A stone weighs 500 N in air and 200N in water. Determine the volume of stone and its specific gravity.

Buoyancy and Specific Gravity of a Stone

Problem Statement

A stone weighs:

500 N in air
200 N in water

Determine:

The volume of the stone.
The specific gravity of the stone.

Solution

1. Calculate the Buoyant Force

\[ F_B = \text{Weight in air} – \text{Weight in water} \] \[ = 500 – 200 \] \[ = 300 \text{ N} \]

2. Calculate the Volume of the Stone

\[ F_B = \gamma_{\text{water}} V_{\text{displaced water}} \] \[ 300 = 9810 \times V_{\text{displaced water}} \] \[ V_{\text{displaced water}} = \frac{300}{9810} \] \[ = 0.0306 \text{ m}^3 \]

3. Calculate the Specific Weight of the Stone

\[ \gamma_{\text{stone}} = \frac{\text{Weight in air}}{V} \] \[ = \frac{500}{0.0306} \] \[ = 16340 \text{ N/m}^3 \]

4. Calculate the Specific Gravity of the Stone

\[ S = \frac{\gamma_{\text{stone}}}{\gamma_{\text{water}}} \] \[ = \frac{16340}{9810} \] \[ = 1.66 \]

Final Results:

Volume of the stone: 0.0306 m³
Specific gravity of the stone: 1.66

Explanation

1. Understanding the Buoyant Force:
When the stone is submerged in water, it experiences an upward buoyant force equal to the weight of the water displaced. The difference between the weight in air and the weight in water gives the buoyant force.

2. Volume of the Stone:
The buoyant force equals the weight of the displaced water. Using Archimedes’ principle, we calculate the volume of water displaced, which is equal to the volume of the stone.

3. Specific Weight Calculation:
The specific weight of the stone is obtained by dividing its weight in air by its volume.

4. Specific Gravity of the Stone:
The specific gravity is the ratio of the specific weight of the stone to that of water. Since \( S > 1 \), the stone is denser than water and will sink when placed in water.

Physical Meaning

1. Archimedes’ Principle in Action:
This principle states that an object submerged in a fluid experiences a buoyant force equal to the weight of the displaced fluid. This explains why objects feel lighter in water.

2. Importance of Specific Gravity:
The specific gravity of an object determines whether it will sink or float in a fluid. If \( S > 1 \), the object sinks; if \( S < 1 \), it floats.

3. Applications in Engineering:
Specific gravity calculations are crucial in material selection for construction, underwater structures, and fluid mechanics. Engineers use this to determine the buoyancy of different materials.