Energy of a Hydrogen Flouride molecule in normal mode vibration

[Dixanadu]

Homework Statement

Hi everyone! first post here :)

Basically, the question is as follows:
Consider a hydrogen fluoride 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.

Homework 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 don't have to worry about phase shift)
Restoring force F by Hooke's law = -kx

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 - I am 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 x_H and that of the fluoride atom as x_F then:

E = 1/2 kx² = 1/2 k(x_H-x_F)²

So...here's the confusion now...the signs of the displacements x_H and x_F...what are they? @_@ and is the equation I am using even the right one?

Thanks a lot guys!

 

[haruspex]

Hi Dixanadu, welcome to the forum.

Basically, the question is as follows:
Consider a hydrogen fluoride molecule (atomic mass of H is 1g/mole and of F is 19 g/mole).

Pedantry point: I believe it's "fluoride".

Total energy E = 1/2 kx²
Shouldn't it be ... the sum of 1/2 mv² and 1/2 kx²?

Yes.

secondly - I am a bit confused as to how I'm meant to work with the displacements - since each atom displaces in the opposite direction

If we consider the displacement of the hydrogen atom as x_H and that of the fluoride atom as x_F then:

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.