Problem sets         Physics 9HB      winter 2003
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The latest is listed first!
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You are encouraged to discuss the problems with me, your TA,
or your classmates. However, the solutions that you hand in
must be written by you alone and must be *your own
understanding* of the problem. You may achieve that
understanding on your own or by talking and working with
others, but what goes on the paper you turn in must include
only the understanding that you have reached and not what
someone else has told you is correct. Including on your
paper information that you got from others but do not
understand yourself is a violation of the Code of Academic
Conduct.

You should *not* expect to read a problem and immediately
see how to do it. Expect to spend some time thinking and
trying various approaches until you find one that works. It
usually helps a lot to begin by identifying the fundamental
physical principles involved in the problem.

To get credit for your answers, you must show your work, and
it is to your advantage to make it neat, clear, and
organized. Please keep in mind that you are writing your
solutions for someone else to read---someone who is not a
mind reader and who has limited time to try to figure out
your solution. If we cannot figure out what you have done,
we cannot give you any credit. Write up your solution after
you have figured out how to do it on scratch paper. Whenever
possible, draw a picture and label it with the relevant
physical variables. List the quantities that you are given
with their values and the unknown quantities that you are
trying to find. Then state the most important physical laws
or principles that you are going to use to solve the
problem. If you turn in solutions on problem sets that are
difficult to read and that do not follow this structure, you
will lose points even if your answer is correct!

Usually one or two of the problems will be graded. You will
receive points for turning in solutions to all the problems,
points for the correctness of the problem or problems
that are graded, and points for presentation.

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Cosmology practice problem:

Here is something to do to practice a little cosmology 
calculation. It is not to be handed in.

Suppose that we live in a "critical" universe 
where the energy density is equal to the critical density.
Use recent data on the present value of the Hubble parameter,
the present temperature of the CMB radiation, and the age of 
the universe. 

What is the ratio of the energy density in CMB radiation to 
the critical energy density at present?

For the next questions, do them two ways. First assume a 
matter dominated then a cosmological constant dominated
evolution during the period from the earlier time refered to 
in the questions up until the present.

Show that the CMB energy to critical energy ratio increases 
as we go back in time. 

Consider the time at which the energy density in the 
CMB radiation was 0.1 of the critical density *at that time*. 
What is the ratio of the present value of the scale factor R to the 
value of R at that time?
About what was that time?
What was kT at that time?
Was the matter relativistic or nonrelativistic then?

Hints: Check the document "More cosmology" for useful
relationships. In doing problems like this it is usually 
easiest to work backwards from the present time, where
we do have data like the present values of the CMB 
temperature and the Hubble parameter. Also it is good 
to use ratios whenever possible to avoid the need to know
irrelevant proportionality constants.

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Problem set 8  due 9am, Tuesday,  Mar. 11 
in the locked boxes at the back of our lecture, 55 Ro.

Ch. T8:  S.7 S.8 A.1
Ch. T9:  B.6 S.3 S.10 S.15 R.1

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Problem set 7  due 9 am, Tuesday,  Mar. 4 
in the locked boxes at the back of our lecture, 55 Ro.

Ch. T4:  S.1  S.2  S.3
Ch. T5:  B.7  S.8  A.1
Ch. T6:  S.2  S.5  S.8
Ch. T7:  R.1

Although I am NOT assigning them, the following 
challenging problems are recommended:

T4:  R.1 
T5:  R.1  (This goes with T.4 R.1)
T6:  R.2
T.7  R.2

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Problem set 6  due 9am, Tuesday,  Feb. 25 
in the locked boxes at the back of our lecture, 55 Ro.

Ch. T1:  B.4 S.8 S.9
Ch. T2:  B.5 S.7
Ch. T3:  S.5 R.1 R.2

Although I am NOT assigning them, the following 
problems are recommended:

T1:  S.3 S.5 S.6 R.1 A.1
T2:  S.5 R.2 A.1
T3:  S.9 S.10 A.1 

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Problem set 5  due 9 am, Tuesday, Feb. 11
in the locked boxes at the back of our lecture, 55 Ro.

1. Far out in space away from any masses, you build a small lab to
measure the propagation of light beams. The y-direction is perpendicular
to x. If you shoot a light beam right down the y-axis, it goes into a
detector farther down the y-axis. a) Now bring your lab down and set it 
on the floor in the room here so that the x-axis is up. Do you need to
re-aim the beam to get it to go into the detector? Explain. b) Now lift
your lab up to the ceiling and let it drop. While it is falling, does
the aiming that you used way out in space still work? Explain. (For the
purposes of this problem, the gravitational field in the room is
uniform.)

2. On the surface of the earth, the metric is approximately 
g00=1+2ax/c^2 , g11= -1+2ax/c^2 g22=g33=-1 and all the rest are zero. 
x is height above the surface of the earth, c is the speed of light, and
a=9.8m/s^2 . a) Consider two events at x=0 y=0 z=0 that are separated by
a very small increment  Dt=1ns of coordinate time. a) What is the
spacetime interval separating them?  b) Consider two events at x=500km
y=0 z=0 that are separated by the same Dt of coordinate time. What is
the spacetime interval separating them? Give your answer as the small
deviation from 1ns. The sign of the deviation is important. (It is not
necessary, but  you may simplify the calculation by approximating with
the Taylor series.)

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Problem set 4  due 9am, Tuesday, Feb. 4
in the locked boxes at the back of our lecture, 55 Ro.

Ch. R9:  B.7 S.4 R.1
Ch. R10:  S.7 S.9 A.1

1. Strong evidence for the expansion of the universe 
comes from the observation of the 
Hubble law: the farther a galaxy is from us the faster 
is is going.   v = H d. 
v = speed with which a galaxy is receding from us.
d = distance to the galaxy
The present best estimate for the Hubble parameter
is H=65 (km/s)/Mpc. Mpc stands for a million pc. 
and 1 pc = 1 parsec=3.26 light years. (Astronomers
like these units.)
What are the SR units of H?
What is the value of 1/H in SR units?
If the light from a quasar is received on earth at a
wavelength 5 times what it was in the rest frame of 
quasar, what is the speed of the quasar and how
far is it to the quasar?

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Problem set 3  due 9 am, Tuesday, Jan 28
in the locked boxes at the back of our lecture, 55 Ro.

Ch. R6:  S.1 S.8 R.2
Ch. R7:  S.9 R.3
Ch. R8:  S.10 S.12 A.1
Describe your understanding of the relation between
R7 S.9, R8 S.12, and R8 A.1.

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Problem set 2  due 9 am, Tuesday, Jan 21
in the locked boxes at the back of our lecture, 55 Ro.

Ch. R3:  S.5 R.2
Ch. R4:  S.2 S.5 S.7
Ch. R5:  B.3 B.7 R.1

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Problem set 1  due 9 am, Tuesday, Jan 14
in the locked boxes at the back of our lecture hall, 55 Ro.

Chapter R1:  B.4 S.7 S.9
Chapter R2:  B.4 B.6 B.7 S.5 R.2 R.3

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