Question

A simple pendulum of bob mass m is oscillating with an angular amplitude ${\alpha }_m$ (in radian). The maximum tension in the string is

• A. $mg$
• B. $mg{\alpha }_m$
• C. $mg{\alpha }_m^2$
• D. $mg(1+{\alpha }_m^2)$

Answer 1

The correct option is D $mg(1+{\alpha }_m^2)$

The tension is maximum when the bob is in the mean position and is given by $T=mg+\frac{m{v}^2}{l}$ where l is the length of the string.
Now, $v_{max}=|A\omega |=\left(l{\alpha }_m\right)\omega =l{\alpha }_m\sqrt{\frac{g}{l}}$
$\Rightarrow v_{max}^2=l^2{\alpha }_m^2\times \frac{g}{l}=lg{\alpha }_m^2$
$\therefore T_{max}=mg+\frac{m}{l}\times lg{\alpha }_m^2=mg(1+{\alpha }_m^2)$
Hence, the correct choice is (d).