An aluminium cube 20 cm on an edge floats on mercury. How much farther will the block sink when the temperature rises from 270 to 320 K? (The coefficient of volumeexpansion of mercury is 1.8 × 10^-4/Cº.)

To determine how much farther the aluminum cube will sink when the temperature of mercury rises from 270 K to 320 K, we need to consider the effects of thermal expansion on both the mercury and the cube. Let's break this down step by step.

Understanding the Problem

The aluminum cube has a volume that can be calculated since it is a cube with each edge measuring 20 cm. The volume (V) of the cube is given by the formula:

• V = side³ = 20 cm × 20 cm × 20 cm = 8000 cm³

Next, we need to understand how the volume of mercury changes with temperature. The coefficient of volume expansion (β) for mercury is given as 1.8 × 10^-4 /°C. This means that for every degree Celsius increase in temperature, the volume of mercury expands by this fraction of its original volume.

Calculating the Temperature Change

The temperature change from 270 K to 320 K is:

• ΔT = 320 K - 270 K = 50 K

Since the temperature change in Kelvin is equivalent to the change in degrees Celsius, we can use this value directly in our calculations.

Volume Expansion of Mercury

Now, we can calculate the change in volume of mercury due to this temperature increase:

• ΔV = β × V_initial × ΔT
• ΔV = (1.8 × 10^-4 /°C) × V_initial × 50°C

To find V_initial, we need to know the initial volume of mercury displaced by the aluminum cube. Since the cube is floating, the volume of mercury displaced is equal to the volume of the cube:

• V_initial = 8000 cm³

Now substituting the values:

• ΔV = (1.8 × 10^-4) × (8000 cm³) × (50) = 7.2 cm³

Determining the New Volume of Mercury

The new volume of mercury after the temperature increase will be:

• V_new = V_initial + ΔV = 8000 cm³ + 7.2 cm³ = 8007.2 cm³

Finding the New Height of the Cube in Mercury

Since the cube is still floating, we can determine how much deeper it sinks. The volume of mercury displaced must equal the volume of the cube submerged. The new submerged volume will be equal to the new volume of mercury displaced:

• V_displaced = V_new = 8007.2 cm³

To find the new height submerged, we can use the formula for the volume of a cube:

• V = side² × height

Let’s denote the new height submerged as h. We know the base area of the cube (20 cm × 20 cm = 400 cm²), so:

• 8007.2 cm³ = 400 cm² × h
• h = 8007.2 cm³ / 400 cm² = 20.018 cm

Calculating the Increase in Submersion Depth

The original height of the cube is 20 cm. The new height submerged is approximately 20.018 cm. Therefore, the increase in submersion depth is:

• Increase = h - original height = 20.018 cm - 20 cm = 0.018 cm

In summary, when the temperature of mercury rises from 270 K to 320 K, the aluminum cube will sink approximately 0.018 cm deeper into the mercury. This demonstrates how thermal expansion can affect buoyancy and displacement in fluids.