Question

Two identical dumbbells are formed by welding two small identical particles each of mass $m$ at the ends of massless rod each of length $l$. The dumbbells are moving in free space with equal speed $u$ without rotating towards each other as shown in the figure. Collision between the particles are elastic. Then

• A. Angular velocity of each dumbbell just after first collision is $\frac{u}{l}$.
• B. Angular velocity of each dumbbell just after first collision is $\frac{2u}{l}$.
• C. Speed of centre of mass of any one of the dumbbells just after collision is zero.
• D. Speed of centre of mass of any one of the dumbbells just after collision is $u$.

Answer 1

Answer: (B), (C)

The correct options are
B Angular velocity of each dumbbell just after first collision is $\frac{2u}{l}$.
C Speed of centre of mass of any one of the dumbbells just after collision is zero.

Let the four particles be $A,B,C$ and $D$.

As observed from the figure only particle $B$ and $C$ collide as they are in the line of motion. Since the collision is elastic and the masses are same, they will simply exchange their velocities after collision. But $A$ and $C$ will continue to move in their inital direction with velocity $u$, due to which the will be rotation of $AB$ and $CD$


Angular velocity after first collision, $\omega =\frac{\text{relative velocity}}{\text{distance between the particles}}$
$\omega =\frac{2u}{l}$

Immediately after collision speed of center of mass,
$V_{CM}=\frac{mu-mu}{2m}=0$