Question

Ratio among linear expansion coefficient $\left( \alpha \right)$ , area expansion coefficient $\left( \beta \right)$ , and the volume expansion coefficient $\left( \gamma \right)$ is:
A. $1:2:3$
B. $3:2:1$
C. $4:3:2$
D. All of these

Answer 1

Hint: Linear expansion, area expansion, and volume expansion are the type of thermal expansions. Linear expansion is the expansion (change) in the length of the material due to the external force. Coefficients are the ability of the material to change.

Complete answer:
Linear expansion coefficient: The property of the material characterizes the ability of a plastic (material) to expand under the effect of temperature elevation. It tells you how much the developed part will remain stable in case of temperature variation. It is denoted by $\alpha $. It is mathematically given as,
$\alpha =\dfrac{\Delta L}{{{L}_{0}}\Delta T}$
Area expansion coefficient: The change in the area of the material concerning the temperature is known as the area expansion coefficient. It is the fractional change in the area per degree of the temperature change. It is denoted by $\beta $.
Mathematically,
$\beta =\dfrac{\Delta A}{A\Delta T}$
As we know that $\dfrac{\Delta A}{A}=2\dfrac{\Delta L}{L}$ the value of area expansion coefficient can be given as,
$\beta =2\alpha \quad .......\left( 1 \right)$
Volume expansion coefficient: The change in the volume of the material concerning the temperature is known as the volume expansion coefficient. It is the fractional change in the volume per degree of the temperature change. It is denoted by $\gamma $.
Mathematically,
$\gamma =\dfrac{\Delta V}{V\Delta T}$
As we know that $\dfrac{\Delta V}{V}=3\dfrac{\Delta L}{L}$ the value of the volume expansion coefficient can be given as,
$\gamma =3\alpha \quad ........\left( 2 \right)$
So, from the equations (1), and (2) the ratio of linear expansion coefficient $\left( \alpha \right)$ , area expansion coefficient$\left( \beta \right)$ , and the volume expansion coefficient $\left( \gamma \right)$ will be,
$\begin{align}
  & \alpha :2\alpha :3\alpha \\
 & \therefore 1:2:3 \\
\end{align}$

Thus, the correct option which satisfies the question is Option A.

Note:
Expansion coefficients are the thermal properties of the material which shows the capability of the material to resist the temperature. These are known as thermal expansion coefficients. It is necessary to track these properties of the material for the application of them in different productions.