Question 82 of 194

A simple pendulum with length $100cm$ and bob of mass $250g$ is executing S.H.M of amplitude $10cm$. The maximum tension in the string is found to be $\dfrac{x}{40}N$. The value of $x$ is _____.

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Answer: 99

From the given figure we can see that
$\sin {\theta }_{0}=\dfrac{A}{l}=\dfrac{10}{100}=\dfrac{1}{10}$
From conservation of energy $\dfrac{m{v}^{2}}{2}=mgl(1-\cos {\theta }_{0})$
Maximum tension occurs at mean position
$T=mg+\dfrac{m{v}^{2}}{l} \Rightarrow T=mg+2mg(1-\cos {\theta }_{0})$
Substituting the values,
$T=mg(1+2(1-\sqrt{1-{\sin }^{2}{\theta }_{0}}) \Rightarrow T=mg(3-2\sqrt{1-\dfrac{1}{100}}) \Rightarrow T=0.25\times 9.8(3-2(1-\dfrac{1}{200})) \Rightarrow T=0.25\times 9.8\times 1.01=2.4745N$
It is given that
$T=\dfrac{x}{40} \Rightarrow x=40\times 2.4745=98.98\approx 99$