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Physics / Classical Mechanics 0 / Chapter 13 / Problem 13.5

Classical Mechanics | 0th Edition | ISBN: 9781891389221 | Authors: John R Taylor

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A bead of mass m is threaded on a frictionless wire that

Chapter 13, Problem 13.5

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QUESTION:

A bead of mass m is threaded on a frictionless wire that is bent into a helix with cylindrical polar coordinates (p, 0, z) satisfying z = c0 and p = R, with c and R constants. The z axis points vertically up and gravity vertically down. Using 0 as your generalized coordinate, write down the kinetic and potential energies, and hence the Hamiltonian g-C as a function of 0 and its conjugate momentum p. Write down Hamilton's equations and solve for and hence E. Explain your result in terms of Newtonian mechanics and discuss the special case that

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QUESTION:

A bead of mass m is threaded on a frictionless wire that is bent into a helix with cylindrical polar coordinates (p, 0, z) satisfying z = c0 and p = R, with c and R constants. The z axis points vertically up and gravity vertically down. Using 0 as your generalized coordinate, write down the kinetic and potential energies, and hence the Hamiltonian g-C as a function of 0 and its conjugate momentum p. Write down Hamilton's equations and solve for and hence E. Explain your result in terms of Newtonian mechanics and discuss the special case that

ANSWER:

Step 1 of 7

The kinetic energy of the system in polar co-ordinates can be given as,

For , , and .

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