A solid metallic cube having total surface area 24 ${\mathrm{m}}^2$ is uniformly heated. If its temperature is increased by $10°\mathrm{C}$, calculate the increase in volume of the cube (Given : $\mathrm{\alpha }=5.0\times {10}^{-4}{{}^{\circ }\mathrm{C}}^{-1}$ )

To determine the increase in volume of a metallic cube when heated, we follow these steps:

Step 1: Calculate the Side Length of the Cube

Given that the total surface area A of the cube is 24 m², we can use the formula for the total surface area of a cube:

A = 6a²

Where a is the length of one side of the cube. Rearranging this equation:

a² = A / 6

Substituting the given value of the surface area:

a² = 24 / 6 = 4

Now, solving for a:

a = √4 = 2 meters

Step 2: Calculate the Initial Volume of the Cube

The initial volume V₀ of the cube can be calculated using the formula:

V₀ = a³

Substitute the value of a = 2 meters:

V₀ = (2)³ = 8 m³

Step 3: Calculate the Coefficient of Volumetric Expansion

The coefficient of volumetric expansion γ is related to the coefficient of linear expansion α by the formula:

γ = 3α

Given that α = 5.0 × 10⁻⁴ °C⁻¹, we can calculate γ as:

γ = 3 × (5.0 × 10⁻⁴) = 1.5 × 10⁻³ °C⁻¹

Step 4: Calculate the Increase in Volume

The increase in volume ΔV can be calculated using the formula:

ΔV = V₀ × γ × ΔT

Where ΔT = 10°C is the change in temperature. Substituting the values:

ΔV = 8 m³ × (1.5 × 10⁻³) × 10

Now, calculate the increase in volume:

ΔV = 8 × 1.5 × 10⁻³ × 10 = 0.12 m³

Step 5: Convert the Increase in Volume to Cubic Centimeters

Since the problem may require the answer in cubic centimeters (cm³), we need to convert the volume from cubic meters to cubic centimeters. The conversion factor is:

1 m³ = 1 × 10⁶ cm³

Thus, we can convert the increase in volume:

ΔV = 0.12 m³ × 1 × 10⁶ cm³/m³

ΔV = 0.12 × 10⁶ cm³ = 1.2 × 10⁵ cm³

Final Answer

The increase in volume of the cube is ΔV = 1.2 × 10⁵ cm³.