# Energy of a Hydrogen Flouride molecule in normal mode vibration

1. Nov 18, 2012

1. The problem statement, all variables and given/known data
Hi everyone! first post here :)

Basically, the question is as follows:
Consider a hydrogen flouride molecule (atomic mass of H is 1g/mole and of F is 19 g/mole).
1. Write the energy of the system in terms of the displacements of both atoms.

There are other questions but if i can get this one I think the others will be fine.

2. Relevant equations
This is a coupled oscillator problem, so we're gona model it as two masses with a spring in between, that has a stiffness k. the equations are:
Total energy E = 1/2 kx2
General solution to the coupled oscillator differential equation: x = A cos(ωt) (we dont have to worry about phase shift)
Restoring force F by Hooke's law = -kx

3. The attempt at a solution

What I'm having trouble with is 2 things:

1st of all - my professor has written in his reference notes that the total energy E of another similar problem is 1/2 kx2. Shouldn't it be 1/2 kA2; the sum of 1/2 mv2 and 1/2 kx2?

secondly - Im a bit confused as to how I'm meant to work with the displacements - since each atom displaces in the opposite direction (I don't think there is any normal mode other than this one) how do i construct the problem? Anyway, here's what I got for a solution but I have no way of telling if its right since its an assessed assignment:

If we consider the displacement of the hydrogen atom as xH and that of the fluoride atom as xF then:

E = 1/2 kx2 = 1/2 k(xH-xF)2

So...here's the confusion now...the signs of the displacements xH and xF...what are they? @_@ and is the equation im using even the right one?

Thanks a lot guys!

2. Nov 18, 2012

### haruspex

Hi Dixanadu, welcome to the forum.
Pedantry point: I believe it's "fluoride".
Yes.
You know that the mass centre of the system won't change. That gives you a relationship between the two displacements. The extension of the 'spring' will be their sum.