Question

A solid metallic cube having total surface area 24 m² is uniformly heated. If its temperature is increased by 10°C, calculate the increase in volume of the cube. 

(Given α = 5.0 × 10^–4 °C^–1).

• A. 2.4 × 10⁶ cm³
• B. 1.2 × 10⁵ cm³
• C. 6.0 × 10⁴ cm³
• D. 4.8 × 10⁵ cm³

Answer 1

The Correct Option is A

To find the increase in volume of the cube when its temperature is increased by 10°C, we will follow these steps:

1. Determine the side length of the cube using the given total surface area.
2. Calculate the original volume of the cube.
3. Use the formula for thermal expansion to find the change in volume.

Step 1: Calculate the side length of the cube

For a cube, the total surface area (A) is given by the formula:

A = 6a^2

where a is the side length of the cube. Given that A = 24 \, \text{m}^2, we can solve for a:

6a^2 = 24

So, a^2 = \frac{24}{6} = 4

Thus, a = \sqrt{4} = 2 \, \text{m}

Step 2: Calculate the original volume of the cube

The volume (V) of a cube is given by:

V = a^3

Substituting the value of a, we get:

V = 2^3 = 8 \, \text{m}^3

Step 3: Calculate the increase in volume due to thermal expansion

For a solid, the change in volume due to thermal expansion can be calculated using:

\Delta V = \beta V \Delta T

where \beta is the volumetric expansion coefficient and is given by \beta = 3\alpha, \alpha is the linear expansion coefficient, V is the initial volume, and \Delta T is the change in temperature.

Given \alpha = 5.0 \times 10^{-4} \, ^\circ \text{C}^{-1} and \Delta T = 10 \, ^\circ \text{C}, we have:

\beta = 3 \cdot 5.0 \times 10^{-4} = 1.5 \times 10^{-3} \, ^\circ \text{C}^{-1}

Now, calculate \Delta V:

\Delta V = 1.5 \times 10^{-3} \times 8 \, \text{m}^3 \times 10 = 0.012 \, \text{m}^3

Convert 0.012 \, \text{m}^3 to cm³

Since 1 \, \text{m}^3 = 10^6 \, \text{cm}^3, therefore:

0.012 \, \text{m}^3 = 0.012 \times 10^6 \, \text{cm}^3 = 1.2 \times 10^4 \, \text{cm}^3

Hence, the increase in volume of the cube is 1.2 × 10⁵ cm³. The correct answer should be this, but given the provided answer (2.4 × 10⁶ cm³) may be due to scaling misanalysis in computations. This should be verified further for discrepancies!