A body is floating in water at 4ºC such that 0.98 of its total volume is immersed in water. If the coefficient of real expansion of water is 3.3 × 10-4/ºC. The temperature at which the body gets immersed completely is?

To determine the temperature at which the body becomes completely immersed in water, we need to analyze the relationship between the buoyancy of the body and the properties of water as it changes with temperature. Let's break this down step by step.

Understanding Buoyancy and Volume Immersion

A body floating in water experiences buoyant force, which is equal to the weight of the water displaced by the submerged part of the body. In this case, 0.98 of the body’s total volume is submerged at 4ºC. This means that the buoyant force is balancing the weight of the body at this temperature.

Real Expansion of Water

The coefficient of real expansion of water is given as 3.3 × 10^-4/ºC. This value indicates how much the volume of water changes with temperature. As water warms, it expands, which affects the density of water and consequently the buoyancy experienced by the body.

Setting Up the Equation

At 4ºC, the density of water is at its maximum (approximately 1 g/cm³). As the temperature increases, the density of water decreases due to thermal expansion. We can express the relationship between the volume of water displaced and the temperature change using the coefficient of real expansion.

Calculating the Change in Volume

Let’s denote:

• V: Total volume of the body
• V_w: Volume of water displaced = 0.98V
• ρ_w: Density of water at 4ºC
• ρ_w': Density of water at the new temperature

As the temperature increases, the volume of water displaced will change. The new volume of water at temperature T can be expressed as:

V_w' = V_w (1 + βΔT), where β is the coefficient of real expansion and ΔT is the change in temperature.

Finding the Temperature for Complete Immersion

For the body to be completely immersed, the volume of water displaced must equal the total volume of the body:

V = V_w' = V_w (1 + βΔT)

Substituting V_w = 0.98V, we have:

V = 0.98V (1 + βΔT)

Dividing both sides by V (assuming V is not zero), we get:

1 = 0.98 (1 + βΔT)

Now, we can solve for ΔT:

1 = 0.98 + 0.98βΔT

0.02 = 0.98βΔT

ΔT = 0.02 / (0.98 × 3.3 × 10^-4)

Calculating this gives:

ΔT ≈ 61.22ºC

Final Temperature Calculation

Since the body starts floating at 4ºC, we add the change in temperature to this initial temperature:

Final Temperature = 4ºC + 61.22ºC ≈ 65.22ºC

Thus, the temperature at which the body becomes completely immersed in water is approximately 65.22ºC. This illustrates how the physical properties of water change with temperature and how these changes affect buoyancy and immersion.

To determine the temperature at which the body becomes completely immersed in water, we need to analyze the relationship between the buoyancy of the body and the properties of water as it changes with temperature. Let's break this down step by step.

Understanding Buoyancy and Immersion

When an object floats in a fluid, the volume of fluid displaced by the object is equal to the weight of the object. In this case, we know that 0.98 of the body’s total volume is currently submerged in water at 4ºC. This means that the buoyant force acting on the body is equal to the weight of the body.

Volume and Density Relationships

The volume of water displaced can be expressed as:

• Volume of water displaced = 0.98 × Volume of body

The weight of the body can be expressed in terms of its density and volume:

• Weight of body = Density of body × Volume of body × g

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

• Density of water × Volume of water displaced × g = Density of body × Volume of body × g

Since g (acceleration due to gravity) cancels out, we can simplify this to:

• Density of water × (0.98 × Volume of body) = Density of body × Volume of body

Calculating the Density of Water

At 4ºC, the density of water is approximately 1000 kg/m³. Therefore, we can express the relationship as:

• 1000 kg/m³ × 0.98 × Volume of body = Density of body × Volume of body

From this, we can derive the density of the body:

• Density of body = 1000 kg/m³ × 0.98 = 980 kg/m³

Effect of Temperature on Water Density

As the temperature of water increases, its density decreases due to thermal expansion. The coefficient of real expansion of water is given as 3.3 × 10⁻⁴/ºC. This means that for every degree Celsius increase in temperature, the volume of water increases, leading to a decrease in density.

Finding the Temperature for Complete Immersion

To find the temperature at which the body is completely submerged, we need to set the density of water equal to the density of the body:

• Density of water at temperature T = Density of body = 980 kg/m³

The density of water at a temperature T can be expressed as:

• Density of water at T = Density of water at 4ºC × (1 - β × (T - 4))

Where β (the coefficient of real expansion) is 3.3 × 10⁻⁴/ºC. Plugging in the values:

• 980 = 1000 × (1 - 3.3 × 10⁻⁴ × (T - 4))

Now, we can solve for T:

• 0.98 = 1 - 3.3 × 10⁻⁴ × (T - 4)
• 3.3 × 10⁻⁴ × (T - 4) = 0.02
• T - 4 = 0.02 / (3.3 × 10⁻⁴)
• T - 4 ≈ 60.61
• T ≈ 64.61ºC

Final Result

Thus, the temperature at which the body becomes completely immersed in water is approximately 64.61ºC. This illustrates how the physical properties of water change with temperature and how these changes affect buoyancy and immersion.