Question

A rigid horizontal smooth $\operatorname{rod} A B$ of mass $0.75\, kg$ and length $40 \,cm$ can rotate freely about a fixed vertical axis through its midpoint $O$. Two rings each of mass $1\, kg$ initially at rest are placed at a distance of $10\, cm$ from $O$ on either side of the rod. The rod is set in rotation with an angular velocity of $30$ radian per sec and when the rings reach the ends of the rod, the angular velocity in rad/sec is

• A. 5
• B. 10
• C. 15
• D. 20

Answer 1

Answer: (B)

According to the conservation of angular momentum
$I_{1} \omega_{1}=I_{2} \omega_{2}$
$\therefore \quad\left[\frac{M L^{2}}{12}+2 m r^{2}\right] \omega_{1}=\left[\frac{M L^{2}}{12}+2 m\left(\frac{L}{2}\right)^{2}\right] \omega_{2}$
or $\left[\frac{0.75 \times(0.4)^{2}}{12}+2 \times 1 \times(0.1)^{2}\right] \times 30$
$=\left[\frac{0.75 \times(0.4)^{2}}{12}+2 \times 1 \times(0.2)^{2}\right] \times \omega_{2}$
or $\omega_{2}=\frac{(0.01+0.02) 30}{(0.01+0.08)}=\frac{0.03 \times 30}{0.09}=10\, rad / sec$