To tackle this problem, we need to understand how temperature affects the volume of both the iron block and the mercury in which it floats. The key concepts here are the coefficients of volume expansion for both materials and how they relate to buoyancy.

Understanding Buoyancy and Volume Expansion

When an object floats in a fluid, the fraction of its volume that is submerged is determined by the principle of buoyancy, which states that the weight of the fluid displaced by the submerged part of the object equals the weight of the object itself. In this case, we have an iron block floating in mercury.

Volume Expansion Coefficients

The coefficient of volume expansion (denoted as β) indicates how much a material's volume changes with temperature. For our scenario:

• β_Fe is the coefficient of volume expansion for iron.
• β_Hg is the coefficient of volume expansion for mercury.

As temperature increases, the volume of both the iron block and the mercury will change according to their respective coefficients of volume expansion.

Calculating the Submerged Volume Fractions

Let’s denote:

• V₀ as the initial volume of the iron block at 0°C.
• V_Fe as the volume of the iron block at 60°C, which can be expressed as:

V_Fe = V₀ (1 + 60β_Fe)

Similarly, for mercury, we denote the initial volume as V_Hg and the volume at 60°C as:

V_Hg = V₀ (1 + 60β_Hg)

Submerged Volume Fractions

At 0°C, the fraction of the iron block submerged in mercury is:

K₁ = V_{iron submerged} / V_Hg

At 60°C, the fraction submerged becomes:

K₂ = V_{iron submerged at 60°C} / V_{Hg at 60°C}

Finding the Ratio K1/K2

To find the ratio K₁/K₂, we can substitute the expressions for the volumes at different temperatures:

K₁ = V₀ / V_Hg

K₂ = V₀ (1 + 60β_Fe) / V₀ (1 + 60β_Hg)

Now, simplifying the ratio gives us:

K₁ / K₂ = (V₀ / V_Hg) / [(V₀ (1 + 60β_Fe)) / (V₀ (1 + 60β_Hg))]

This simplifies to:

K₁ / K₂ = (1 + 60β_Hg) / (1 + 60β_Fe)

Final Expression

Thus, the ratio of the submerged fractions at the two temperatures can be expressed as:

K₁ / K₂ = 1 + (60β_Fe) / (1 + 60β_Hg)

In conclusion, the ratio K₁/K₂ is indeed given by:

K₁ / K₂ = 1 + (60β_Fe) / (1 + 60β_Hg)

This relationship illustrates how the differing expansion rates of iron and mercury affect the buoyancy of the iron block as the temperature changes. Understanding these principles helps us grasp the fundamental concepts of thermodynamics and fluid mechanics.

To tackle the problem of how the fraction of a block of iron submerged in mercury changes with temperature, we need to consider the principles of buoyancy and thermal expansion. The key here is to understand how both the iron and the mercury respond to temperature changes.

Understanding Buoyancy

When an object is placed in a fluid, it experiences an upward force known as buoyancy, which is equal to the weight of the fluid displaced by the submerged part of the object. For our scenario, the fraction of the iron block submerged in mercury can be expressed using Archimedes' principle.

Initial Conditions

At 0 degrees Celsius, let’s denote the fraction of the volume of the iron block that is submerged in mercury as K1. The weight of the displaced mercury equals the weight of the iron block. Mathematically, this can be expressed as:

• Weight of iron block = Volume of iron block × Density of iron
• Weight of displaced mercury = Volume submerged × Density of mercury

Effect of Temperature on Volume

As the temperature increases to 60 degrees Celsius, both the iron and the mercury will expand. The volume expansion can be described by the formula:

ΔV = V0 × β × ΔT

Where:

• ΔV = change in volume
• V0 = original volume
• β = coefficient of volume expansion
• ΔT = change in temperature

Calculating New Volumes

For the iron block, the new volume at 60 degrees Celsius becomes:

V_iron(60) = V_iron(0) × (1 + 60 × β_Fe)

For mercury, the new volume is:

V_mercury(60) = V_mercury(0) × (1 + 60 × β_Hg)

Finding the Ratio K1/K2

At the new temperature, we denote the fraction of the iron block submerged in mercury as K2. The relationship between the submerged volumes at the two temperatures can be expressed as:

K1 × V_mercury(0) × ρ_Hg = V_iron(0) × ρ_Fe

And for the second condition:

K2 × V_mercury(60) × ρ_Hg = V_iron(60) × ρ_Fe

Setting Up the Ratio

To find the ratio K1/K2, we can divide the two equations:

K1/K2 = (V_iron(0) × ρ_Fe) / (V_mercury(0) × ρ_Hg) × (V_mercury(60) / V_iron(60))

Substituting the expressions for the volumes at 60 degrees Celsius, we get:

K1/K2 = (1 + 60 × β_Fe) / (1 + 60 × β_Hg)

Final Expression

Thus, the ratio of the fractions submerged at the two temperatures is:

K1/K2 = 1 + 60 × β_Fe / (1 + 60 × β_Hg)

This result shows how the different coefficients of volume expansion for iron and mercury affect the buoyancy and, consequently, the fraction of the iron block submerged in mercury at different temperatures.

To tackle the problem of how the fraction of a block of iron submerged in mercury changes with temperature, we need to consider the principles of buoyancy and thermal expansion. The key here is to understand how both the iron and mercury respond to temperature changes and how that affects their respective volumes.

Understanding Buoyancy

When an object is placed in a fluid, it experiences an upward force known as buoyancy, which is equal to the weight of the fluid displaced by the object. The fraction of the object submerged in the fluid can be expressed as:

• K: The fraction of the volume of the object submerged.
• V_submerged: The volume of the object that is submerged.
• V_total: The total volume of the object.

For a block of iron floating in mercury, the buoyant force must equal the weight of the iron block. Hence, we can write:

K1 = V_submerged / V_total = (Weight of iron) / (Weight of displaced mercury)

Volume Expansion with Temperature

As temperature increases, both the iron and mercury will expand. The volume expansion can be described by the formula:

ΔV = V0 * β * ΔT

Where:

• ΔV: Change in volume
• V0: Original volume
• β: Coefficient of volume expansion
• ΔT: Change in temperature

For our scenario, at 0°C, the volume of iron is V0 and at 60°C, its volume becomes:

V_iron(60°C) = V0 * (1 + 60 * β_Fe)

Similarly, for mercury, the volume at 60°C becomes:

V_mercury(60°C) = V_mercury(0°C) * (1 + 60 * β_Hg)

Calculating the Ratios

At 0°C, the fraction submerged is:

K1 = V_iron(0°C) / V_mercury(0°C)

At 60°C, the fraction submerged becomes:

K2 = V_iron(60°C) / V_mercury(60°C)

Substituting the expanded volumes into the equation gives:

K2 = (V0 * (1 + 60 * β_Fe)) / (V_mercury(0°C) * (1 + 60 * β_Hg))

Finding the Ratio K1/K2

Now, we can find the ratio of K1 to K2:

K1/K2 = (V_iron(0°C) / V_mercury(0°C)) / ((V_iron(0°C) * (1 + 60 * β_Fe)) / (V_mercury(0°C) * (1 + 60 * β_Hg)))

This simplifies to:

K1/K2 = (1) / (1 + 60 * β_Fe) / (1 + 60 * β_Hg)

Thus, we arrive at the final ratio:

K1/K2 = (1 + 60 * β_Hg) / (1 + 60 * β_Fe)

Final Result

In conclusion, the ratio of the fractions submerged at different temperatures is:

K1/K2 = 1 + 60 * β_Fe / (1 + 60 * β_Hg)

This relationship illustrates how the thermal expansion of both materials affects their buoyancy in a fluid, providing a clear understanding of the interplay between temperature and volume in this context.