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Two masses m1 and m2 are joined by a massless spring
Chapter 13, Problem 13.21(choose chapter or problem)
Two masses m1 and m2 are joined by a massless spring (force constant k and natural length /0) and are confined to move in a frictionless horizontal plane, with CM and relative positions R and r as defined in Section 8.2. (a) Write down the Hamiltonian g-C using as generalized coordinates X, Y, r, 0, where (X, Y) are the rectangular components of R, and (r, 0) are the polar coordinates of r. Which coordinates are ignorable and which are not? Explain. (b) Write down the 8 Hamilton equations of motion. (c) Solve the r equations for the special case that p0 = 0 and describe the motion. (d) Describe the motion for the case that p0 A 0 and explain physically why the r equation is harder to solve in this case.
Questions & Answers
QUESTION:
Two masses m1 and m2 are joined by a massless spring (force constant k and natural length /0) and are confined to move in a frictionless horizontal plane, with CM and relative positions R and r as defined in Section 8.2. (a) Write down the Hamiltonian g-C using as generalized coordinates X, Y, r, 0, where (X, Y) are the rectangular components of R, and (r, 0) are the polar coordinates of r. Which coordinates are ignorable and which are not? Explain. (b) Write down the 8 Hamilton equations of motion. (c) Solve the r equations for the special case that p0 = 0 and describe the motion. (d) Describe the motion for the case that p0 A 0 and explain physically why the r equation is harder to solve in this case.
ANSWER:Step 1 of 5
(a)
Assume M and
Here R and r are the position of center of mass and the relative position of these masses respectively.