Question

A rigid body is made of three identical thin rods, each of length L fastened together in the form of letter H. The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of H is horizontal. What is the angular speed of the body when the plane of H is vertical ?

• A. $\sqrt{\frac{g}{L}}$
• B. $\frac{1}{2} \, \sqrt{\frac{g}{L}}$
• C. $\frac{3}{2}\sqrt{\frac{g}{L}}$
• D. $2\sqrt{\frac{g}{L}}$

Answer 1

Answer: (C)

$I=0+\left[\frac{M L^{2}}{12} + M \left(\frac{L}{2}\right)^{2}\right]+\left[0 + M L^{2}\right]=\frac{4}{3}ML^{2}$
( using parallel axis theorem )


$\because \frac{1}{2}I\omega ^{2}=mgh_{c m}$
$\therefore \frac{1}{2}\left(\frac{4}{3} M L^{2}\right)\left(\omega \right)^{2}=3Mg\left(\frac{L}{2}\right)$
$\Rightarrow \omega =\frac{3}{2}\sqrt{\frac{g}{L}}$