Question

Two uniform rods of equal lengths but different masses are rigidly joined to form an $L$- shaped body, which is then pivoted about $\text{O}$ as shown in the figure. Find the net torque about point $\text{O}$. Given $M=m\sqrt{3}$

• A. $0$
• B. $mgL\left(\frac{\sqrt{3}}{4}\right)$
• C. $mgl\left(2\sqrt{3}\right)$
• D. $mgL\left(\frac{\sqrt{3}}{2}\right)$

Answer 1

The correct option is A $0$

Assume
$⨀$ direction out of the plane
$⨂$ direction into the plane
Let $L$ be the length of each rod.


About point $\text{O}$, $\tau =r_{\perp }.F$
Because of $m$ mass rod
${\tau }_1=\left(\frac{L}{2}\sin {60}^{\circ }\right).mg=mgL.\frac{\sqrt{3}}{4}⨀$
Because of $M$ mass rod
${\tau }_2=\left(\frac{L}{2}\sin {30}^{\circ }\right).\sqrt{3}mg=mgL.\frac{\sqrt{3}}{4}⨂$

Both torques are in opposite directions.
Net torque $\overrightarrow{\tau }=\overrightarrow{{\tau }_1}+\overrightarrow{{\tau }_2}$
$\overrightarrow{\tau }=mgL\frac{\sqrt{3}}{4}+(-mgL\frac{\sqrt{3}}{4})=0$