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_2`, 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

Options

A. 1
B. 3
C. 5
D. `1//5`

Explanation

To solve the problem, we need to analyze the motion of the two bobs and the conditions for them to complete a full vertical circle.

### Step-by-Step Solution:

1. **Understanding the Minimum Velocity for Circular Motion**:
- For a bob to complete a full vertical circle, the minimum velocity at the topmost point is given by:
\[ v_{\text{min}} = \sqrt{gL} \]
- At the bottom of the circle, the minimum velocity required is:
\[ v_{\text{min}} = \sqrt{5gL} \]
- Here, \(L\) is the length of the string.

2. **Initial Conditions for the First Bob**:
- The first bob (mass \(m\), length \(l_1\)) is given a minimum velocity to complete the circle. At the bottom, its velocity is:
\[ v_1 = \sqrt{5g l_1} \]
- At the top of the circle, its velocity will be:
\[ v_{\text{top}} = \sqrt{g l_1} \]

3. **Collision with the Second Bob**:
- The second bob (mass \(m\), length \(l_2\)) is initially at rest. When the first bob collides elastically with the second bob at the top of its path, the velocities will exchange due to the elastic collision of two equal masses.
- After the collision, the first bob will have a velocity of \(0\) (as it transfers its momentum), and the second bob will have a velocity of:
\[ v_2 = \sqrt{g l_1} \]

4. **Minimum Velocity for the Second Bob**:
- For the second bob to complete a full vertical circle, after the collision, it must have a minimum velocity at the top given by:
\[ v_{\text{min}} = \sqrt{g l_2} \]
- Setting the velocity of the second bob after the collision equal to the minimum required for it to complete the circle gives us:
\[ \sqrt{g l_1} = \sqrt{5g l_2} \]

5. **Solving for the Ratio**:
- Squaring both sides to eliminate the square roots:
\[ g l_1 = 5g l_2 \]
- Dividing both sides by \(g\) (which is non-zero):
\[ l_1 = 5 l_2 \]
- Thus, the ratio of the lengths of the strings is:
\[ \frac{l_1}{l_2} = 5 \]

### Final Answer:
The ratio \(\frac{l_1}{l_2}\) is \(5\).

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