The length of a steel wire is `l_(1)` when the stretching force is `T_(1)` and `l_(2)` when the stretching force is `T_(2)`. The natural length of the wire is

Options

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

Explanation

To find the natural length of the steel wire when subjected to different stretching forces, we can use Hooke's Law, which relates the force applied to the elongation of the wire. Here’s a step-by-step solution:

### Step 1: Understand Hooke's Law
According to Hooke's Law, the force \( T \) applied to a wire is proportional to the elongation \( \Delta x \) of the wire:
\[ T = K \Delta x \]
where \( K \) is the spring constant (stiffness of the wire).

### Step 2: Define Variables
Let:
- \( L \) be the natural length of the wire.
- \( L_1 \) be the length of the wire when the force \( T_1 \) is applied.
- \( L_2 \) be the length of the wire when the force \( T_2 \) is applied.

The elongation in each case can be defined as:
- For force \( T_1 \):
\[ \Delta x_1 = L_1 - L \]
- For force \( T_2 \):
\[ \Delta x_2 = L_2 - L \]

### Step 3: Write Equations for Each Case
Using Hooke's Law for both cases, we can write:
1. For \( T_1 \):
\[ T_1 = K (L_1 - L) \quad \text{(Equation 1)} \]
2. For \( T_2 \):
\[ T_2 = K (L_2 - L) \quad \text{(Equation 2)} \]

### Step 4: Divide the Two Equations
Now, we can divide Equation 1 by Equation 2:
\[ \frac{T_1}{T_2} = \frac{L_1 - L}{L_2 - L} \]

### Step 5: Cross Multiply
Cross-multiplying gives us:
\[ T_1 (L_2 - L) = T_2 (L_1 - L) \]

### Step 6: Expand and Rearrange
Expanding both sides:
\[ T_1 L_2 - T_1 L = T_2 L_1 - T_2 L \]
Rearranging terms yields:
\[ T_1 L_2 - T_2 L_1 = T_1 L - T_2 L \]
Factoring out \( L \) from the right side:
\[ T_1 L_2 - T_2 L_1 = L (T_1 - T_2) \]

### Step 7: Solve for Natural Length \( L \)
Now, we can solve for \( L \):
\[ L = \frac{T_1 L_2 - T_2 L_1}{T_1 - T_2} \]

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