The length of a metal wire is `l_(1)` when the tension in it is `T_(1)` and is `l_(2)` when the tension is `T_(2)`. Then natural length of the wire is

Options

A. `(l_(1)+l_(2))/(2)`
B. `sqrt((l_(1)l_(2))`
C. `(l_(1)T_(2)-l_(2)T_(1))/(T_(2)-T_(1))`
D. `(l_(1)T_(2)-l_(2)T_(1))/(T_(1)+T_(2))`

Explanation

To find the natural length of the metal wire when given two different tensions and their corresponding lengths, we can use the relationship defined by Young's modulus. Here’s a step-by-step solution:

### Step 1: Understand the relationship
We know that Young's modulus (Y) is defined as the ratio of stress to strain. Stress is given by \( \frac{T}{A} \) (where T is the tension and A is the cross-sectional area), and strain is given by \( \frac{l - L}{L} \) (where l is the extended length and L is the natural length).

### Step 2: Set up equations for both tensions
For the first tension \( T_1 \) and length \( l_1 \):
\[ Y = \frac{T_1/A}{(l_1 - L)/L} \]
This can be rearranged to:
\[ Y = \frac{T_1 \cdot L}{A \cdot (l_1 - L)} \quad \text{(Equation 1)} \]

For the second tension \( T_2 \) and length \( l_2 \):
\[ Y = \frac{T_2/A}{(l_2 - L)/L} \]
This can be rearranged to:
\[ Y = \frac{T_2 \cdot L}{A \cdot (l_2 - L)} \quad \text{(Equation 2)} \]

### Step 3: Equate the two expressions for Young's modulus
Since both expressions equal Y, we can set them equal to each other:
\[ \frac{T_1 \cdot L}{A \cdot (l_1 - L)} = \frac{T_2 \cdot L}{A \cdot (l_2 - L)} \]
We can cancel A and L (assuming L is not zero) from both sides:
\[ \frac{T_1}{l_1 - L} = \frac{T_2}{l_2 - L} \]

### Step 4: Cross-multiply to solve for L
Cross-multiplying gives:
\[ T_1 \cdot (l_2 - L) = T_2 \cdot (l_1 - L) \]
Expanding both sides:
\[ T_1 \cdot l_2 - T_1 \cdot L = T_2 \cdot l_1 - T_2 \cdot L \]

### Step 5: Rearranging to isolate L
Rearranging the equation to isolate L:
\[ T_1 \cdot l_2 - T_2 \cdot l_1 = T_1 \cdot L - T_2 \cdot L \]
Factoring out L from the right side:
\[ T_1 \cdot l_2 - T_2 \cdot l_1 = L \cdot (T_1 - T_2) \]

### Step 6: Solve for L
Finally, we can solve for L:
\[ L = \frac{T_1 \cdot l_2 - T_2 \cdot l_1}{T_1 - T_2} \]

### Conclusion
The natural length of the wire is given by:
\[ L = \frac{T_1 \cdot l_2 - T_2 \cdot l_1}{T_1 - T_2} \]