Physics 318: First prelim exam
Thursday, February 22, 2007 Duration: 90 minutes
Instructions:
Answer all questions. This is a closed book exam. There are 15 points for Q1, 15 for Q2 and
10 for Q3.
1.
A pendulum which is exposed to gravity consists of a mass
m
which is suspended by means of a massless
ideal elastic string of unextended length
l
and spring constant
k
. The ideal elastic string follows Hooke’s law
like an ideal spring. The string is constrained by a frictionless fixture which still allows the string to extend,
as shown in the figure.
a. Calculate the kinetic and potential energies of the mass
m
, using the coordinates
r
,
θ
and
φ
indicated
in the figure. Find the Lagrangian for this system.
b. Find the equations of motion and quote them in the form:
m
¨
r
=
. . .
m
¨
θ
=
. . .
m
¨
φ
=
. . .
c. For this system there are two conserved quantities, functions of
r
,
θ
,
φ
, ˙
r
,
˙
θ
and
˙
φ
. Derive expressions
for these conserved quantities and explain their physical interpretation.
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2.
A particle of mass
m
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 Spring '08
 FLANAGAN
 mechanics, Gravity, General Relativity, spherical polar coordinates, sin sin ey

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