Question

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 $T_{\max }=2T_{\min }$ 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

• A. $\frac{3mg}{2}$
• B. $mg$
• C. $\frac{3mg}{4}$
• D. $3mg$

Answer 1

Answer: (A)

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