A sphere with a hole drilled through its center is threaded onto a wire. The wire...

A sphere with a hole drilled through its center is threaded onto a wire. The wire is attached to an electric motor, causing the wire to revolve around the vertical axis as shown in the figure. The angular velocity (ω) does not change. The wire’s shape is given by z=b(r^2) where r is the distance from the z axis and b is a positive constant. Treat the sphere as a point mass; you may neglect friction in this problem.

a/Write down the kinetic and potential energy of the system and the Lagrangian in terms of one generalized coordinate, r, and its derivative.

b/ Use the Euler-Lagrange equation to obtain an equation of motion for this system.

c/If the sphere moves in a circle of constant radius R (i.e. it stays in one place on the wire), determine the value of the constant b.

d/What is the generalized momentum of the system?

e/Determine the Hamiltonian for this system using the full definition of the Hamiltonian as given in equation 13.4 in your book.

f/ Determine whether this Hamiltonian is equal to T+U.

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Expert Solution

PLEASE GIVE A THUMBS UP.

genius_generous answered 1 year ago

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