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 1: Understand the Problem
We have a piece of steel with:
- Weight in air: \( w \)
- Weight when immersed in water: \( w_1 \)
- Weight when immersed in an unknown liquid: \( w_2 \)

### Step 2: Apply Archimedes' Principle
According to Archimedes' principle, the buoyant force acting on the steel when it is immersed in a fluid is equal to the weight of the fluid displaced by the steel.

- When the steel is immersed in water, the buoyant force can be expressed as:
\[ B_w = w - w_1 \]
where \( B_w \) is the buoyant force in water.

- When the steel is immersed in the unknown liquid, the buoyant force can be expressed as:
\[ B_l = w - w_2 \]
where \( B_l \) is the buoyant force in the unknown liquid.

### Step 3: Relate Buoyant Forces to Densities
The buoyant force can also be expressed in terms of the volume of the steel and the density of the liquid:
- For water:
\[ B_w = V \cdot \rho_w \cdot g \]
- For the unknown liquid:
\[ B_l = V \cdot \rho_l \cdot g \]
where:
- \( V \) is the volume of the steel,
- \( \rho_w \) is the density of water,
- \( \rho_l \) is the density of the unknown liquid,
- \( g \) is the acceleration due to gravity.

### Step 4: Set Up the Equations
From the expressions for buoyant forces, we can write:
1. \( w - w_1 = V \cdot \rho_w \cdot g \)
2. \( w - w_2 = V \cdot \rho_l \cdot g \)

### Step 5: Eliminate Volume and Gravity
We can eliminate \( V \) and \( g \) from these equations by dividing them:
\[ \frac{w - w_2}{w - w_1} = \frac{V \cdot \rho_l \cdot g}{V \cdot \rho_w \cdot g} \]
This simplifies to:
\[ \frac{w - w_2}{w - w_1} = \frac{\rho_l}{\rho_w} \]

### Step 6: Find Relative Density
The relative density (specific gravity) of the liquid is defined as:
\[ \text{Relative Density} = \frac{\rho_l}{\rho_w} \]
Thus, we can conclude:
\[ \text{Relative Density} = \frac{w - w_2}{w - w_1} \]

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