms}^{-1}$

Solution

Tension is maximum at bottom of vertical circle and it is $ T_1=\frac{m v^2}{r}+m g $
Tension is minimum at top and it is $ T_2=\frac{m u^2}{r}-5 m g $ Given, ratio $\frac{T_1}{T_2}=4 \Rightarrow 21 m g=3 \frac{m u^2}{r}$ $ \Rightarrow \quad u^2=7 \mathrm{rg} $ Now, velocity at top, $ v=\sqrt{u^2-4 r g}=\sqrt{7 r g-4 r g}=\sqrt{3 r g}=\sqrt{50} \mathrm{~m} / \mathrm{s} $

Asked in: AP EAMCET 2018 (24 Apr Shift 1)

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