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[answered] A problem in Analytical Mechanics. intermediate level physi


A problem in Analytical Mechanics. intermediate level physics (3rd or 4th year mechanics).


If you have three bodies of equal mass connnected by two springs arranged in a line,

the potential energy is k/2[ (x1-x2)^2 + (x2-x3)^2 ].


The equations of motion are

m x1'' = -k (x1-x2)

m x2'' = -k (x2-x1 + x2-x3)

m x3'' = -k (x3-x1)


Express the right hand side as a matrix operating on the column vector

(x1, x2, x3). That is, the transpose of the row vector that starts this line.


Find the eigenvalues and eigenvectors that correspond to this matrix.


?Describe the normal modes of the system in text.
The given equations of motion are

 

&

 

&1 k x1 x 2 mx &

 

&2 k x1 2x 2 x 3 mx

 

&

 

&3 k x 3 x 2 mx Let assume a solution of the form x1 A1 i t x 2 A 2 e x A 3 3

 

x&1 & A1...

 


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