Question

A bob of mass $m$, suspended by a string of length $l_1$ is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass $m$ suspended by a string of length $l_1$, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $l_1/l_2$ is

• A. $1$
• B. $3$
• C. $5$
• D. $1/5$

Answer 1

The correct option is C $5$

Velocity of the first bob at $A=\sqrt{5g{l}_1}$
Velocity of the first bob at $B=\sqrt{g{l}_1}$
At point $B$ is collides elastically with another bob of same mass $m$ suspended by a string of length $l_1$ as shown in figure.
When two bodies of equal masses undergoes an elastic collision, their velocities are interchanged.
$\therefore$ Velocity of the second bob at $B=\sqrt{g{l}_1}$
But to complete the vertical circle, the velocity of the second bob at $B=\sqrt{5g{l}_2}$
$\therefore \sqrt{g{l}_1}=\sqrt{5g{l}_2}$ or $\frac{l_1}{l_2}=5$.