Question

A thin uniform rod of length l and 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. $\frac{1}{6}\frac{l\omega }{g}$
• B. $\frac{1}{2}\frac{l^2{\omega }^2}{g}$
• C. $\frac{1}{6}\frac{l^2{\omega }^2}{g}$
• D. $\frac{1}{3}\frac{l^2{\omega }^2}{g}$

Answer 1

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

At lowest height

$E=\frac{1}{2}I{\omega }^2$

At maximum weight $\omega =0$

$E=mgh+0$ (his height from lower point)

By conservation of energy

$\frac{1}{2}I{\omega }^2=mghh=\frac{I{\omega }^2}{2mg}h=\frac{m{l}^2{\omega }^2}{6mg}=\frac{l^2{\omega }^2}{6g}$