Question

A thin uniform rod of length $l$ is hinged at the lower end to the level floor and stands vertically. If the rod is allowed to fall, its upper end will strike the floor with a velocity:

• A. $\sqrt{3gl}$
• B. $\sqrt{5gl}$
• C. $\sqrt{gl}$
• D. $\sqrt{2gl}$

Answer 1

The correct option is A $\sqrt{3gl}$

When the rod is standing vertically, its center of mass is $\frac{l}{2}$ distance above the ground. So it has a potential energy. When it falls to the ground, its potential energy is converted into kinetic energy.

So using law of conservation of energy. we have
$mg\frac{l}{2}=\frac{1}{2}I{\omega }^2\Rightarrow mg\frac{l}{2}=\frac{1}{2}\left(\frac{1}{3}m{l}^2\right){\left(\frac{v}{l}\right)}^2\Rightarrow v=\sqrt{3gl}$