Determine the specific weight and volume of an object that weighs 10N in water and 12N in oil of specific gravity 0.8.

Specific Weight and Volume of an Object

Problem Statement

An object weighs:

10 N in water
12 N in oil (specific gravity = 0.8)

Determine:

The volume of the object.
The specific weight of the object.

Solution

1. Define the Equilibrium Condition

The weight of the object is equal to the sum of its apparent weight in a fluid and the buoyant force exerted by the fluid.

2. Buoyant Force in Water

\[ W – 10 = \gamma_{\text{water}} V \] \[ W – 10 = 9810 V \] \[ (a) \]

3. Buoyant Force in Oil

\[ W – 12 = \gamma_{\text{oil}} V \] Since \( \gamma_{\text{oil}} = 0.8 \times 9810 \), we substitute: \[ W – 12 = 7848 V \] \[ (b) \]

4. Solving for Volume

Subtract equation (b) from equation (a): \[ (W – 10) – (W – 12) = 9810 V – 7848 V \] \[ 2 = 1962 V \] \[ V = \frac{2}{1962} \] \[ V = 0.001019 \text{ m}^3 \]

5. Solving for Weight of Object

Substituting \( V = 0.001019 \) into equation (a): \[ W = 10 + (9810 \times 0.001019) \] \[ W = 19.996 \text{ N} \]

6. Calculate Specific Weight of Object

\[ \gamma_b = \frac{W}{V} \] \[ = \frac{19.996}{0.001019} \] \[ = 19623 \text{ N/m}^3 \]

Final Results:

Volume of the object: 0.001019 m³
Specific weight of the object: 19623 N/m³

Explanation

1. Archimedes’ Principle:
When an object is submerged in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced. The difference between the actual weight and the apparent weight in a fluid gives the buoyant force.

2. Volume of the Object:
The buoyant force is proportional to the volume of displaced fluid. By equating the difference in weights in water and oil, we solve for the volume of the object.

3. Specific Weight Calculation:
The specific weight of the object is determined using its actual weight and volume. Since the object is denser than water (as \( \gamma_b > \gamma_{\text{water}} \)), it sinks.

4. Importance in Fluid Mechanics:
This principle is widely used in designing floating objects, measuring densities, and determining material buoyancy in different fluids.

Physical Meaning

1. Density and Buoyancy:
Objects with densities higher than the fluid they are in will sink, while those with lower densities will float. The depth of submersion depends on the fluid’s density.

2. Industrial Applications:
– Used in ship and submarine design.
– Helps in fluid density measurement.
– Used in oil-water separation techniques.

3. Real-World Examples:
– Metal objects sink in water but may float in mercury.
– Objects that sink in water may float in denser liquids like glycerin.