A steel wire is suspended vertically from a rigid support. When loaded with a weight in air, it expands by and when the weight is immersed completely in water, the extension is reduced to `L_(w)`. Then relative done of the material of weight is

Options

A. `(L_(a))/(L_(a) - L_(w))`
B. `(L_(w))/(L_(a))`
C. `(L_(a))/(L_(w))`
D. `(L_(w))/(L_(a) - L_(w))`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a steel wire suspended vertically. When a weight is attached to it, the wire extends by \( L_A \) when in air and by \( L_W \) when the weight is submerged in water. We need to find the relative density of the material of the weight.

### Step 2: Write the Expression for Extension in Air
The extension of the wire in air can be expressed using Young's modulus:
\[ L_A = \frac{F L}{Y A} \]
where:
- \( L_A \) = extension in air
- \( F \) = force applied (weight)
- \( L \) = original length of the wire
- \( Y \) = Young's modulus of the material
- \( A \) = cross-sectional area of the wire

### Step 3: Write the Expression for Extension in Water
When the weight is submerged in water, the effective force acting on the wire is reduced due to the buoyant force. The extension in water can be expressed as:
\[ L_W = \frac{(W - F_b) L}{Y A} \]
where:
- \( W \) = weight of the object
- \( F_b \) = buoyant force = \( \rho_{water} \cdot V \cdot g \)
- \( V \) = volume of the object
- \( \rho_{water} \) = density of water

The buoyant force can also be expressed as:
\[ F_b = \frac{\rho_{water}}{\rho_{object}} W \]
Thus, we can write:
\[ L_W = \frac{W - \frac{\rho_{water}}{\rho_{object}} W}{Y A} L \]
This simplifies to:
\[ L_W = \frac{W \left(1 - \frac{\rho_{water}}{\rho_{object}}\right) L}{Y A} \]

### Step 4: Relate the Two Extensions
Now, we can relate the two extensions \( L_A \) and \( L_W \):
\[ \frac{L_A}{L_W} = \frac{F L}{(W \left(1 - \frac{\rho_{water}}{\rho_{object}}\right) L)} \cdot \frac{Y A}{Y A} \]
This simplifies to:
\[ \frac{L_A}{L_W} = \frac{F}{W \left(1 - \frac{\rho_{water}}{\rho_{object}}\right)} \]

### Step 5: Find the Relative Density
The relative density \( \rho_{relative} \) is defined as:
\[ \rho_{relative} = \frac{\rho_{object}}{\rho_{water}} \]
From the previous equation, we can rearrange to find:
\[ \rho_{relative} = \frac{L_A}{L_A - L_W} \]

### Final Expression
Thus, the relative density of the material of the weight is given by:
\[ \rho_{relative} = \frac{L_A}{L_A - L_W} \]