Practice Exam 5 - practicework 05 BAUTISTA ALDO Due May 8...

practicework 05 – BAUTISTA, ALDO – Due: May 8 2006, 4:00 am 1
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V1:1, V2:4, V3:2, V4:4, V5:3.
Question 1
Part 1 of 6. 10 points.
Asteroids X , Y , and Z have equal mass.
They orbit in the same angular direction
around a planet with much greater mass (lo-
cated at the black dot, at one of the foci of
each of the two ellipsical orbits and at the
center of the circular orbit).
3
2
1
X
Y
Z
Figure 1: The orbits of the aster-
oids are in the plane of the figure
and are drawn to scale.
Let the subscripts denote the asteroid’s po-
sition. The magnitude of the angular momen-
tum ‘ of the asteroid X at positions 2 and 1
are related by
1. ‘
2
X
< ‘
1
X
2. Not enough information is available.
3. ‘
2
X
> ‘
1
X
4. ‘
2
X
= ‘
1
X
correct
Explanation:
The net force on asteroid X is central; e.g. ,
~
F = G
M m
r 2
ˆ r , therefore angular momentum
is conserved; i.e., ‘ = constant .
Angular momentum
~
‘ = m~v × ~r = m r
2
~ω
is conserved; thus ‘
2
X
= ‘
1
X
.
Question 2
Part 2 of 6. 10 points.
The magnitude of the angular velocity ω of
the asteroid X at positions 2 and 1 are related
by
1. ω
2
X
= ω
1
X
2. Not enough information is available.
3. ω
2
X
< ω
1
X
4. ω
2
X
> ω
1
X
correct
Explanation:
Angular momentum
~
‘ = m r
2
~ω is con-
served. Since r
2
X
< r
1
X
, then ω
2
X
> ω
1
X
.
Question 3
Part 3 of 6. 10 points.
Hint:
1) The semi-major axis a
X
of the orbit of
asteroid X is equal to the radius r
Y
of the
orbit of asteroid Y .
2) The semi-minor axis b
X
of asteroid X is
equal to the semi-minor axis b
Z
of asteroid Z .
3) And the area of the orbit of asteroid Y
( A = π r
2
Y
) is equal to the area of the orbit of
asteroid Z ( A = π a
Z
b
Z
) .
X
Y
Z
Figure 2: The orbits of the aster-
oids are shown for comparison pur-
poses. The elliptical orbits of the as-

practicework 05 – BAUTISTA, ALDO – Due: May 8 2006, 4:00 am 2
teroids are relocated and rotated in
order that they have a common cen-
ter and their major axes are aligned.
The period T of asteroids X and Y are
related by
1. T
X
< T
Y
2. T
X
> T
Y
3. T
X
= T
Y
correct
4. Not enough information is available.
Explanation:
Geometry of Ellipses:
r
1
r
2
a
b
a a
F
1
F
2
C
Figure 3: The geometrical ele-
ments of an ellipse
An ellipse is defined as the curve traced
by a particle moving so that the sum of its
distances from two fixed points F
1
and F
2
is
constant. The points F
1
and F
2
are called the
foci of the ellipse and C is the center of the
ellipse. The planet is located at one of these
foci and the orbit of the asteroid is designated
by the ellipse. Using the notation indicated,
we have r
1
+ r
2
= 2 a , where a is the semi-
major axis, b is the semi-minor axis, ε is the
eccentricity, ε a is the distance from the center
of the ellipse to a focus of the ellipse, the ratio
b
a
=
p
1 - ε 2 , and the area A = π a b .