A piece of steel has a weight `w` in air, `w_(1)` when completely immersed in water and `w_(2)` when completely immersed in an unknown liquid. The relative density (specific gravity) of liquid is

Options

A. `(w - w_(1))/(w - w_(2))`
B. `(w - w_(2))/(w - w_(1))`
C. `(w_(1) - w_(2))/(w - w_(1))`
D. `(w_(1) - w_(2))/(w - w_(2))`

Answer

To find the relative density (specific gravity) of the unknown liquid, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a piece of steel with a weight \( w \) in air.
- When immersed in water, its weight is \( w_1 \).
- When immersed in an unknown liquid, its weight is \( w_2 \).
- We need to find the relative density of the unknown liquid.

2. **Buoyancy Force in Water**:
- The loss of weight when the steel is immersed in water is given by:
\[ \text{Loss in weight in water} = w - w_1 \]
- This loss in weight is equal to the buoyant force, which can be expressed as:
\[ \text{Buoyant force} = \text{Density of water} \times g \times V \]
- Here, \( V \) is the volume of the steel piece.

3. **Buoyancy Force in Unknown Liquid**:
- The loss of weight when the steel is immersed in the unknown liquid is given by:
\[ \text{Loss in weight in unknown liquid} = w - w_2 \]
- This loss in weight is also equal to the buoyant force in the unknown liquid:
\[ \text{Buoyant force} = \text{Density of liquid} \times g \times V \]

4. **Setting Up the Equations**:
- From the buoyancy in water:
\[ w - w_1 = \rho_{water} \cdot g \cdot V \]
- From the buoyancy in the unknown liquid:
\[ w - w_2 = \rho_{liquid} \cdot g \cdot V \]

5. **Taking the Ratio**:
- Taking the ratio of the two equations:
\[ \frac{w - w_2}{w - w_1} = \frac{\rho_{liquid} \cdot g \cdot V}{\rho_{water} \cdot g \cdot V} \]
- The \( g \) and \( V \) cancel out:
\[ \frac{w - w_2}{w - w_1} = \frac{\rho_{liquid}}{\rho_{water}} \]

6. **Finding Relative Density**:
- The relative density (specific gravity) of the liquid is defined as:
\[ \text{Relative Density} = \frac{\rho_{liquid}}{\rho_{water}} \]
- Therefore, we can express it as:
\[ \text{Relative Density} = \frac{w - w_2}{w - w_1} \]

### Final Answer:
The relative density (specific gravity) of the unknown liquid is given by:
\[ \text{Relative Density} = \frac{w - w_2}{w - w_1} \]