Download PS119 maths review

PS119: Laboratory #1 Review of Mathematics
“Mathematics is just organized reasoning”- Richard Feynman
“There is something fascinating about science. One gets such wholesale returns of
conjecture out of such a trifling investment of fact” – Mark Twain
1.INTRODUCTION
You cannot do physics and astronomy without mathematics but the mathematics you
need to make progress is usually quite modest. The mathematics you need for the course
will be reviewed in the first half of this lab.
The second part of the lab introduces you to the elements of mathematical reasoning.
Enrico Fermi (who built the first nuclear reactor in the world at this University) used to
set problems to his students such as “How many piano tuners are there in Chicago?”
These problems require some knowledge (for example, the number of houses in
Chicago); some estimation (for example, the length of time it takes to tune a piano) and
some guess work (for example, the fraction of houses that have pianos). By writing down
all the assumptions explicitly, you can see what parts of the problem involve knowledge,
what parts estimation, what parts guessing and how much extra work and research you to
get a reasonable answer. Quite often you can make surprisingly good estimations of the
solutions to very difficult problems by simplifying the problem or thinking about how
knowledge that you do have sheds light on the problem in hand.
To illustrate this point further I have appended a paper by two astronomers who used
simple calculations to show that you can predict the size of almost anything in the
universe from a small set of numbers based on the ratios of physical constants that can be
measured in the laboratory.
2. NUMBERS AND GRAPHS
(2.1)
Evaluate the following using a calculator.
(a) 3.72 x 102 + 1.6 x 10-2 + 4.4
(b)
3.1415
8.36 10 6  4.365 10 5 97485


0.0245
2.47 1010
9.4 10 2
(2.2) Using logarithms

In dealing with the extremely large and extremely small numbers met in
astronomy, it is often useful to use exponents or powers of ten. For example the number
1
10000000 is represented by the notation 107 and 0.001 by the notation 10-3. The number 1
is expressed as 100. Numbers are often expressed in “scientific notation” in which the
actual number, say 23456.3 is written as a number between 1 and 10 times an appropriate
power of ten; i.e. this number is written as 2.34563 x 104.This notation is related to the
logarithm of a number. The logarithm of a number, x, is another number, y, defined so
that y= 10x. Since you all have calculators you have a special button that converts
numbers into logs. The log of the number 3 is about 0.5; the log of the number 3000 is
about 3.5.
Because the sizes and physical conditions of objects in the universe are so
different, we often plot the logarithms of numbers in a graph to allow us to represent
different types of object in a single diagram. For instance in figure 1 of the Nature paper
appended to this lab is plotted the masses and sizes of all objects that can exist in the
Universe, which vary by a factor of 1060. As another example, because the range of
brightness of stars in the sky is huge they are usually expressed in a logarithmic scale
with the bright star Vega being given a magnitude of 0.
A further convenience arises because multiplication of two numbers corresponds
to addition of their logarithms, division of two numbers by subtraction.
The rules of logarithms are:
log(a x b)
log(a/b)
log ab
=
=
=
log a +log b
log a – log b
b x log a
=
=
=
10a + b
10a -b
10a x b
The rules of exponents are
10a x 10b
10a/10b
(10a)b
Thus 1047 x 1036 = 1083 and the logarithm of the product is 45 + 36 =83.
Using your calculator write down the logarithms of the following numbers:
(a)
256453.2
(b)
0.00000123
(c)
What is the number whose logarithm is 4.56253 ?
2
(2.3)
Plotting graphs
Plotting data in a graph allows us to see if there are trends, implying perhaps
some deeper physical connection, and gives us a qualitative feel for the scatter, which
measures the “goodness” of the data. Being able to drawing graphs to represent data is
an essential skill in science. Sometimes you plot the data on a linear scale (suitable for
cases in which there is a linear relationship between the two parameters) and sometimes
logarithmic when the relationship may be more of a power law [ y = x m implies that log y
= m log x]
(a)
Plot a graph of the following data that gives the distance of an aircraft from a
fixed point as a function of time.
Time
7:00
8:00
9:00
10:00
11:00
Distance (miles
750
1300
1850
2400
2950
Draw a straight line through the data and determine:
(i)
When did the aircraft take off?
(ii)
How fast is the aircraft traveling?
(b)
The following data is gives the intrinsic brightness of a star (sun =1) against its
temperature. Try and plot a graph of this data on a linear scale and then on a log scale.
Temperature (0K)
2,500
3,200
3,850
4,450
5,150
5,700
6,400
8,700
9,600
15,200
29,000
Luminosity
0.0008
0.013
0.075
0.18
0.36
1.0
4.0
24
63
750
16,000
Log (Temperature)
Log(Luminosity)
3
What is the approximate relationship between Luminosity and temperature?
(2.3)
Small angle approximation
You were probably taught at school to measure angles in degrees; thus there are 360
degrees in a circle and the angles within a triangle add up to 180 degrees. You were also
taught that the distance BC in a right angle triangle is given by
BC = Sin  x AB = Tan  x AC
B

A
C
In astronomy we often deal with small angles within a triangle and it is easier to work in
radians, rather than degrees. There are 2 radians in a circle; if we have a coin 1 inch in
diameter at a distance of 100 inches we find that the angle between the edges of the coin
is 1/100 radian. For small angle then, Sin ≈ Tan  ≈  provided  is measured in radians
and this approximation considerably simplifies the mathematics.
(a)
Astronomers frequently use the arcsecond as a unit of angular measurement.
There are 60 arcseconds in an arcminute and 60 arcminutes in a degree. How many
arcseconds are there in a radian?
(b)
A unit of distance we shall use in PS119 is the parsec. This is the distance at
which, if we looked back to earth, the orbit of the earth around the sun would subtend one
arcsecond [this funny definition comes about because the distances to stars is measured
using parallax, which is the subject of the next lab].
1 parsec
1
A.U.
1 arcsecond
4
The radius of the orbit of the earth is called an Astronomical Unit (A.U.) and has been
measured by radar as 149,597,870.691 km.
(i)
How many kilometers are there in a light year
(ii)
How many light years are there in a parsec?
(3) Fermi questions
Write down your assumptions and simplifications clearly as well as your calculations.
(1)
How much does each lecture you attend at the university of Chicago cost if you
pay the full tuition fees and complete the required courses?
(2)
How many drops of water are there in Lake Michigan?
(3)
How many times you expect your heart to beat?
(4)
McDonalds claims to have sold over 100 billion hamburgers. How many cows
have been used to make this number of hamburgers?
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