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## Homework Statement

Consider the CO2 molecule as a system made of a central mass [tex]m_2[/tex] connected by equal springs of spring constant k to two masses [tex]m_1[/tex] and [tex]m_3[/tex]

a) set up and solve the equations for the two normal modes in which the masses oscillate along the line joining their centers (the x-axis).

b) putting [tex]m_1[/tex] = [tex]m_3[/tex] = 16 units and [tex]m_2[/tex] = 12 units, what would be the ratio of the frequencies of the two normal modes?

## The Attempt at a Solution

I made x_1,x_2 and x_3 the displacement to the right from equilibrium position.

a)

[tex]m_3\frac{d^2x_3}{dt^2} = -k(x_3-x_2)[/tex]

[tex]m_2\frac{d^2x_2}{dt^2} = -k(x_2-x_1) -k(x_2-x_3)[/tex]

[tex]m_1\frac{d^2x_1}{dt^2} = -k(x_1-x_2)[/tex]

Assuming [tex]x_1 = C_1cos(wt), x_2=C_2cos(wt)[/tex] etc...

solving for the double time derivatives and plugging them in above gives:

[tex]w^2C_1m_1=k(x_1-x_2) [/tex]

[tex]w^2C_2m_2=k(x_2-x_1) + k(x_2-x_3) [/tex]

[tex]w^2C_3m_3=k(x_3-x_2)[/tex]

Where do I go from here?