1: (easy) Use the method of constrained multipliers to determine the Lagrangian and Lagranges
equations for a particle of mass m, that is constrained to roll along the surface of a paraboloid with
az = x2 + y2 = r2. The only external force is gravity. You should get a system of 4 equations, 3
equations of motion and one equation of constraint.
2: (Potentially tricky) Using the results of problem 1
A) Prove that the particle will undergo a circular path at a constant heigh h if it is given an
angular velocity = ! = p
2g/a Hint in this case both the height and the radius are constant.B) Show that a particle displaced from this path will undergo small oscillations with a frequency
given by 1
p
2g/a and a period given by P =
q a
2g .C) Plot Solutions to problem 1, include a plot that verifies A) and B) are correct.
Hint use the equation for conservation of angular momentum derived in problem 1 and assume
angular velocity as in part A
Hint assume oscillations take place almost in the plane z = h so we can leave = mg
a .Finally Derive an equation for the radial motion from the equations in problem 1 then assume
small perturbations about r given by r0 + u where u are small perturbations about a fixed value.Derive the governing equation of motion for u then taylor expand your equation to first order in u
recalling u << r0. This procedure should give you a integrable dierential equation.
3: 7-22 (Easy) By discuss, explain whether or not the energy of the system is conserved and
whether or not the Hamiltonian represents the total energy.
4: 7-26 (Medium) You can just write the lagrangians down if you know them. You may not
ignore the pulley mass here.
5: 7-28 (Easy)
6: 7-34 (Medium) Use lag ranges method. See Figure Below for hint. Note the reaction asked
for is the normal force from the generalized force of constraint (eq 7.66). Assume all the potential
is in the sliding mass. Your equation of constraint is r=R or r-R=0. SKIP part B too annoying.
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