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Consider two particles of equal masses, m1 = m2, attached
Chapter 8, Problem 8.9(choose chapter or problem)
Consider two particles of equal masses, m1 = m2, attached to each other by a light straight spring (force constant k, natural length L) and free to slide over a frictionless horizontal table. (a) Write down the Lagrangian in terms of the coordinates r1 and r2, and rewrite it in terms of the CM and relative positions, R and r, using polar coordinates (r, 0) for r. (b) Write down and solve the Lagrange equations for the CM coordinates X, Y. (c) Write down the Lagrange equations for r and 0. Solve these for the two special cases that r remains constant and that remains constant. Describe the corresponding motions. In particular, show that the frequency of oscillations in the second case is co = '2k/ml
Questions & Answers
QUESTION:
Consider two particles of equal masses, m1 = m2, attached to each other by a light straight spring (force constant k, natural length L) and free to slide over a frictionless horizontal table. (a) Write down the Lagrangian in terms of the coordinates r1 and r2, and rewrite it in terms of the CM and relative positions, R and r, using polar coordinates (r, 0) for r. (b) Write down and solve the Lagrange equations for the CM coordinates X, Y. (c) Write down the Lagrange equations for r and 0. Solve these for the two special cases that r remains constant and that remains constant. Describe the corresponding motions. In particular, show that the frequency of oscillations in the second case is co = '2k/ml
ANSWER:Step 1 of 8
(a)
The central force is given as,
Here, 
The potential energy of the particle in the central force field is given as,
Here, 
Substitute 
The force in the vector form,
If 
