1. Tardigrade
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  3. Physics
  4. A simple pendulum, consisting of a small ball of mass m attached to a massless string hanging vertically from the ceiling, is oscillating with an amplitude such that T max =2 T min where T max and T min are the maximum and minimum tension in the string respectively. The value of maximum tension T max in the string is

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Q. A simple pendulum, consisting of a small ball of mass $m$ attached to a massless string hanging vertically from the ceiling, is oscillating with an amplitude such that $T_{\max }=2 T_{\min }$ where $T_{\max }$ and $T_{\min }$ are the maximum and minimum tension in the string respectively. The value of maximum tension $T_{\max }$ in the string is

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Solution:

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$T_{\min }=m g \cos \theta$
$T_{\max }=m g+\frac{m v^{2}}{l}=m g+\frac{m v^{2}}{l}$
$\frac{1}{2} m v^{2}=m g l(1-\cos \theta)$
$T_{\max }=2 T_{\min }$
$m g+\frac{m v^{2}}{l}=2 m g \cos \theta$
$\therefore \cos \theta=\frac{3}{4}$
$v^{2}=2 g\left(1-\frac{3}{4}\right)=\frac{g l}{2}$
$T_{\max }=m g+\frac{m}{l} \times \frac{g l}{2}=\frac{3 m g}{2}$