A model of a CO₂ molecules is made of two spheres, each of mass m₁, representing oxygen atoms and a sphere of mass m₂, in between, representing the carbon atom. The spheres are interconnected by springs of force constant k. Such a system can vibrate freely in two the ways shown by arrows.

In the first case, the two oxygen atoms vibrate with the same frequency with the carbon atom remaining motionless
$\therefore {\mathrm{\omega }}_1=\sqrt{\frac{\mathrm{k}}{{\mathrm{m}}_1}}$
Let x₁ be the instantaneous leftward displacement of the right hand oxygen atom, x₂ (rightward) of the carbon atom and x₃ (leftward) of the left oxygen atom. Since the system is free from external force, its centre of mass remains stationary.
$\therefore \mathrm{O}=\frac{{\mathrm{m}}_1\left({\mathrm{\ell }}_0-{\mathrm{x}}_1\right)+{\mathrm{m}}_2{\mathrm{x}}_2-{\mathrm{m}}_1\left({\mathrm{\ell }}_0+{\mathrm{x}}_3\right)}{{\mathrm{m}}_1+{\mathrm{m}}_2+{\mathrm{m}}_3}$
${\mathcal{l}}_0=$natural length of each spring.
$\begin{array}{l}\mathrm{or} {\mathrm{m}}_1{\mathrm{x}}_1={\mathrm{m}}_2{\mathrm{x}}_2-{\mathrm{m}}_1{\mathrm{x}}_3\\ \mathrm{or} {\mathrm{m}}_1\left({\mathrm{x}}_1+{\mathrm{x}}_3\right)={\mathrm{m}}_2{\mathrm{x}}_2\end{array}$
Let us consider motion of m₂.
Force on carbon atom m₂ to the right $=-\mathrm{k}\left({\mathrm{x}}_1+{\mathrm{x}}_2\right)-\mathrm{k}\left({\mathrm{x}}_2+{\mathrm{x}}_3\right)$
$$\begin{gathered} ={\mathrm{m}}_2\frac{{\mathrm{d}}^2{\mathrm{x}}_2}{{\mathrm{dt}}^2} \mathrm{or} \\ \begin{array}{l}\frac{{\mathrm{d}}^2{\mathrm{x}}_2}{{\mathrm{dt}}^2}=-\frac{\mathrm{k}}{{\mathrm{m}}_2}\left({\mathrm{x}}_1+{\mathrm{x}}_3+2{\mathrm{x}}_2\right)=-\frac{\mathrm{k}}{{\mathrm{m}}_2}\left(\frac{{\mathrm{m}}_2{\mathrm{x}}_2}{{\mathrm{m}}_1}+2{\mathrm{x}}_2\right)\\ =-\frac{\mathrm{k}}{{\mathrm{m}}_1{\mathrm{m}}_2}\left({\mathrm{m}}_2+2{\mathrm{m}}_1{\mathrm{x}}_2\right)\end{array} \end{gathered}$$
Hence motion is simple harmonic and its $\mathrm{\omega }$ is given by
${\mathrm{\omega }}_2^2=\mathrm{k}\left(\frac{1}{{\mathrm{m}}_1}+\frac{2}{{\mathrm{m}}_2}\right)\text{ or }{\mathrm{\omega }}_2=\sqrt{\mathrm{k}\left(\frac{1}{{\mathrm{m}}_1}+\frac{2}{{\mathrm{m}}_2}\right)}$