Normal Modes of a Triangle Shaped Molecule

[NewNuNeutrino]

Homework Statement

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.

Homework Equations

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

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!

 

[TSny]

Hello, NewNuNeutrino.

Which line of equation 10.110 are you asking about?

 

[NewNuNeutrino]

Hello, NewNuNeutrino.

Which line of equation 10.110 are you asking about?

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!

 

[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.