Subsection 10.2.4 Relation between \(\alpha, \,\beta,\) and \(\gamma\)
Consider a solid cube of side \(l_{o}\) at temperature \(T_{o}\) and \(l\) at temperature \(T.\) Hence from linear expansion, we have
\begin{equation*}
l=l_{o}\left[1+\alpha \left(T-T_{o}\right)\right]
\end{equation*}
The volume of a cube is therefore,
\begin{equation*}
\text{or,}\quad V=l^{3} =\left[l_{o}(1+\alpha \Delta T)\right]^{3} = l_{o}^{3}\left[1+\alpha \Delta T)\right]^{3}
\end{equation*}
\begin{equation*}
= V_{o}\left[1+3\alpha \Delta T+3\alpha^{2}\Delta T^{2}+\alpha^{3}\Delta T^{3})\right]
\end{equation*}
Now since \(\alpha \) is very small, \(\alpha^{2}\) is very very small. Hence the quantities \(\alpha^{2}\Delta T^{2}\) and \(\alpha^{3}\Delta T^{3}\) are neglected.
\begin{equation}
\therefore\quad V=V_{o} \left[1+3\alpha\Delta T\right] \tag{10.2.1}
\end{equation}
\begin{equation}
\text{but,}\quad V=V_{o} \left[1+\gamma\Delta T\right] \tag{10.2.2}
\end{equation}
\begin{equation}
\gamma = 3\alpha \tag{10.2.3}
\end{equation}
Similarly the area of a cube is given by -
\begin{equation*}
\text{or,}\quad A=l^{2} =\left[l_{o}(1+\alpha \Delta T)\right]^{2}
\end{equation*}
\begin{equation*}
= l_{o}^{2}\left[1+\alpha \Delta T\right]^{2} = A_{o}\left[1+2\alpha \Delta T+\alpha^{2}\Delta T^{2}\right]
\end{equation*}
\begin{equation}
\therefore\quad A=A_{o} \left[1+2\alpha\Delta T\right] \tag{10.2.4}
\end{equation}
\([\because \alpha \) is very small.
\begin{equation}
\text{but,}\quad A=A_{o} \left[1+\beta\Delta T\right] \tag{10.2.5}
\end{equation}
\begin{equation}
\beta = 2\alpha \tag{10.2.6}
\end{equation}
\begin{equation*}
\therefore \alpha:\beta:\gamma= 1:2:3
\end{equation*}
