First slide

Circular motion

Question

A stone of mass 1 kg tied to a light inextensible string of length L = 10/3 m 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

Moderate

A

$10 {ms}^{- 1}$
B

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

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

$20 {ms}^{- 1}$

Solution

$\begin{array}{l} m = 1 kg \Rightarrow L = \frac{10}{3} m \\ \frac{T_{max}}{T_{min}} = 4 . . . . . . . . . (i) \end{array}$
and $V_{L} = \sqrt{V_{H}^{2} + 4 gL}$
Tension at highest point, $T_{min} = \frac{{mv}_{H}^{2}}{K} - mg$
Tension at lowest point,
$T_{max} = {mv}_{L}^{2} / L + mg = \frac{m (v_{H}^{2} + 4 gL)}{L} + mg$
Now Eq. (i) can be written as
$\begin{array}{l} m [\frac{v_{H}^{2} + 4 gL}{L} + g] = 4 \times m (\frac{v_{H}^{2}}{L} - g) \\ \Rightarrow v_{H} = \sqrt{3 gL} = 10 {ms}^{- 1} \end{array}$

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