A solid whose volume does not change with temperature floats in a liquid. For two different t₁ and t₂ of liquid, fraction f₁ and f₂ of volume of solid remain submerged in the liquid. The coefficient of volume expansion is equal to

To understand the relationship between the volume expansion of a solid and its buoyancy in a liquid, we need to delve into the concepts of volume expansion and buoyancy. When a solid is placed in a liquid, it experiences buoyant forces that depend on the density of both the solid and the liquid. The fraction of the solid submerged in the liquid can change with temperature due to the thermal expansion of the liquid. Let's break this down step by step.

Key Concepts

First, let's clarify some important terms:

• Volume Expansion: This refers to the increase in volume of a substance when its temperature rises. For liquids, this is quantified by the coefficient of volume expansion (β).
• Buoyancy: This is the upward force exerted by a fluid that opposes the weight of an object immersed in it. The amount of an object submerged in a fluid is determined by the balance of forces acting on it.

Understanding the Problem

In your scenario, we have a solid that does not change its volume with temperature, meaning its coefficient of volume expansion is zero (α = 0). However, the liquid does expand with temperature, affecting its density and, consequently, the buoyancy experienced by the solid.

Let’s denote:

• V_s = Volume of the solid
• ρ_s = Density of the solid
• ρ_l(t) = Density of the liquid at temperature t
• f_1 = Fraction of the solid submerged in the liquid at temperature t_1
• f_2 = Fraction of the solid submerged in the liquid at temperature t_2

Buoyancy and Submersion

The buoyant force acting on the solid can be expressed as:

F_b = ρ_l(t) * V_s * g

Where g is the acceleration due to gravity. The weight of the solid is:

F_g = ρ_s * V_s * g

For the solid to float, the buoyant force must equal the weight of the solid:

ρ_l(t) * V_s * g = ρ_s * V_s * g

From this, we can derive the condition for floating:

ρ_l(t) = ρ_s

Temperature Effects

As the temperature of the liquid changes from t_1 to t_2, the density of the liquid will also change. This can be expressed using the coefficient of volume expansion:

ρ_l(t) = ρ_l(t_0) * (1 - β * (t - t_0))

Where ρ_l(t_0) is the density at a reference temperature t_0. Thus, we can express the fractions submerged at different temperatures as:

f_1 = ρ_s / ρ_l(t_1)

f_2 = ρ_s / ρ_l(t_2)

Finding the Coefficient of Volume Expansion

To find the coefficient of volume expansion (β) in relation to the fractions submerged, we can set up the following relationship:

f_1 = ρ_s / [ρ_l(t_0) * (1 - β * (t_1 - t_0))]

f_2 = ρ_s / [ρ_l(t_0) * (1 - β * (t_2 - t_0))]

By rearranging these equations, we can express β in terms of f_1 and f_2:

β = (ρ_l(t_0) * (f_1 - f_2)) / (ρ_s * (t_2 - t_1))

This equation shows how the coefficient of volume expansion relates to the changes in the fraction of the solid submerged as the temperature of the liquid changes.

Conclusion

In summary, the coefficient of volume expansion of the liquid can be determined by observing how the fraction of the solid submerged changes with temperature. This relationship highlights the interplay between thermal expansion and buoyancy, illustrating fundamental principles of fluid mechanics and thermodynamics.