Question

What is normal modes of linear triatomic molecule, when is rotating around its center of mass with angular velocity omega?
The middle molecule mass is M
and the other molecules mass is m
and the spring constant is k
normal modes of linear triatomic molecule is soleved in Herbert Goldstein's classical mechanics book, when is not rotating.

Answer 1

As it is rotating with the angular velocity $\omega$

Let the radius of the molecule is r,

And due to rotation of the molecules, spring is getting elongate.

$mx\omega^2=kx$

$\Rightarrow x=\dfrac{mr\omega^2}{k}$

So, both springs will elongate $x=\dfrac{mr\omega^2}{k}$

$\omega=\sqrt{\dfrac{k}{m}}$

new angular velocity $\omega_1=\sqrt{\dfrac{k}{m}(1+\dfrac{2m}{M})}$

now, $\omega_1=\omega\sqrt{(1+\dfrac{2m}{M})}$