Each side of a box made of metal sheet in cubic shape is 'a' at room temperature 'T', the coefficient of linear expansion of the metal sheet is ' $\alpha^{\prime}$. The metal sheet is heated uniformly, by a small temperature $\Delta T ,$ so that its new temperature is $T +\Delta T$. Calculate the increase in the volume of the metal box.
- $3 a ^{3} \alpha \Delta T$
- $4 a ^{3} \alpha \Delta T$
- $4 \pi a ^{3} \alpha \Delta T$
- $\frac{4}{3} \pi a ^{3} \alpha \Delta T$
The Correct Option is A
Solution and Explanation
To understand the problem, we need to calculate the increase in volume of a metal box when it is heated by a small temperature $\Delta T$. The box is in the shape of a cube with side length 'a' at the initial temperature 'T'. The coefficient of linear expansion of the metal sheet is given as $\alpha'$.
First, we need to know how thermal expansion works in a three-dimensional object like a cube:
- The linear expansion in solids is given by the formula: $L = L_0 (1 + \alpha' \Delta T)$, where $L_0$ is the original length, and $\alpha'$ is the coefficient of linear expansion.
- For a cube, each side will expand, and its new side length will be: $a_{\text{new}} = a (1 + \alpha' \Delta T)$
Now, the original volume of the cube is $V_0 = a^3$ . The volume of the cube after heating, $V_{\text{new}}$, will be:
$V_{\text{new}} = (a_{\text{new}})^3 = (a(1 + \alpha' \Delta T))^3$
Expanding this expression using binomial theorem, we get:
- $V_{\text{new}} = a^3 (1 + \alpha' \Delta T)^3$
- Assuming $\alpha' \Delta T$ is small, we approximate using binomial expansion up to the first order:
$(1 + \alpha' \Delta T)^3 \approx 1 + 3 \alpha' \Delta T$
Next, substituting back to find the new volume:
$V_{\text{new}} = a^3 (1 + 3 \alpha' \Delta T)$
The increase in volume $\Delta V$ is given by:
$\Delta V = V_{\text{new}} - V_0 = a^3(1 + 3 \alpha' \Delta T) - a^3$
This simplifies to:
$\Delta V = 3 a^3 \alpha' \Delta T$
Thus, the correct answer is $3 a^3 \alpha' \Delta T$, which corresponds to the first option.
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Concepts Used:
Thermal Expansion
Thermal expansion is the tendency of matter to change its shape, area, and volume in response to a change in temperature. Temperature is a monotonic function of the average molecular kinetic energy of a substance.
The expansion of the solid material is taken to be the linear expansion coefficient, as the expansion takes place in terms of height, thickness and length. The gaseous and liquid expansion takes the volume expansion coefficient. Normally, if the material is fluid, we can explain the changes in terms of volume change.
The bonding force among the molecules and atoms differs from material to material. These characteristics of the compounds and elements are known as the expansion coefficient.
