Description
1. (20 points) Is the constraint
(x
2 + y
2 + z
2
)dx + 2(xdx + ydy + zdz) = 0 (1)
a holonomic constraint or a non-holonomic one?
2. (30 points) A particle is subjected to the potential V (x) = −F x, where F is a constant.
The particle travels from x = 0 to x = a in a time interval t0. Assume the motion of the particle
can be expressed in the form x(t) = A + Bt + Ct2
. Find the values of A, B, and C such that the
action is a minimum.
3. (25 points) A bead of mass m slides down a frictionless wire bent into the form of a
parabola y = Ax2
, determine the constraint force acting on the bead.
4. (25 points) A uniform hoop of mass m and radius r rolls under gravity from rest without
slipping on top of a fixed cylinder of radius R, use the method of Lagrange multipliers to find
the point where the hoop falls off the cylinder.
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