Question

A thin uniform rod of length l and mass m is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega .$ Its centre of mass rises to a maximum height of

• A. $\frac{1}{3}\frac{l^2{\omega }^2}{g}$
• B. $\frac{1}{6}\frac{l\omega }{g}$
• C. $\frac{1}{2}\frac{l^2{\omega }^2}{g}$
• D. $\frac{1}{6}\frac{l^2{\omega }^2}{g}$

Answer 1

The correct option is D $\frac{1}{6}\frac{l^2{\omega }^2}{g}$

The uniform rod of length l and mass m can swing about an axis passing through the end. When it is given angular velocity $\omega$ it starts rotating in vertical plane. When the centre of mass rises through height h, the kinetic energy decreases.
Using conservation of mechanical energy $△K+△U=0$
$(0-\frac{1}{2}I{\omega }^2)+\left(mgh\right)=0$

$\Rightarrow mgh=\frac{1}{2}\left(m\frac{l^2}{3}\right).{\omega }^2$
$\therefore h=\frac{l^2.{\omega }^2}{6g}$