A pendulum consists of point mass m is on the end of a massless spring (spring c

Question

A pendulum consists of point mass m is on the end of a massless spring (spring constant k, unstretched length b). Assume

The pendulum support does not move, and the mass is constrained to oscillate in the vertical plane that includes the support.

There is no friction anywhere.

The spring is wrapped around a massless rod, so it remains in a straight line.

g=gj,andisconstant.

The origin is at the point of support, and U(0) = 0.

For generalized coordinates: use x (the amount the spring is stretched or compressed) and ? (the angle between the rod and the vertical).

a) i. Starting from expressions for (xm,ym) (the Cartesian coordinates for m), write an expression for the Lagrangian L(x,x',?,?')

ii. Write the Euler-Lagrange equations. Simplify, but do not solve.

b) Write the Hamiltonian H(x,px,?,p?) and the Hamilton equations of motion. Show that these equations reduce to the same equations you found with the Euler-Lagrange equation

Remember: H must be in terms of the generalized coordinates and the generalized momenta!

c) For each: give a reason, not just a calculation!
i. Is H = Etot?

ii. Is p? conserved?

iii. Is ETOT conserved?

iv. Is H conserved?

Explanation / Answer

please refer this pdf, it explains the problem very good. www.people.fas.harvard.edu/~djmorin/chap6.pdf?

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.