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ballzac
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Homework Statement
______|equillibrium position________
______|__i_____________________
m^^^^^m^^^^^m^^^^^m^^^^^m
____k_|qi|____k______ k________kA collection of particles each of mass m separated by springs with spring constant k. The displacement of the ith mass from its equilibrium position is q_i=q_i(t). Write the Langrangian for this one dimensional chain of masses.
Homework Equations
[tex]K=\frac{1}{2}mv^2[/tex]
[tex]V=\frac{1}{2}kx^2[/tex]
[tex]L=K-V[/tex]
The Attempt at a Solution
So we have [tex]K=\frac{1}{2}mv^2=\frac{1}{2}m(\frac{dx}{dt})^2[/tex]
Can I now say
[tex]K=\frac{1}{2}m\frac{dq_i}{dt}^2[/tex]
?
(I think I have to because I then have to differentiate wrt q_i
Also, the extension of the springs to begin with must matter because if they are taught then the potential energy must be higher. Am I right? Does it matter that there is a spring on either side? I guess it does, but as the mass is just moving between its equilibrium point maybe they kind of cancel out. There are N particles, so am I right in thinking that the energy required is the sum of all functions of [tex]q_i[/tex] for every [tex]i\leq N[/tex]?
I haven't been given very long for this, and I've never come across this stuff before, so I'm having issues figuring it out. Thanks for the help.
EDIT: Yeah, so I'm thinking [tex]V=\frac{1}{2}kq_i ^2[/tex] or [tex]V=kq_i ^2[/tex] but then I'm still not sure if the original taughtness of the spring comes into it.
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