A solid of densit `5000 kgm^(-3)` weights 0.5 kgf in air. It is completely immersed in water of density `1000kgm^(-3)`. Calculate the apparent weight of the solid in water.

Answer

To calculate the apparent weight of a solid when it is completely immersed in water, we can follow these steps:

### Step 1: Determine the weight of the solid in air.
The weight of the solid is given as \( 0.5 \, \text{kgf} \).

### Step 2: Calculate the volume of the solid.
We know the density of the solid and its mass. The formula for density (\( \rho \)) is:
\[ \rho = \frac{\text{mass}}{\text{volume}} \]
Rearranging this gives us:
\[ \text{volume} = \frac{\text{mass}}{\text{density}} \]
Given:
- Mass of the solid, \( m = 0.5 \, \text{kg} \)
- Density of the solid, \( \rho_s = 5000 \, \text{kg/m}^3 \)

Substituting the values:
\[ \text{volume} = \frac{0.5 \, \text{kg}}{5000 \, \text{kg/m}^3} = \frac{0.5}{5000} = 1 \times 10^{-4} \, \text{m}^3 \]

### Step 3: Calculate the buoyant force acting on the solid.
The buoyant force (\( F_b \)) is equal to the weight of the fluid displaced by the solid. The formula for buoyant force is:
\[ F_b = \rho_l \cdot V \cdot g \]
Where:
- \( \rho_l \) is the density of the liquid (water), \( 1000 \, \text{kg/m}^3 \)
- \( V \) is the volume of the solid, \( 1 \times 10^{-4} \, \text{m}^3 \)
- \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \)

Substituting the values:
\[ F_b = 1000 \, \text{kg/m}^3 \cdot 1 \times 10^{-4} \, \text{m}^3 \cdot 9.8 \, \text{m/s}^2 = 0.98 \, \text{N} \]

### Step 4: Convert the buoyant force to kgf.
To convert Newtons to kgf, we use the conversion:
\[ 1 \, \text{kgf} = 9.8 \, \text{N} \]
Thus:
\[ F_b = \frac{0.98 \, \text{N}}{9.8 \, \text{N/kgf}} = 0.1 \, \text{kgf} \]

### Step 5: Calculate the apparent weight of the solid in water.
The apparent weight (\( W_a \)) is given by:
\[ W_a = W - F_b \]
Where:
- \( W \) is the weight of the solid in air, \( 0.5 \, \text{kgf} \)
- \( F_b \) is the buoyant force, \( 0.1 \, \text{kgf} \)

Substituting the values:
\[ W_a = 0.5 \, \text{kgf} - 0.1 \, \text{kgf} = 0.4 \, \text{kgf} \]

### Final Answer:
The apparent weight of the solid in water is \( 0.4 \, \text{kgf} \).

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