A body weighs `W_(1)` gf in air and when immersed in a liquid, it weighs `W_(2)` gf, while it weighs `W_(3)` gf on immersing it in water. Find (i) Volume of the body (ii) upthrust due to liquid (iii) relative density of the solid and (iv) relative density of the liqudi.

Answer

To solve the problem step by step, we will use the principles of buoyancy and relative density.

### Given:
- Weight of the body in air: \( W_1 \) gf
- Weight of the body in liquid: \( W_2 \) gf
- Weight of the body in water: \( W_3 \) gf

### Step 1: Find the Volume of the Body
The volume of the body can be calculated using the loss of weight when it is immersed in water. The loss of weight in water is given by:

\[ \text{Loss of weight in water} = W_1 - W_3 \]

Using Archimedes' principle, this loss of weight is equal to the weight of the water displaced, which is related to the volume of the body:

\[ \text{Volume of the body} = \frac{\text{Loss of weight in water}}{\text{Density of water}} \]

Assuming the density of water is \( 1 \) gf/cm³ (since we are using grams and the unit is in gf), we can simplify this to:

\[ \text{Volume of the body} = W_1 - W_3 \quad \text{(in cm}^3\text{)} \]

### Step 2: Find the Upthrust due to the Liquid
The upthrust (or buoyant force) acting on the body when it is immersed in the liquid can be calculated using the loss of weight in the liquid:

\[ \text{Upthrust due to liquid} = W_1 - W_2 \]

### Step 3: Find the Relative Density of the Solid
Relative density (or specific gravity) of the solid can be calculated using the formula:

\[ \text{Relative Density of Solid} = \frac{\text{Weight in air}}{\text{Loss of weight in water}} = \frac{W_1}{W_1 - W_3} \]

### Step 4: Find the Relative Density of the Liquid
Relative density of the liquid can be calculated using the ratio of the loss of weight in the liquid to the loss of weight in water:

\[ \text{Relative Density of Liquid} = \frac{\text{Loss of weight in liquid}}{\text{Loss of weight in water}} = \frac{W_1 - W_2}{W_1 - W_3} \]

### Summary of Results:
1. **Volume of the body**: \( V = W_1 - W_3 \) cm³
2. **Upthrust due to liquid**: \( U = W_1 - W_2 \) gf
3. **Relative Density of Solid**: \( RD_s = \frac{W_1}{W_1 - W_3} \)
4. **Relative Density of Liquid**: \( RD_l = \frac{W_1 - W_2}{W_1 - W_3} \)