Hey,(adsbygoogle = window.adsbygoogle || []).push({});

I've been trying to solve this question from Goldstein's Classical Mechanics.

The picture I have of the question is from a later edition and the hint was removed from the question, the hint was let

η_{3}=ζ_{3}

η_{1}=[itex]\frac{ζ_{1}+ζ_{5}}{\sqrt{2}}[/itex]

η_{5}=[itex]\frac{ζ_{1}-ζ_{5}}{\sqrt{2}}[/itex]

What I have done is first let each particle be represented by a displacement x1...x5,

Then wrote out T = 1/2m([itex]x^{2}_{1}+x^{2}_{3}+x^{2}_{5}[/itex]) + 1/2M([itex]x^{2}_{2}+x^{2}_{4}[/itex])

and V = k/2 *( [itex] x_{i}-x_{j}-b[/itex] ) i = 2..5, j = 1..4 i≠j

so V = k/2 *( [itex]x_{2}-x_{1}-b[/itex] ) + k/2(....) up to i = 5 j = 4

then Since η = x - dx the system is at equilibrium when

b = dx_{2}- dx_{1}= dx_{3}- dx_{2}= .... up to i = 5 j = 4

then V = 1/2k (η_{2}- η_{1}) + ... up to i = 5 j = 4

Then I subbed in the hints it provided and also as one of the hints says treat the normal co-ords of 2 and 4 as symetric I let η_{2}=ζ_{2}=-η_{4}

Some stuff cancelled and I ended up with

V = k/2 *( [itex]ζ^{2}_{1}+ζ^{2}_{2}+ζ^{2}_{5}-2\sqrt{2}ζ_{2}ζ_{5}[/itex] )

I turned it into a matrix which was (this is where I start stuffing up I think)

...............1.....0......0 (sorry had to use the ... to make the matrix look kind of like a matrix)

V = k/2.....0.....4....sqrt2

...............0..-sqrt2..1

Then since there were only varibles of 1, 2 and 5 I turned T into

..............m 0 0

T = 1/2...0 M 0

..............0 0 m

Then did the usual thing for eigenvalues |V-ω^{2}T|=0

One was pretty ugly, one was sqrt(k/m), and the last one I had trouble finding because it was a mess of a cubic.

I decided to put the question at the bottom so the add didn't squish it,

http://img193.imageshack.us/img193/7747/asdasddh.jpg [Broken]

Is what I did alright? The part I'm not confident about at all is when i turn V into a matrix, and the book they also drop the 1/2 for both V and T which I didn't really understand why.

Thanks in advanced for any help.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Coupled Oscillator

**Physics Forums | Science Articles, Homework Help, Discussion**