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The maximum value attained by the tension in the string of a swinging pendulum is four times the minimum value it attains. There is no slack in the string. The angular amplitude of the pendulum is
$90^{\circ}$
$60^{\circ}$
$45^{\circ}$
$30^{\circ}$
Solution
(b)
For a pendulum with angular amplitude $\theta$
$\cos \theta=\frac{l-h}{l} \Rightarrow h=l(1-\cos \theta)$
When pendulum is released at mean position. If its speed is $v$, then by energy conservation, we have
$\frac{1}{2} m v^2=m g h \Rightarrow v^2=2 g l(1-\cos \theta)$
Now, maximum tension occurs at mean position and it is given by
$T_{\max }=m g+\frac{m v^2}{l}=m g+m \cdot 2 g(1-\cos \theta)$
And minimum tension occurs at extreme position, its value is
$T_{\min }=m g \cos \theta$
$\text { Given, } T_{\max }=4 T_{\min }$
$\Rightarrow m g+2 m g(1-\cos \theta)=4 m g \cos \theta$
$\Rightarrow 3 m g=6 m g \cos \theta$
$\Rightarrow \cos \theta=\frac{1}{2} \Rightarrow \theta=60^{\circ}$
