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Buoyant force is a fascinating concept from physics, especially relevant when we discuss objects submerged in fluids like liquids or gases. When an object is submerged in a fluid, it experiences an upward force called buoyant force. This force is what makes hot-air balloons rise and ships float. The origin of the buoyant force is a fundamental principle known as Archimedes' Principle.
Archimedes' Principle states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid that the object displaces. Therefore, the greater the volume of the object submerged, the more fluid it displaces, and the stronger the buoyant force it experiences.
In mathematical terms, if an object with a volume of \(V\) is submerged in a fluid with density \(\rho\), then the buoyant force \(F_b\) can be calculated as:

• \(F_b = V \rho g\)

where \(g\) is the acceleration due to gravity. This upward force acts in the opposite direction to the gravitational force that pulls the object down, reducing the apparent weight of the object.

Specific gravity is a dimensionless quantity that is used to compare the density of a substance to the density of a reference substance, usually water. For example, if the specific gravity of a liquid is 2, it means the liquid is twice as dense as water.
In simple terms, specific gravity helps to determine how heavy or light a fluid is compared to water. This is crucial when studying buoyancy because it affects how much fluid an object displaces and therefore the buoyant force the object experiences.
Specific gravity \(SG\) is defined mathematically as:

• \(SG = \frac{\text{Density of the substance}}{\text{Density of reference substance (water)}}\)

Since it is a ratio, specific gravity has no units. It is simply a helpful tool to understand and analyze the behavior of substances when they interact with water or other fluids.

When an object is submerged in a fluid, its apparent weight changes due to the effect of the buoyant force. The apparent weight is what we perceive or measure when an object is in a fluid compared to when it is in air.
In a fluid, the weight of the object is reduced by the buoyant force. If an object has a true weight \(w\) when outside any fluid and apparent weight \(w_1\) and \(w_2\) when inside two different fluids with specific gravities \(\rho_1\) and \(\rho_2\), the reduction in its weight due to the buoyant force can be described as follows:

• In fluid 1: \(w = w_1 + V \rho_1 g\)
• In fluid 2: \(w = w_2 + V \rho_2 g\)

Through the step-by-step derivation, we know that the relation among these weights is given by:

• \(w = \frac{w_1 \rho_2 - w_2 \rho_1}{\rho_2 - \rho_1}\)

This equation highlights the interplay between the density differences of the fluids and the apparent weight change of the object. Understanding this relationship helps us predict how objects behave in different fluids, which is essential in engineering, fluid mechanics, and various real-world applications.