Answer to Question #101422 in Classical Mechanics for Ehsan

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