Physics 3221
Fall Term 2003
Test 1, October 3, 2003
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This is an open book and notes test lasting 50 minutes.
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There are two problems, divided into subsections. Problem 1 has 2 parts
and problem 2 has 4 parts. The points for each part are marked.
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Begin each problem on a fresh sheet of paper. Use only one side of a
sheet of paper.
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Put your name, the problem number, and the page number in the upper
right hand corner of each sheet.
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To receive partial credit you must explain what you are doing. Carefully
labeled figures are important. Randomly scrawled equations aren't helpful.
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Draw a box around important results.
Some constants which may be useful:
Gravitational constant (G) = 6.67 × 10-11 N.m2/kg2
Solar mass (Msun) = 1.989 × 1030 kg
Mass of the earth = 5.98 × 1024 kg
Distance between earth and sun = 1 astronomical unit (AU) = 1.496 × 1011 m
There are 3 pages including this page. Do not forget to look at all parts of the
problems.
Page 1 of 3
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1. Since the sun is much more massive than any of the planets in the solar system (it
is about 1000 times heavier than Jupiter), the time required for any of the planets
to complete one revolution around the sun (orbital period) depends upon the
following parameters:
(i)
the mass of the sun (Msun)
(ii)
Gravitational constant (G)
(iii)
Distance of the planet from the sun (a).
a) Using dimensional analysis, find a formula for the orbital period (t) of any
planet around the sun as a function of the parameters listed above. (2 points)
b) Earth has an orbital period of about 365 days and Pluto has an orbital period
of about 90500 days. If the earth is 1 astronomical unit (AU) away from the
sun, how far is Pluto from the sun (in AU)? (2 points)
2. The figure shows a cube with side h. The point O is the origin of the coordinate
system.
G
a3
a2
h
a1
a) What are the coordinates of the point P at the body center of the cube, in
terms of h? (The body center can also be defined as the point at the center
of any body diagonal of the cube such as EB and OG). (0.5 points)
b) Find the vectors (with components in terms of h) pointing from (i) O to A
(ii) O to C and (iii) O to P and define them as vectors a1, a2, and a3
respectively. (The directions of the x, y and z axes and the vectors a1, a2,
and a3 are marked in the figure). (0.5 points)
Page 2 of 3
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c) Calculate the vectors defined by: (2 points)
b1 ≡ 2π a2 × a3
V
b2 ≡ 2π a3 × a1
V
b3 ≡ 2π a1 × a2
V
where V is the volume of the parallelepiped defined by the vectors
a1, a2, and a3. Express the components of the vectors b1, b2, and b3
in terms of h.
d) Show that: (3 points)
bi • aj = 2π δij,
where aj represents the vectors a1, a2 and a3 for j=1, 2 and 3
respectively and similarly for bi. (Careful, bi and aj are vectors
and not components of vectors as you encountered in your
homework problems, e.g. for i=1 and j=2, the above equation is:
b1 • a2 = 2π δ12).
Page 3 of 3
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