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.

 

[paranormal]

I would approach this problem by first understanding the concept of normal modes and how they apply to a molecule. Normal modes refer to the different ways a molecule can vibrate or move while maintaining its overall shape. In this case, we are dealing with a triangular molecule with three identical atoms and spring interactions between them.

To find the normal modes, we can use the equations provided in the linked file, specifically equations 10.109 and 10.110. These equations describe the motion of the molecule in terms of its coordinates and displacements. By setting up the coordinates correctly and considering only planar motion, we can simplify the equations and solve for the normal modes.

In this case, we would find that there are six normal modes for this triangle shaped molecule. These modes represent the different ways the molecule can vibrate or move. For example, one normal mode could be the symmetric stretching of all three atoms in the same direction, while another mode could be the asymmetric stretching of two atoms in one direction and one atom in the opposite direction.

It is important to note that these normal modes are not independent of each other, as they are all connected through the spring interactions between the atoms. This means that when one normal mode is excited, it can cause a combination of other modes to also be excited.

In conclusion, the six normal modes for this triangle shaped molecule represent the different ways the molecule can vibrate or move while maintaining its overall shape. They are interconnected and can be described using the equations provided in the linked file.