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Homework Assignment 3
Physics 302, Classical Mechanics
Fall, 2010
A. V. Kotwal
Handed out:
Friday, September 17, 2010
Due in class on:
Friday, September 24, 2010
Problems
(Ten points for each problem unless noted otherwise.)
1. Use the Lagrange multiplier method to solve the following problem: A particle in a uniform grav
itational ﬁeld is free to move without friction on a paraboloid of revolution whose symmetry axis
is vertical (opening upward). Obtain the force of constraint. Prove that for a given energy and
angular momentum about the symmetry axis there is a minimum and a maximum height to which
the particle will go.
2. A bead is constrained to move without friction on a helix whose equation in cylindrical polar coor
dinates is
ρ
=
b, z
=
aφ
, with the potential
V
=
1
2
k
(
ρ
2
+
z
2
). Use the Lagrange multiplier method
to ﬁnd the constraint forces.
3. Consider a threedimensional oneparticle system whose potential energy in cylindrical polar coordi
nates
ρ,θ,z
is of the form
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