- #1
albertrichardf
- 165
- 11
Homework Statement
An ideal spring of relaxed length l and spring constant k is attached to two blocks, A and B of mass M and m respectively. A velocity u is imparted to block B. Find the length of the spring when B comes to rest.
Homework Equations
[tex] ∆K + ∆U = 0 [/tex]
[tex] U = \frac {kx^2}{2} [/tex]
[tex] ∆p = 0 [/tex]
[tex] L = K - U [/tex]
The Attempt at a Solution
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I used energy conservation to find the answer. I got:
[tex] (mu)^2( \frac 1m - \frac1M) = kx^2 [/tex]
Then divide by k and take the square root.
I then tried with the Lagrangian. My coordinates are x and X, where X is the position of B, and x is the position of A. u is the velocity of A and U is the velocity B. And because the solutions will not change if I multiply the Lagrangian, I eliminated the factor of half from the energies. Thus my Lagrangian is:
[tex] mu^2 + MU^2 - k(x - X - l)^2 = L [/tex]
I used the Euler Lagrange equations with respect to x and X. However when I used them I simply end up with Hooke's law. But Hooke's law relates the acceleration to the extension of the spring and I do not have the acceleration. Thus I can't solve for the extension. Plus the solution should depend on the final and initial velocities but it does not.
How could I solve this using the Lagrangian?
Thank you for answering