A stone of mass $1\, kg$ tied to a light inextensible string of length $L = \frac{{10}}{3}m$ 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 in the string is $4$ and if $g$ is taken to be $10m/{\sec ^2}$, the speed of the stone at the highest point of the circle is ....... $m/\sec$
A stone of mass $1\, kg$ tied to a light inextensible string of length $L = \frac{{10}}{3}m$ 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 in the string is $4$ and if $g$ is taken to be $10m/{\sec ^2}$, the speed of the stone at the highest point of the circle is ....... $m/\sec$
$20 $
$10\sqrt 3 $
$5\sqrt 2 $
$10 $
A stone of mass $1\, kg$ tied to a light inextensible string of length $L = \frac{{10}}{3}m$ 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 in the string is $4$ and if $g$ is taken to be $10m/{\sec ^2}$, the speed of the stone at the highest point of the circle is ....... $m/\sec$
Since the maximum tension ${T_B}$ in the string moving in the vertical circle is at the bottom and minimum tension ${T_T}$ is at the top.
${T_B} = \frac{{mv_B^2}}{L} + mg$ and ${T_T} = \frac{{mv_T^2}}{L} - mg$
$\frac{{{T_B}}}{{{T_T}}} = \frac{{\frac{{mv_B^2}}{L} + mg}}{{\frac{{mv_T^2}}{L} - mg}} = \frac{4}{1}$ or $\frac{{v_B^2 + gL}}{{v_T^2 - gL}} = \frac{4}{1}$
or $v_B^2 + gL = 4v_T^2 - 4gL$ but $v_B^2 = v_T^2 + 4gL$
$v_T^2 + 4gL + gL = 4v_T^2 - 4gL$==> $3v_T^2 = 9gL$
$v_T^2 = 3 \times g \times L = 3 \times 10 \times \frac{{10}}{3}$ or ${v_T} = 10\,m/\sec $
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