Question

In the figure shown, the springs are connected to the rod at one end and at the midpoint. The rod is hinged at its lower end. Rotational SHM of the rod (Mass $m$, length $L$) will occur only if

• A. $k>mg/3L$
• B. $k>2mg/3L$
• C. $k>2mg/5L$
• D. $k>0$

Answer 1

Answer: (C)

$k>2mg/5L$

The rod performs SHM if

• the angular displacement of the rod is small (assumed)

• Restoring torque > Disturbing torque

also for a small angular displacement we have
$y=\frac{L}{2}\varphi x=L\varphi$
Applying the condition
Restoring torque > Disturbing torque
$ky\frac{L}{2}cos\varphi +kxLcos\varphi >mg\frac{L}{2}sin\varphi$

taking $\varphi$ to be small...

$ky\frac{L}{2}+kxL>mg\frac{L}{2}\varphi$
$k\frac{L^2}{4}\varphi +k{L}^2\varphi >mg\frac{L}{2}\varphi$
$\frac{5}{4}k{L}^2\varphi >mg\frac{L}{2}\varphi$
$k>\frac{2}{5L}mg$