Homework Assignment 1
Physics 55
Handed out:
Due in class on:
Monday, August 24, 2005
Friday, September 2, 2005
Problem 1: An E-mail Introduction
Please introduce yourself by sending me an e-mail message to hsg@phy.duke.edu with the following information:
1. Your major or intended major at Duke (ok to leave this out if you don’t know).
2. Your hobbies (what you like to do outside of academics).
3. Your longer term professional interests (if you know them).
4. What you would like to get out of this course.
5. Whether you have had a previous exposure to astronomy.
6. Whether you would be interested in doing homework problems that involve calculus or that assume
that you have had a calculus-based physics course. (The official prerequisite for this course is high
school algebra but many students who take this know calculus or more.)
7. Any suggestions that you might have about how you would like this course to be taught.
Problem 2: Email Your PRS Transmitter Code to the Course TA
As soon as you have purchased your PRS transmitter, please email your transmitter number (which should be
on the back of the transmitter) to the course TA, Mr. Botao Jia, whose email address is bj9@phy.duke.edu.
This is important so that I can keep track of your PRS responses starting next week.
Problem 3: Warm-up Problems
Before doing the following problems, please read carefully the part of the course syllabus
http://www.phy.duke.edu/~hsg/55/55-syllabus.html#grading
that discusses the grading of homework assignments. You will lose credit if you do not follow the guidelines
for writing a homework assignment. In particular, please write clearly (the TA and I have to be able to
understand your answers easily) and it is important that you justify your answers by showing details of your
derivations and by writing some sentences or phrases that explain what you are doing and why. It is not
sufficient to just write down an answer such as “yes” or “2.42” since neither the TA or myself can then tell
if you understand how the answer was obtained or what is the significance of the answer.
The following short problems are designed to encourage you to practice some of the mathematics that will
be used during this course. Note: The official math level of this course is modest, about the Algebra I high
school level, but I will occasionally include optional more challenging problems for those students who have
stronger math and science backgrounds than the minimum prerequisites.
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1. The following two problems should be straightforward if you remember how to arrange a calculation
by “cancelling units”. If you don’t remember how to do this or want to see some examples, please
come to this week’s recitation or make an appointment to see the TA or myself.
(a) A certain professor began her talk by saying that she was going to speak for the duration of one
microcentury. Given that a “micro” means one-millionth of something, determine how long is
a microcentury in minutes to three significant digits. Is this a reasonable amount of time for a
person to give a lecture?
Note: Since Physics 55 is a science course, you should always try to see if your final answer makes
sense or not scientifically or in terms of common sense.
(b) On the alien planet Floog in orbit around the star Mizar B (Mizar B is a real star and you will
soon learn where to find it in the sky), the citizens use several different lengths that are related
as follows:
1 blorck = 2 × 10−3 hlars,
1 hlar = 7 × 105 flisks,
1.4 × 107 flisks = 1 phorp.
If the distance of Floog to Mizar B is 1000 blorcks, what is that distance in units of phorps?
(Make sure to include the details of how you got your answer!)
2. Although many of you might think of equations as boring recipes that tell you how to calculate some
new number from some known numbers, equations can be much more valuable in science since they
can give you qualitative intuition about how some physical quantity influences some other physical
quantity. As a first step toward learning (or reviewing) how to obtain such qualitative insight, please
consider the following equation that we will discuss later in the semester and that was first discovered
and used by Isaac Newton as he tried to understand the patterns of the planets and of moons orbiting
planets:
4π 2 a3
.
(1)
p2 =
G (M1 + M2 )
Here the symbols p, a, M1 , and M2 are all positive numbers that denote properties of physical objects
(the time p for a mass M1 to orbit another mass M2 , the average width a of a mass’s orbit, and the
masses of two objects orbiting each other), while the symbol π (a lower case Greek pi) is the famous
number associated with the circumference of a circle with approximate value 3.14, and G is a constant
whose value can be obtained from experiments.
Please answer the following questions (don’t forget to justify your answers):
(a) In Eq. (1), assuming that the two masses M1 and M2 remain constant in value, if the parameter a
becomes bigger in value, does the parameter p become bigger, stay the same, or become smaller?
Scientifically, you are asking how the time p for a mass to orbit around some other mass depends
qualitatively on the average size a of its orbit.
(b) In Eq. (1), assuming that the parameters a and M2 remain constant in value, if the mass M1
becomes bigger in value, does the parameter p become bigger, stay the same, or become smaller?
Scientifically, you are asking qualitatively how the time p for one mass to orbit another changes
if the mass of that object becomes bigger when the orbit does not change its size.
(c) In Eq. (1), assuming that the parameters p and M1 remain constant in value, if the mass M2
becomes bigger, does the parameter a become larger, stay the same, or become smaller? Scientifically, you are asking how the average size a of an orbit changes if a mass becomes bigger,
assuming that the period of the orbit does not change.
(d) Finally, assuming that the parameters p and a remain constant in value, if the mass M 1 becomes
bigger, does the mass M2 become larger, stay the same, or become smaller?
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Problem 4: Alternative Advanced Problems
Students comfortable with calculus or math beyond calculus can optionally do the following problems
instead of the above problems.
(a) If i denotes a square root of -1, what is the value of ii (i raised to the i)?
(b) For x a non-negative number, consider the infinite continued fraction
1
x+
(2)
1
x+
x+
1
x+
1
···
Determine whether this expression is an increasing or decreasing function of x as x increases in
value, and find the value of x for which it increases most rapidly.
(c) From the first lecture and from your reading in Chapter 1 of the text, you will have learned about
the extraordinary range of length and time scales associated with the universe. Is there a smallest
possible length scale, a size below which it would not make sense to have any spatial structure?
The answer is currently believed to be yes (!) and a likely value of that smallest length was
predicted in 1899 (!) by the simple trick of trying to combine some of the fundamental physical
constants that describe nature to see if a unit of length could be achieved.
Reproduce this discovery by using the following information. The fastest possible speed is the
universal speed of light, denoted by the symbol c (from the first letter of the Latin word celeritas
for swift) which has units of meters per second (m/s). (This speed is called universal since, as we
will discuss later in the course, everyone measures the same speed of light, no matter how fast they
are moving.) All quantum phenomena involve the extremely important constant h̄ (pronounced
“h-bar”) which has units of angular momentum or kg × m2 /s. And all gravitational phenomena
involve the universal gravitational constant G which has units of m3 /(kg × s2 ).
i. Find three real numbers u, v, and w such that the product of the three fundamental constants
raised to these powers:
cu h̄v Gw ,
(3)
has a unit of meters, i.e., length.
This is an example of what is called “dimensional analysis” in physics and engineering.
ii. Look up the values of c, h̄, and G in Appendix A of your text and evaluate this smallest
possible length. Compare this size to the diameter of a proton.
iii. Once you have a smallest possible length, you also have the smallest possible time which is
the time it takes light, the fastest possible motion, to travel across that length. What is your
predicted value of the smallest possible time? How old is the universe in units of that time?
(You get a very big integer!)
We will discuss the shortest length and time when we discuss cosmology toward the end of the
course. For now, you may be interested to know that the physical meaning of this shortest length
is the radius of the smallest possible black hole that is consistent with the Heisenberg uncertainty
principle of quantum mechanics. Also just because we can identify a smallest length and time
by guesswork doesn’t mean that they have a physical meaning, and it will be many years in the
future (centuries?) before the human race will have the technology to explore such tiny sizes.
Problem 5: Comments about the Homework and Course
• About how long did this assignment take to complete?
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• Do you feel that you are understanding the course material? If not, what topics have proved difficult
to understand?
• Do you feel that what you are learning is interesting or beneficial?
• Other comments or suggestions about the homework or lectures?
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