A stone is tied to one end of a light inexensible string of length l and made to roate on a vertical circle keeping other end of the spring at the centre. If speed of stone at the highest point is `v(v gt sqrt(gl))` then its speed at the lowest point is

Options

A. `sqrt(4gl)`
B. `sqrt(v^(2)+2gl)`
C. `sqrt(v^(2)+gl)`
D. `sqrt(v^(2)+4gl)`

Answer

To solve the problem, we will use the principle of conservation of mechanical energy. The total mechanical energy (kinetic energy + potential energy) at the highest point of the stone's motion will be equal to the total mechanical energy at the lowest point.

### Step-by-Step Solution:

1. **Identify the Points**:
- At the highest point of the vertical circle, the stone has a speed \( v \) and is at a height of \( 2l \) (where \( l \) is the length of the string).
- At the lowest point, the stone will have a speed \( v_1 \) and is at a height of \( 0 \).

2. **Write the Energy Conservation Equation**:
The total mechanical energy at the highest point is the sum of kinetic energy and potential energy:
\[ \text{Total Energy at Highest Point} = \text{K.E.} + \text{P.E.} = \frac{1}{2} m v^2 + mg(2l) \]
At the lowest point, the potential energy is zero (since we take it as the reference level), and the total mechanical energy is:
\[ \text{Total Energy at Lowest Point} = \text{K.E.} + \text{P.E.} = \frac{1}{2} m v_1^2 + 0 \]

3. **Set the Energies Equal**:
By conservation of energy, we can set the total energy at the highest point equal to the total energy at the lowest point:
\[ \frac{1}{2} m v^2 + mg(2l) = \frac{1}{2} m v_1^2 \]

4. **Cancel the Mass (m)**:
Since \( m \) appears in all terms, we can cancel it out:
\[ \frac{1}{2} v^2 + 2gl = \frac{1}{2} v_1^2 \]

5. **Rearrange the Equation**:
Multiply the entire equation by 2 to eliminate the fraction:
\[ v^2 + 4gl = v_1^2 \]

6. **Solve for \( v_1 \)**:
Rearranging gives us:
\[ v_1^2 = v^2 + 4gl \]
Taking the square root of both sides, we find:
\[ v_1 = \sqrt{v^2 + 4gl} \]

### Final Answer:
The speed of the stone at the lowest point is:
\[ v_1 = \sqrt{v^2 + 4gl} \]