# Normal Modes of a Triangle Shaped Molecule

1. Nov 8, 2013

### NewNuNeutrino

1. The problem statement, all variables and given/known data
A molecule consists of three identical atoms located at the vertices of a 45 degree right triangle.
Each pair of atoms interacts by an eﬀective spring potential, with all spring constants equal
to k. Consider only planar motion of this molecule. What are 6 normal modes and what do they represent?

The real stickler of this problem is the set up.

2. Relevant equations

See section 10.9.1 in this:
http://www-physics.ucsd.edu/students/courses/fall2010/physics110a/LECTURES/CH10.pdf

3. The attempt at a solution
Basically I started off and did the same thing until they took some approximations starting with 10.109.

I don't understand how they got 10.110 in the linked file. I think I understand how they got 10.109 and 10.111, since y2,y1 and x3,x1 are zero if you set your coordinates correctly and don't allow the molecule to spin too much.

The rest of problem is a doable and understandable, I'm just getting stuck on this one part.

Thank you!

2. Nov 9, 2013

### TSny

Hello, NewNuNeutrino.

3. Nov 9, 2013

### NewNuNeutrino

I know where he gets d$_{12}$=$\sqrt{(-a+x_{3}-x_{2})^{2}+(a+y_{3}-y_{2})^{2}}$
but I don't know what approximation he uses to get to
d$_{12}$=$\sqrt{2}a-\frac{1}{\sqrt{2}}(x_{3}-x_{2})+\frac{1}{\sqrt{2}}(y_{3}-y_{2})$

It seems like it should be simple, but I can't figure it out.

Thank you!

4. Nov 9, 2013

### TSny

Try to get the expression into a form that you can use $\sqrt{1+\epsilon} \approx 1+\epsilon/2$

You might let $Δx = x_2-x_3$ and $Δy = y_3-y_2$ (note the order of subscripts). Then you can write the initial expression as $$d_{23}=\sqrt{(a+Δx)^2+(a+Δy)^2}$$
Also note that to first order accuracy, $(a+Δx)^2 \approx a^2+2aΔx$, etc.