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## Homework Statement

Two harmonic oscillators A and B , of mass m and spring constants k

_{A}and k

_{B}are coupled together by a spring of spring constant k

_{C}.Find the normal frequencies ω' and ω'' and describe the normal modes of oscillation if (k

_{C})

^{2}= k

_{A}k

_{B})

## Homework Equations

## The Attempt at a Solution

The system I have drawn looks like this

wall|---k

_{A}---m---k

_{c}---m---k

_{b}---|wall

with axes X

_{A}and X

_{B}at the equilibrium position of the left mass and right masses, the differential equations for the motion are:

mx¨

_{A}+k

_{A}x

_{A}+k

_{C}(x

_{A}-x

_{B})=0

mx¨

_{B}+k

_{B}x

_{B}+k

_{C}(x

_{B}-x

_{A})=0

After rearranging and ω

_{i}=√(k

_{i}/m)

x¨

_{A}+x

_{A}[(ω

_{A})

^{2}+(ω

_{C})

^{2}]-x

_{B}(ω

_{C})

^{2}= 0

and x¨

_{B}+x

_{B}[(ω

_{B})

^{2})+(ω

_{C})

^{2}]-x

_{A}(ω

_{C})

^{2}=0

(I cant find anywhere on the site how to make a proper x double dot, anyone know?)

displacement of each mass with have the form:

x

_{A}=Acos(ωt) and x

_{B}=Bcos(ωt)

plugging into the differential equations and solving for A/B on each and setting them equal to eachother:

ω

_{C}

^{2}/(ω

_{A}

^{2}+ω

_{C}

^{2}-ω

^{2})=(ω

_{b}

^{2}+ω

_{C}

^{2}-ω

^{2})/ω

_{C}

^{2}

and here is where I get stuck. I have not been able to get the quadratic for ω

^{2}into a form I can work with, so I think I have done something wrong, or there is a better way to go about this problem