When the tension in a metal wire is `T_(1)`, its length is `I_(1)`. When the tension is `T_(2)`, its length is `I_(2)`. The natural length of wire is

Options

A. a. `(I_(1)+I_(2))/(2)`
B. b. `sqrt(I_(1)I_(2))`
C. c. `(I_(2)T_(2)-I_(2)T_(1))/(T_(2)-T_(1))`
D. d. `(I_(1)T_(2)+I_(2)T_(1))/(T_(1)+T_(2))`

Answer

To find the natural length of the metal wire when given two different tensions and their corresponding lengths, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two tensions \( T_1 \) and \( T_2 \) in a metal wire, which correspond to lengths \( L_1 \) and \( L_2 \) respectively.
- We need to find the natural length \( L \) of the wire.

2. **Using Hooke's Law**:
- According to Hooke's Law, the extension (or change in length) of a wire is directly proportional to the tension applied to it.
- This can be expressed as:
\[ T \propto \Delta L \]
- Where \( \Delta L \) is the change in length from the natural length \( L \).

3. **Expressing Extensions**:
- The extensions for the two tensions can be expressed as:
\[ \Delta L_1 = L_1 - L \]
\[ \Delta L_2 = L_2 - L \]

4. **Setting up the Proportionality**:
- From the proportional relationship, we can write:
\[ \frac{T_1}{T_2} = \frac{L_1 - L}{L_2 - L} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives us:
\[ T_1 (L_2 - L) = T_2 (L_1 - L) \]

6. **Expanding the Equation**:
- Expanding both sides:
\[ T_1 L_2 - T_1 L = T_2 L_1 - T_2 L \]

7. **Rearranging the Equation**:
- Rearranging to isolate \( L \):
\[ T_1 L_2 - T_2 L_1 = T_1 L - T_2 L \]
- Factoring out \( L \) on the right side:
\[ T_1 L_2 - T_2 L_1 = L (T_1 - T_2) \]

8. **Solving for \( L \)**:
- Finally, we can solve for the natural length \( L \):
\[ L = \frac{T_1 L_2 - T_2 L_1}{T_1 - T_2} \]

### Final Answer:
The natural length of the wire is given by:
\[ L = \frac{T_1 L_2 - T_2 L_1}{T_1 - T_2} \]