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physicsjock
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Hey,
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
η3=ζ3
η1=[itex]\frac{ζ_{1}+ζ_{5}}{\sqrt{2}}[/itex]
η5=[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 = dx2 - dx1 = dx3 - dx2 = ... up to i = 5 j = 4then V = 1/2k (η2 - η1) + ... 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 η2=ζ2=-η4
Some stuff canceled 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-ω2T|=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
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.
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
η3=ζ3
η1=[itex]\frac{ζ_{1}+ζ_{5}}{\sqrt{2}}[/itex]
η5=[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 = dx2 - dx1 = dx3 - dx2 = ... up to i = 5 j = 4then V = 1/2k (η2 - η1) + ... 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 η2=ζ2=-η4
Some stuff canceled 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-ω2T|=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
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.
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