Homework Assignment 7
Physics 302, Classical Mechanics
Fall 2010
A. V. Kotwal
Handed out:
Monday, October 25, 2010
Due time:
Monday, November 1, 2010
Problems
Each problem is out of 20 points.
1. Consider the Kepler problem. The energy conservation equation is given by
E
=
μ
2
dr
dt
2
+
l
2
2
μr
2

α
r
,
where
α
=
GμM
=
Gm
1
m
2
,
l
is the angular momentum, and
E
is the total energy of the system.
(a) Prove the Kepler’s third law for elliptical orbits
T
2
=
4
π
2
G M
a
3
,
where
M
=
m
1
+
m
2
is the total mass,
a
=
1
2
(
r
min
+
r
max
) is the semimajor axis, and
T
is the
period of the orbit.
(b) Show that the semiminor axis is given by
b
=
a
p
1

2
,
where
=
q
1 +
2
El
2
μα
2
is the eccentricity of the orbit. We notice that when
= 0,
b
=
a
, the
orbit is a circle.
2. A uniform distribution of dust in the solar system adds to the gravitional attaction of the Sun on a
planet an additional force
~
F
=

mC~
r
where
m
is the mass of the planet,
C
is a constant proportional to the gravitational constant and
the density of the dust, and
~
r
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 Spring '09
 Physics, Energy, Mass, Work, General Relativity, Kepler's laws of planetary motion, Celestial mechanics

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