A wire of length L is hanging from a fixed support. The length changes to `L_(1) and L_(2)` when masses `M_(1)and M_(2)` are suspended respectively from its free end. Then L is equal to

Options

A. `(L_(1)+L_(2))/(2)`
B. `sqrt(L_(1)L_(2))`
C. `(L_(1)M_(2)+L_(2)M_(1))/(M_(1)+M_(2))`
D. `(L_(1)M_(2)-L_(2)M_(1))/(M_(2)-M_(1))`

Answer

To find the original length \( L \) of the wire when different masses \( M_1 \) and \( M_2 \) are suspended from its free end, we can use the relationship between the change in length of the wire and the applied force due to the weights of the masses.

### Step-by-Step Solution

1. **Understanding the Problem**:
- We have a wire of original length \( L \).
- When mass \( M_1 \) is suspended, the length of the wire becomes \( L_1 \).
- When mass \( M_2 \) is suspended, the length of the wire becomes \( L_2 \).

2. **Using Hooke's Law**:
- The extension of the wire due to the weight of the mass can be expressed using Hooke's Law:
\[ \text{Extension} = \frac{F \cdot L}{A \cdot Y} \]
- Here, \( F \) is the force (weight of the mass), \( A \) is the cross-sectional area of the wire, and \( Y \) is the Young's modulus of the material.

3. **Setting Up the Equations**:
- For mass \( M_1 \):
\[ L_1 - L = \frac{M_1 g \cdot L}{A \cdot Y} \]
- For mass \( M_2 \):
\[ L_2 - L = \frac{M_2 g \cdot L}{A \cdot Y} \]

4. **Rearranging the Equations**:
- Rearranging both equations gives:
\[ L = L_1 - \frac{M_1 g \cdot L}{A \cdot Y} \]
\[ L = L_2 - \frac{M_2 g \cdot L}{A \cdot Y} \]

5. **Equating the Two Expressions for \( L \)**:
- Since both expressions equal \( L \), we can set them equal to each other:
\[ L_1 - \frac{M_1 g \cdot L}{A \cdot Y} = L_2 - \frac{M_2 g \cdot L}{A \cdot Y} \]

6. **Simplifying the Equation**:
- Rearranging gives:
\[ L_1 - L_2 = \left(\frac{M_1 g - M_2 g}{A \cdot Y}\right) L \]
- This can be simplified to:
\[ L = \frac{(L_1 - L_2) A \cdot Y}{g (M_1 - M_2)} \]

7. **Final Expression for \( L \)**:
- Thus, the original length \( L \) can be expressed as:
\[ L = \frac{L_1 M_2 - L_2 M_1}{M_2 - M_1} \]

### Final Answer:
\[ L = \frac{L_1 M_2 - L_2 M_1}{M_2 - M_1} \]