Physics 110, Problem set #4
Due April 28, 2010
1.
A pendulum is suspended from the edge of a rotating disk, as shown.
a.
Using
(
𝜃𝜃
,
𝜙𝜙
,
𝜓𝜓
)
as the generalized coordinates, write down the Lagrangian
of the system (10 pts). Note that
𝜙𝜙
is measured from the projection of
𝜓𝜓
,
as shown.
b.
What are the conserved quantities? There are at least two (6 pts).
c.
Find three equations of motion for the system (6 pts).
*This problem is not hard at all with Mathematica.*
`
2.
Constraints (25 pts): A bead with (mass
m
) is constrained to move along a stiff
parabolic wire described by
α
y
z
=
on the
yz
plane, as shown.
a.
Write down the Lagrangian of the system, using
z
y
x
,
,
as the generalized
coordinates (2 pts).
b.
What are the 2 constraint equations? Express them in the form of
0
;
0
2
1
=
=
G
G
(2 pts).
c.
Find the equations of motion for
z
y
x
,
,
, and the Lagrange multiplier(s) (4
pts).
d.
The same wire is spinning around the
z
axis with angular speed
ω
(counterclockwise).
Repeat parts (b)(c) (6 pts).
e.
Use these equations to find the equilibrium position of the bead. (4 pts)
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 Spring '09
 R
 Cartesian Coordinate System, Mass, pts

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