fix problems with html output

[light-and-matter.git] / cp / ch01 / ch01.rbtex

<%
  require "../eruby_util.rb"
%>
<% figure_in_toc("poolskater") %>
<%
  chapter(
    '01',
    %q{Conservation of Mass and Energy},
    'ch:energy',
    ''
  )
%>


<% begin_sec("Symmetry and Conservation Laws",0) %>

<% marg(0) %>

<%
  fig(
    'startrails',
    %q{%
        Due to the rotation of the earth, everything in the sky appears to spin in circles.
        In this time-exposure photograph, each star appears as a streak.
    }
  )
%>
<% end_marg %>

Even before history began, people must already have noticed certain facts
about the sky. The sun and moon both rise in
the east and set in the west. Another fact that can be settled to a
fair degree of accuracy using the naked eye is that the apparent
sizes of the sun and moon don't change noticeably. (There is an
optical illusion that makes the moon appear bigger when it's near
the horizon, but you can easily verify that it's nothing more
than an illusion by checking its angular size against some standard,
such as your pinkie held at arm's length.) If the sun and moon were
varying their distances from us, they would appear to get bigger and
smaller, and since they don't appear to change in size,
it appears, at least approximately, that they always stay
at the same distance from us.

From observations like these, the ancients constructed a scientific
\emph{model},\index{model} in which the sun and moon traveled around
the earth in perfect circles. Of course, we now know that the earth isn't
the center of the universe, but that doesn't mean the model wasn't
useful. That's the way science always works. Science never aims to
reveal the ultimate reality. Science only tries to make models of
reality that have predictive power.

Our modern approach to understanding physics revolves around the
concepts of \emph{symmetry}\index{symmetry} and \emph{conservation laws}, both
of which are demonstrated by this example.

The sun and moon were believed to move in circles, and a circle
is a very symmetric shape. If you rotate a circle about its center,
like a spinning wheel, it doesn't change. Therefore, we say that
the circle is \emph{symmetric} with respect to rotation about its
center. The ancients thought it was beautiful that the universe
seemed to have this type of symmetry built in, and they became very
attached to the idea.

A \emph{conservation law}\index{conservation law} is a statement that some number stays the
same with the passage of time. In our example, the distance between
the sun and the earth is conserved, and so is the distance between
the moon and the earth. (The ancient Greeks were even able to determine
that earth-moon distance.)

<%
  fig(
    'noether',
    %q{%
Emmy Noether (1882-1935). The daughter of a prominent German mathematician, she did not show
any early precocity at mathematics --- as a teenager she was more interested in music and dancing.
She received her doctorate in 1907 and rapidly built a world-wide reputation, but the University
of G\"{o}ttingen refused to let her teach, and her colleague Hilbert had to advertise her courses in the
university's catalog under his own name. A long controversy ensued, with her opponents asking
what the country's soldiers would think when they returned home and were expected to learn
at the feet of a woman. Allowing her on the faculty would also mean letting her vote
in the academic senate. Said Hilbert, ``I do not see that the sex of the candidate is against
her admission as a privatdozent [instructor]. After all, the university senate is not a bathhouse.''
She was finally admitted to the faculty in 1919. A Jew, Noether fled Germany in 1933 and
joined the faculty at Bryn Mawr in the U.S.
    },
    {
      'width'=>'wide',
      'narrowfigwidecaption'=>true,
      'float'=>false
    }
  )
%>

In our example, the symmetry and the conservation law both give the
same information. Either statement can be satisfied only by a circular
orbit. That isn't a coincidence. Physicist
Emmy Noether\index{Noether, Emmy} showed on very general mathematical grounds that for
physical theories of a certain type, every symmetry leads to
a corresponding conservation law. Although the precise
formulation of Noether's theorem,\index{Noether's theorem} and its proof, are too mathematical for this
book, we'll see many examples like this one, in which the physical
content of the theorem is fairly straightforward.

The idea of perfect circular orbits seems very beautiful and intuitively appealing. It came as a great disappointment, therefore, when
the astronomer Johannes Kepler\index{Kepler, Johannes} discovered, by the painstaking analysis of
precise observations, that orbits such as the moon's were actually
ellipses, not circles. This is the sort of thing that led the biologist
Huxley to say, ``The great tragedy of science is the slaying of a beautiful theory by an ugly fact.''
The lesson of the story, then, is that symmetries are important and beautiful,
but we can't decide which symmetries are right based only on common sense or
aesthetics; their validity has to be determined based on observations and
experiments.

<% marg(0) %>
<% fig(
     'swan-lake-symmetry',
     %q{In this scene from Swan Lake, the choreography has a symmetry with respect to left and right.}
   )
%>
\spacebetweenfigs
<% fig(
     'c-s-wu-with-beamline',
     %q{C.S.~Wu at Columbia University in 1963.}
   )
%>
<% end_marg %>

As a more modern example, consider the symmetry between right and left. For example,
we observe that a top spinning clockwise has exactly the same behavior as a top
spinning counterclockwise. This kind of observation led physicists to believe,
for hundreds of years, that the laws of physics were perfectly symmetric with
respect to right and left. This mirror symmetry appealed to physicists' common sense.
However, experiments by Chien-Shiung Wu\index{Wu, Chien-Shiung} et al. in 1957 showed that right-left
symmetry was violated in certain types of nuclear reactions.
Physicists were thus forced to change their opinions about what constituted
common sense.

<% end_sec() %>
<% begin_sec("Conservation of Mass",0) %>\index{mass!conservation of}\index{conservation!of mass}
We intuitively feel that matter shouldn't appear or disappear out of nowhere: that
the amount of matter should be a conserved quantity.
If that was to happen, then it seems as though atoms would have to be created or destroyed,
which doesn't happen in any physical processes that are familiar from everyday life, such
as chemical reactions. On the other hand, I've already cautioned you against believing that
a law of physics must be true just because it seems appealing. The laws of physics have
to be found by experiment, and there seem to be experiments that are exceptions
to the conservation of matter. A log weighs more than its
ashes. Did some matter simply disappear when the log was burned?

The French chemist Antoine-Laurent Lavoisier was the first scientist to realize\index{Lavoisier!Antoine-Laurent}
that there were no such exceptions. Lavoisier hypothesized that
when wood burns, for example, the supposed loss of weight is actually accounted for by the
escaping hot gases that the flames are made of.
Before Lavoisier, chemists had almost never
weighed their chemicals to quantify the amount of each substance that was undergoing
reactions.
They also didn't completely understand that gases were just another
state of matter, and hadn't tried performing reactions in sealed chambers to determine
whether gases were being consumed from or released into the air. For this they had at least one
practical excuse, which is that if you perform a gas-releasing reaction in a sealed chamber
with no room for expansion, you get an explosion! Lavoisier invented a balance that was capable
of measuring milligram masses, and figured out how to do reactions in an upside-down
bowl in a basin of water, so that the gases could expand by pushing out some of the water.
In one crucial experiment, Lavoisier heated a red mercury compound, which we would now
describe as mercury oxide (HgO), in such a sealed chamber.
A gas was produced (Lavoisier later named
it ``oxygen''), driving out some of the water, and the red compound was transformed into
silvery liquid mercury metal. The crucial point was that the total mass of the entire
apparatus was exactly the same before and after the reaction. Based on many observations
of this type, Lavoisier proposed a general law of nature, that matter is always conserved.

<% self_check('conservation',<<-'SELF_CHECK'
In ordinary speech, we say that you should ``conserve'' something, because if
you don't, pretty soon it will all be gone. How is this different from the meaning of
the term ``conservation'' in physics?
  SELF_CHECK
  ) %>

<% marg(150) %>

<%
  fig(
    'lavoisier',
    %q{%
        Portrait of Monsieur Lavoisier and His Wife,
        by Jacques-Louis David, 1788. Lavoisier invented the
        concept of conservation of mass. The husband is depicted with his scientific apparatus,
        while in the background
        on the left is the portfolio belonging to Madame Lavoisier, who is thought to have
        been a student of David's.
    }
  )
%>
<% end_marg %>

Although Lavoisier was an honest and energetic public official, he was caught up
in the Terror and sentenced to death in 1794. 
He requested a fifteen-day
delay of his execution so that he could complete some experiments that he thought might
be of value to the Republic. 
The judge, Coffinhal, infamously replied that ``the state has no need of scientists.''
As a scientific
experiment, Lavoisier decided to try to determine how long his consciousness would continue
after he was guillotined, by blinking his eyes for as long as possible. He blinked twelve times
after his head was chopped off. Ironically, Judge Coffinhal 
was himself executed only three months later, falling victim to the same chaos.


\begin{eg}{A stream of water}\label{eg:faucet}
The stream of water is fatter near the mouth of the faucet, and skinnier lower down. This can be understood
using conservation of mass. Since water is being neither created nor destroyed, the mass of the
water that leaves the faucet in one second must be the same as the amount that flows past a lower
point in the same time interval. The water speeds up as it falls, so the two quantities of water can
only be equal if the stream is narrower at the bottom.
\end{eg}
<% marg(100) %>
<%
  fig(
    'faucet',
    %q{%
      Example \ref{eg:faucet}.
    }
  )
%>
<% end_marg %>

Physicists are no different than plumbers or ballerinas in that they have a technical vocabulary
that allows them to make precise distinctions. A pipe isn't
just a pipe, it's a PVC pipe. A jump isn't just a jump, it's a grand jet\'{e}.
We need to be more precise now about what we really mean by ``the amount of matter,'' which is
what we're saying is conserved.
Since physics is a mathematical science, definitions in physics are usually definitions
of numbers, and we define these numbers \emph{operationally}. An operational definition
is one that spells out the steps  required in order to \emph{measure}
that quantity. For example, one way that an electrician knows that current and voltage are two different
things is that she knows she has to do completely different things in order to measure them
with a meter.

<% marg(0) %>
<%
  fig(
    'mass_on_spring',
    %q{%
      The time for one cycle of vibration is related to the object's 
                       mass.
    }
  )
%>%
                \spacebetweenfigs%
                %
<%
  fig(
    'bodymass',
    %q{%
      Astronaut Tam\-ara Jer\-nigan measures her mass
                      aboard the Space Shuttle. She is strapped into a chair attached to a spring,
                like the mass in figure \figref{mass_on_spring}. \photocredit{NASA}
    }
  )
%>%
        
<% end_marg %>%

If you ask a room full of  ordinary people to define what is meant by mass, they'll probably
propose a bunch of different, fuzzy ideas, and speak as if they all pretty much meant the same
thing: ``how much space it takes up,'' ``how much it weighs,'' ``how much matter is in it.''
Of these, the first two can be disposed of easily. If we were to define mass as a measure of how
much space an object occupied, then mass wouldn't be conserved when we squished a piece of foam
rubber. Although Lavoisier did use weight in his experiments, weight also won't quite work as
the ultimate, rigorous definition, because weight is a measure of how hard gravity pulls on
an object, and gravity varies in strength from place to place. Gravity is measurably weaker on the top
of a mountain that at sea level, and much weaker on the moon. The reason this didn't matter to
Lavoisier was that he was doing all his experiments in one location. The third proposal
is better, but how exactly should we define ``how much matter?'' To make it into an operational
definition, we could do something like figure \figref{mass_on_spring}. 
A larger mass is harder to
whip back and forth --- it's harder to set into motion, and harder to stop once it's started.
For this reason, the vibration of the mass on the spring will take a longer time if the mass is
greater. If we put two different masses on the spring, and they both take the same time to complete
one oscillation, we can define them as having the same mass.

Since I started this chapter by highlighting the relationship between conservation laws and
symmetries, you're probably wondering what symmetry is related to conservation of mass. I'll
come back to that at the end of the chapter.

When you learn about a new physical quantity, such as mass, you need to know what units are used
to measure it. This will lead us to a brief digression on the metric system, after which we'll
come back to physics.

<% end_sec() %>
<% begin_sec("Review of the Metric System and Conversions",0) %>

<% begin_sec("The Metric System") %>

Every country in the world besides the U.S. has adopted a
system of units known colloquially as the ``metric system.''
Even in the U.S., the system is used universally by scientists, and also by many engineers.
This system is entirely decimal, thanks to the same
eminently logical people who brought about the \index{French
Revolution}French Revolution. In deference to France, the
system's official name is the Syst\`{e}me International, or SI,
meaning International System. (The phrase ``SI system'' is
therefore redundant.)\index{Syst\`{e}me International}

The metric system works with a
single, consistent set of \index{metric system!prefixes}\index{mega-
(metric prefix)}\index{kilo- (metric prefix)}\index{centi-
(metric prefix)}\index{milli- (metric prefix)}\index{micro-
(metric prefix)}\index{nano- (metric prefix)}prefixes
(derived from Greek) that modify the basic units. Each
prefix stands for a power of ten, and has an abbreviation
that can be combined with the symbol for the unit. For
instance, the meter is a unit of distance. The prefix kilo-
stands for $1000$, so a kilometer, 1 km, is a thousand meters.

In this book, we'll be using a flavor of the metric system, the SI, in which there are three basic units,
measuring distance, time, and mass. The basic unit of distance is the meter (m), the one for time
is the second (s), and for mass the kilogram (kg). Based on these units, we can define others,
e.g., m/s (meters per second) for the speed of a car, or kg/s for the rate at which water flows
through a pipe. It might seem odd that we consider the basic
unit of mass to be the kilogram, rather than the gram. The reason for doing this is that when we
start defining other units starting from the basic three, some of them come out to be a more
convenient size for use in everyday life. For example, there is a metric unit of force, the newton (N),
which is defined as the push or pull that would be able to change a 1-kg object's velocity by 1 m/s,
if it acted on it for 1 s. A newton turns out to be about the amount of force you'd use to pick up
your keys. If the system had been based on the gram instead of the kilogram, then the newton would
have been a thousand times smaller, something like the amount of force required in order to pick
up a breadcrumb.

The following are the most common metric prefixes. You
should memorize them.

\begin{tabular}{llllp{60mm}}
    \multicolumn{2}{c}{prefix} & meaning & \multicolumn{2}{c}{example} \\
    kilo-  &   k   & $1000$     &   60 kg     & =  a person's mass  \\
    centi- &   c   & $1/100$  &   28 cm     & =  height of a piece of paper  \\
    milli- &   m   & $1/1000$  &   1 ms     & =  time for one vibration of a guitar string playing the note D
\end{tabular}

The prefix centi-, meaning $1/100$, is only used in the
centimeter; a hundredth of a gram would not be written as 1
cg but as 10 mg. The centi- prefix can be easily remembered
because a cent is $1/100$ of a  dollar. The official SI
abbreviation for seconds is ``s'' (not ``sec'') and grams
are ``g'' (not ``gm'').

You may also encounter the prefixes mega- (a million) and
micro- (one millionth).

<% end_sec() %>
<% begin_sec("Scientific Notation",nil,'scientific-notation') %>\index{scientific notation}

Most of the interesting phenomena in our universe
are not on the human scale. It would take about 1,000,000,000,000,000,000,000
bacteria to equal the mass of a human body. When the
physicist Thomas Young discovered that light was a wave, scientific
notation hadn't been invented, and
he was obliged to write that the time required for one
vibration of the wave was 1/500 of a millionth of a
millionth of a second. Scientific notation is a less awkward
way to write very large and very small numbers such as
these.  Here's a quick review.

Scientific notation means writing a number in terms of a
product of something from 1 to 10 and something else that is
a power of ten. For instance,
\begin{align*}
& 32 = 3.2 \times 10^1\\
& 320 =  3.2 \times 10^2\\
& 3200 = 3.2 \times 10^3  \quad\ldots
\end{align*}
Each number is ten times bigger than the last.

Since $10^1$  is ten times smaller than $10^2$ , it makes
sense to use the notation $10^0$  to stand for one, the
number that is in turn ten times smaller than $10^1$ .
Continuing on, we can write $10^{-1}$  to stand for 0.1, the
number ten times smaller than $10^0$ . Negative exponents
are used for small numbers:

\begin{align*}
&3.2 =  3.2 \times 10^0\\
&0.32 = 3.2 \times 10^{-1}\\
&0.032 = 3.2 \times 10^{-2} \quad\ldots
\end{align*}

A common source of confusion is the notation used on the
displays of many calculators. Examples:

\vspace{8mm}

\hspace{10mm}\begin{tabular}{ll}
$3.2 \times 10^6$  &     (written notation)\\
3.2E+6             &     (notation on some calculators)\\
$3.2^6$            &     (notation on some other calculators)
\end{tabular}

\vspace{8mm}

\noindent The last example is particularly unfortunate, because
$3.2^6$ really stands for the number 
$3.2 \times 3.2 \times 3.2 \times 3.2 \times 3.2 \times 3.2= 1074$, a totally different number from $3.2 \times 10^6=3200000$.
The calculator notation should never be used in writing.
It's just a way for the manufacturer to save money by
making a simpler display.

<% self_check('bacteria-queue',<<-'SELF_CHECK'
A student learns that $10^4$  bacteria, standing in line to
register for classes at Paramecium Community College, would
form a queue of this size:

\anonymousinlinefig{sc-bacteria-queue-1}

\noindent The student concludes that $10^2$  bacteria would form a
line of this length:

\anonymousinlinefig{sc-bacteria-queue-2}

\noindent Why is the student incorrect?
  SELF_CHECK
  ) %>

<% end_sec() %>
<% begin_sec("Conversions") %>\index{conversions of units}\index{units, conversion of}

I suggest you avoid memorizing lots of conversion factors
between SI units and U.S. units. Suppose the United Nations
sends its black helicopters to invade California (after all
who wouldn't rather live here than in New York City?), and
institutes water fluoridation and the SI, making the use of
inches and pounds into a crime punishable by death. I think
you could get by with only two mental conversion factors:

\begin{indentedblock}
\noindent{}1 inch =  2.54 cm

\noindent An object with a weight on Earth of 2.2 pounds-force has a mass of 1 kg.
\end{indentedblock}

\noindent The first one is the present definition of the inch, so it's
exact. The second one is not exact, but is good enough for
most purposes. (U.S. units of force and mass are confusing,
so it's a good thing they're not used in science. In U.S.
units, the unit of force is the pound-force, and the best
unit to use for mass is the slug, which is about 14.6 kg.)

More important than memorizing conversion factors is
understanding the right method for doing conversions. Even
within the SI, you may need to convert, say, from grams to
kilograms. Different people have different ways of thinking
about conversions, but the method I'll describe here is
systematic and easy to understand. The idea is that if 1 kg
and 1000 g represent the same mass, then we can consider a fraction like
\begin{equation*}
  \frac{10^3\ \gunit}{1\ \kgunit}
\end{equation*}
to be a way of expressing the number one. This may bother
you. For instance, if you type 1000/1 into your calculator,
you will get 1000, not one. Again, different people have
different ways of thinking about it, but the justification
is that it helps us to do conversions, and it works! Now if
we want to convert 0.7 kg to units of grams, we can multiply
kg by the number one:
\begin{equation*}
  0.7\ \kgunit \times \frac{10^3\ \gunit}{1\ \kgunit}
\end{equation*}
If you're willing to treat symbols such as ``kg'' as if they
were variables as used in algebra (which they're really
not), you can then cancel the kg on top with the kg on the
bottom, resulting in
\begin{equation*}
  0.7\ \cancel{\kgunit} \times \frac{10^3\ \gunit}{1\ \cancel{\kgunit}}  = 700\ \gunit   \qquad   .
\end{equation*}
To convert grams to kilograms, you would simply flip the
fraction upside down.

One advantage of this method is that it can easily be
applied to a series of conversions. For instance, to convert
one year to units of seconds,

\begin{multline*}
1\ \cancel{\text{year}} \times
\frac{365\ \cancel{\text{days}}}{1\ \cancel{\text{year}}} \times
\frac{24\ \cancel{\text{hours}}}{1\ \cancel{\text{day}}} \times
\frac{60\ \cancel{\text{min}}}{1\ \cancel{\text{hour}}} \times
\frac{60\ \sunit}{1\ \cancel{\text{min}}} = \\
= 3.15 \times 10^7\ \sunit   \qquad   .
\end{multline*}

<% begin_sec("Should that exponent be positive or negative?") %>

A common mistake is to write the conversion fraction
incorrectly. For instance the fraction
\begin{equation*}
  \frac{10^3\ \kgunit}{1\ \gunit} \qquad \text{(incorrect)}
\end{equation*}
does not equal one, because $10^3$  kg is the mass of a car,
and 1 g is the mass of a raisin. One correct way of
setting up the conversion factor would be
\begin{equation*}
  \frac{10^{-3}\ \kgunit}{1\ \gunit} \qquad \text{(correct)} \qquad .
\end{equation*}
You can usually detect such a mistake if you take the time
to check your answer and see if it is reasonable.

If common sense doesn't rule out either a positive or a
negative exponent, here's another way to make sure you get
it right. There are big prefixes, like kilo-, and small ones,
like milli-.
In the example above, we want the
top of the fraction to be the same as the bottom. Since $k$
is a big prefix, we need to \emph{compensate} by putting a
small number like $10^{-3}$  in front of it, not a big
number like $10^3$.

\startdq

\begin{dq}
Each of the following conversions contains an error.  In
each case, explain what the error is.

(a) $1000\ \kgunit \times \frac{1\ \kgunit}{1000\ \gunit}  = 1\ \gunit$

(b) $50\ \munit \times \frac{1\ \zu{cm}}{100\ \munit}    =  0.5\ \zu{cm}$

\end{dq}


<% end_sec() %>
<% end_sec() %>
<% end_sec %>
<% begin_sec("Conservation of Energy",0) %>

<% begin_sec("Energy") %>

<% marg(10) %>
<%
  fig(
    'hockey-puck',
    %q{%
       A hockey puck is released at rest. If it spontaneously scooted off in some
       direction, that would violate the symmetry of all directions in space.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'joule',
    %q{%
       James Joule (1818-1889) discovered the law of conservation of energy.
    }
  )
%>
<% end_marg %>%

\label{rotational-symmetry}\index{symmetry!rotational}\index{rotational symmetry}
Consider the hockey puck in figure \figref{hockey-puck}. If we release it at rest, we expect
it to remain at rest. If it did start moving all by itself, that would be strange: it would
have to pick some direction in which to move, and why would it pick that direction rather
than some other one? If we observed such a phenomenon, we would have to conclude that that
direction in space was somehow special. It would be the favored direction in which hockey
pucks (and presumably other objects as well) preferred to move. That would violate our
intuition about the symmetry of space, and this is a case where our intuition is right:
a vast number of experiments have all shown that that symmetry is a correct one. In other
words, if you secretly pick up the physics laboratory with a crane, and spin it around
gently with all the physicists inside, all their experiments will still come out the same,
regardless of the lab's new orientation. If they don't have windows they can look out of,
or any other external cues (like the Earth's magnetic field), then they won't notice anything
until they hang up their lab coats for the evening and walk out into the parking lot.

Another way of thinking about it is that a moving hockey puck would have some
\emph{energy}, whereas a stationary one has none. I haven't given you an operational
definition of energy yet, but we'll gradually start to build one up, and it will
end up fitting in pretty well with your general idea of what energy means from
everyday life. Regardless of the mathematical details of how you would actually
calculate the energy of a moving hockey puck, it makes sense that a puck
at rest has zero energy. It starts to look like energy is conserved. A puck that
initially has zero energy must continue to have zero energy, so it can't start
moving all by itself.

You might conclude from this discussion that we have a new example of Noether's
theorem: that the symmetry of space with respect to different directions must be
equivalent, in some mysterious way, to conservation of energy. Actually that's not
quite right, and the possible confusion is related to the fact that we're not
going to deal with the full, precise mathematical statement of Noether's theorem.
In fact, we'll see soon that conservation of energy is really more closely related
to a different symmetry, which is symmetry with respect to the passage of time.

<% end_sec() %>
<% begin_sec("The principle of inertia",nil,'principle-of-inertia') %>\index{inertia, principle of}

Now there's one very subtle thing about the example of the hockey puck, which wouldn't
occur to most people. If we stand on the ice and watch the puck, and we don't see it
moving, does that mean that it really is at rest in some absolute sense? Remember,
the planet earth spins once on its axis every 24 hours. At the latitude where I live,
this results in a speed of about 800 miles per hour, or something like 400 meters per
second. We could say, then that the puck wasn't really staying at rest. We could
say that it was really in motion at a speed of 400 m/s, and remained in motion at that
same speed. This may be inconsistent with our earlier description, but it is still
consistent with the same description of the laws of physics. Again, we don't need to
know the relevant formula for energy in order to believe that if the puck keeps the same
speed (and its mass also stays the same), it's maintaining the same energy.

In other words, we have two different \emph{frames of reference}, both equally valid.
The person standing on the ice measures all velocities relative to the ice, finds
that the puck maintained a velocity of zero, and says that energy was conserved.
The astronaut watching the scene from deep space might measure the velocities
relative to her own space station; in her frame of reference, the puck is moving
at 400 m/s, but energy is still conserved.

<% marg(50) %>
<%
  fig(
    'aristotle',
    %q{%
      Why does Aristotle look so sad? Is it because he's realized that his entire
      system of physics is wrong?
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'jets-in-formation-over-ny',
    %q{The jets are at rest. The Empire State Building is moving.}
  )
%>
<% end_marg %>

This probably seems like common sense, but it wasn't common sense to one of the
smartest people ever to live, the ancient Greek philosopher Aristotle.\index{Aristotle}
He came up with an entire system of physics based on the premise that
there is one frame of reference that is special: the frame of reference defined
by the dirt under our feet. He believed that all motion had a tendency to slow
down unless a force was present to maintain it. Today, we know that Aristotle
was wrong. One thing he was missing was that he didn't understand the concept
of friction as a force. If you kick a soccer ball, the reason it eventually comes
to rest on the grass isn't that it ``naturally'' wants to stop moving. The reason
is that there's a frictional force from the grass that is slowing it down. (The energy
of the ball's motion is transformed into other forms, such as heat and sound.)
Modern people may also have an easier time seeing his mistake, because we
have experience with smooth motion at high speeds. For instance, consider a passenger
on a jet plane who stands up in the aisle and inadvertently drops his bag of peanuts.
According to Aristotle, the bag would naturally slow to a stop, so it would become
a life-threatening projectile in the cabin! From the modern point of view,
the cabin can just as well be considered to be at rest.\index{Galileo Galilei}
<%
  fig(
    'galileo-trial',
    %q{Galileo Galilei was the first physicist to state the principle of inertia (in a somewhat different formulation than the one given here).
       His contradiction of Aristotle had serious consequences. He was interrogated by the Church authorities and
       convicted of teaching that the earth went around the sun as a matter of fact and not, as he had promised previously,
       as a mere mathematical hypothesis. He was placed under permanent house arrest, and forbidden to write about or
       teach his theories. Immediately after being forced to recant his claim that the earth revolved around the sun, the old
       man is said to have muttered defiantly ``and yet it does move.''},
    {
      'width'=>'wide',
      'narrowfigwidecaption'=>true,
      'float'=>false
    }
  )
%>

The \emph{principle of inertia} says, roughly, that
all frames of reference are equally valid:

\begin{important}[The principle of inertia]
The results of experiments don't depend on the straight-line,
constant-speed motion of the apparatus. 
\end{important}

\noindent Speaking slightly more precisely, the
principle of inertia says that if frame B moves at constant speed, in a straight line,
relative to frame A, then frame B is just as valid as frame A, and in fact an observer
in frame B will consider B to be at rest, and A to be moving. The laws of physics
will be valid in both frames. The necessity for the
more precise formulation becomes evident if you think about examples in which the
motion changes its speed or direction. For instance, if you're in a car that's
accelerating from rest, you feel yourself being pressed back into your seat.
That's very different from the experience of being in a car cruising at constant
speed, which produces no physical sensation at all. A more extreme example of this
is shown in figure \figref{rocket-sled} on page  \pageref{fig:rocket-sled}.

A frame of reference moving at constant speed in a straight line is known as an
inertial frame of reference. A frame that changes its speed or direction of motion
is called noninertial. The principle of inertia applies only to inertial frames.
The frame of reference defined by an accelerating car is noninertial, but the one
defined by a car cruising at constant speed in a straight line is inertial.\index{frame of reference}\index{frame of reference!inertial}\index{frame of reference!noninertial}

<%
  fig(
    'rocket-sled',
    %q{%
      This Air Force doctor volunteered to ride a rocket sled as 
      a medical experiment. The obvious effects on
      his head and face are not because of the sled's speed but because of its rapid changes in speed: increasing
      in 2 and 3, and decreasing in 5 and 6.
      In 4 his speed is greatest, but because his speed is not
      increasing or decreasing very much at this moment, there is little effect on him.
    },
    {
      'width'=>'wide'
    }
  )
%>

<% marg(0) %>
<%
  fig(
    'foucault',
    %q{%
      Foucault demonstrates his pendulum to an audience at
                      a lecture in 1851.
    }
  )
%>
<% end_marg %>



\begin{eg}{Foucault's pendulum}
        Earlier, I spoke as if a frame of reference attached to the surface of the rotating earth was
        just as good as any other frame of reference. Now, with the more exact formulation of the
        principle of inertia, we can see that that isn't quite true. A point on the earth's surface
        moves in a circle, whereas the principle of inertia refers only to motion in a straight line.
        However, the curve of the motion is so gentle that under
        ordinary conditions we don't notice that the local dirt's frame of reference isn't
        quite inertial. The first demonstration of the noninertial nature of the earth-fixed
        frame of reference was by L\'{e}on Foucault\index{Foucault, L\'{e}on}
         using a very massive pendulum (figure \figref{foucault}) whose oscillations
        would persist for many hours without becoming imperceptible. Although Foucault did
        his demonstration in Paris, it's easier to imagine what would happen at the north pole:
        the pendulum would keep swinging in the same plane, but the earth would spin underneath
        it once every 24 hours. To someone standing in the snow, it would appear that the
        pendulum's plane of motion was twisting. The effect at latitudes less than 90
        degrees turns out to be slower, but otherwise similar. The Foucault pendulum was
        the first definitive experimental proof that the earth really did spin on its axis,
        although scientists had been convinced of its rotation for a century based on more
        indirect evidence about the structure of the solar system.
\end{eg}

People have a strong intuitive belief that there is a state of absolute rest,\index{Copernicus}
and that the earth's surface defines it. But Copernicus proposed as a mathematical
assumption, and Galileo argued as a matter of physical reality, that the earth spins
on its axis, and also circles the sun. Galileo's opponents objected that this was
impossible, because we would observe the effects of the motion. They said, for example,
that if the earth was moving, then you would never be able to jump up in the air and
land in the same place again --- the earth would have moved out from under you.
Galileo realized that this wasn't really an argument about the earth's motion but
about physics. In one of his books, which were written in the form of dialogues, he has
the three characters debate what would happen if a ship was cruising smoothly across
a calm harbor and a sailor climbed up to the top of its mast and dropped a rock.
Would it hit the deck at the base of the mast, or behind it because the ship had moved out from
under it? This is the kind of experiment referred to in the principle of
inertia, and Galileo knew that it would come out the same regardless of the ship's
motion. His opponents' reasoning, as represented by the dialog's stupid character
Simplicio, was based on the assumption that once the rock lost contact with the sailor's
hand, it would naturally start to lose its forward motion. In other words, they didn't
even believe in the idea that motion naturally continues unless a force acts to stop it.

But the principle of inertia says more than that. It says that motion isn't even real:
to a sailor standing on the deck of the ship, the deck and the masts and the rigging are not even
moving. People on the shore can tell him that the ship and his own body are moving in a straight
line at constant speed. He can reply, ``No, that's an illusion. I'm at rest. The only reason you think I'm moving is
because you and the sand and the water are moving in the opposite direction.''
The principle of inertia says that straight-line, constant-speed motion is a matter of opinion.
Thus things can't ``naturally'' slow down and stop moving, because we can't even agree on which things
are moving and which are at rest.

If observers in different frames of reference disagree on velocities, it's natural to
want to be able to convert back and forth. For motion in one dimension, this can be
done by simple addition.\index{velocity!addition of}

\begin{eg}{A sailor running on the deck}\label{eg:runtobow}
\egquestion
A sailor is running toward the front of a ship, and the other sailors say that in
their frame of reference, fixed to the deck, his velocity is $7.0$ m/s. The ship
is moving at $1.3$ m/s relative to the shore. How fast does an observer on the beach
say the sailor is moving?

\eganswer
They see the ship moving at $7.0$ m/s, and the sailor moving even faster than that because
he's running from the stern to the bow. In one second, the ship moves $1.3$ meters,
but he moves $1.3+7.0$ m, so his velocity relative to the beach is $8.3$ m/s. 
\end{eg}

The only way to make this rule give consistent results is if we define velocities in
one direction as positive, and velocities in the opposite direction as negative.

\begin{eg}{Running back toward the stern}
\egquestion
The sailor of example \ref{eg:runtobow} turns around and runs back toward the stern
at the same speed relative to the deck. How do the other sailors describe this
velocity mathematically, and what do observers on the beach say?

\eganswer
Since the other sailors described his original velocity as positive, they have to call
this negative. They say his velocity is now $-7.0$ m/s. A person on the shore says
his velocity is $1.3+(-7.0)=-5.7$ m/s.
\end{eg}



<% end_sec() %>
<% begin_sec("Kinetic and gravitational energy") %>\index{energy!kinetic}\index{energy!gravitational}\index{kinetic energy}\index{gravitational energy}

Now suppose we drop a rock. The rock is initially at rest, but then begins moving.
This seems to be a violation of conservation of energy, because a moving rock
would have more energy. But actually this is a little like the example of the
burning log that seems to violate conservation of mass. Lavoisier realized that
there was a second form of mass, the mass of the smoke, that wasn't being accounted
for, and proved by experiments that mass \emph{was}, after all, conserved once the
second form had been taken into account. In the case of the falling rock, we have two
forms of energy. The first is the energy it has because it's moving, known as
\emph{kinetic energy}. The second form is a kind of energy that it has because it's
interacting with the planet earth via gravity. This is known as \emph{gravitational
energy}.\footnote{You may also see this referred to in some books as gravitational potential energy.}
The earth and the rock attract each other gravitationally, and the greater the distance
between them, the greater the gravitational energy --- it's a little like stretching a spring.

<% marg(70) %>
<%
  fig(
    'pool-skater',
    %q{%
      The skater has converted all his kinetic
      energy into gravitational energy on the
      way up the side of the pool.
      Photo by J.D. Rogge,\linebreak[4]www.sonic.net/$\sim$shawn.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'pool-skater-line-art',
    %q{%
      As the skater free-falls, his gravitational energy is
      converted into kinetic energy.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'seesaw',
    %q{Example \ref{eg:lever}.}
  )
%>
<% end_marg %>

The SI unit of energy is the joule (J),\index{joule (unit)} and in those units, we find that lifting a 1-kg
mass through a height of 1 m requires 9.8 J of energy. This number, 9.8 joules per meter
per kilogram, is a measure of the strength of the earth's gravity near its surface.
We notate this number, known as the gravitational field, as $g$, and often round it off to 10 for convenience in rough calculations.
If you lift a 1-kg rock to a height of 1 m above the ground, you're giving up 9.8 J of the energy you got from eating
food, and changing it into gravitational energy stored in the rock. If you then release
the rock, it starts transforming the energy into kinetic energy, until finally when the
rock is just about to hit the ground, all of that energy is in the form of kinetic energy. That
kinetic energy is then transformed into heat and sound when the rock hits the ground.\label{gravitational-field}\label{gravitational field}\index{field!gravitational}

Stated in the language of algebra, the formula for gravitational energy is
\begin{equation*}
  GE = mgh \qquad ,
\end{equation*}
where $m$ is the mass of an object, $g$ is the gravitational field, and $h$ is the
object's height.

\begin{eg}{A lever}\label{eg:lever}\index{lever}
Figure \figref{seesaw} shows two sisters on a seesaw. The one on the left has twice as
much mass, but she's at half the distance from the center. No energy input is needed in
order to tip the seesaw. If the girl on the left goes up a certain distance, her gravitational
energy will increase. At the same time, her sister on the right will drop twice the distance,
which results in an equal decrease in energy, since her mass is half as much. In symbols, we have
\begin{equation*}
  (2m)gh
\end{equation*}
for the gravitational energy gained by the girl on the left, and
\begin{equation*}
  mg(2h)
\end{equation*}
for the energy lost by the one on the right. Both of these equal $2mgh$, so the amounts gained and
lost are the same, and energy is conserved.

Looking at it another way, this can be thought of as an example of the kind of experiment that
you'd have to do in order to arrive at the equation $GE=mgh$ in the first place. If we didn't
already know the equation, this experiment would make us suspect that it involved the product
$mh$, since that's what's the same for both girls.
\end{eg}

Once we have an equation for one form of energy, we can establish equations for other forms of
energy. For example, if we drop a rock and measure its final velocity, $v$, when it hits the ground,
we know how much GE it lost, so we know that's how much KE it must have had when it was at that
final speed. Here are some imaginary results from such an experiment.

\begin{tabular}{|c|c|c|} \hline
\textbf{m} (kg) & \textbf{v} (m/s) & \textbf{energy} (J) \\ \hline
1.00 & 1.00 & 0.50 \\ \hline
1.00 & 2.00 & 2.00 \\ \hline
2.00 & 1.00 & 1.00 \\ \hline
\end{tabular}

Comparing the first line with the second, we see that
doubling the object's velocity doesn't just double its
energy, it quadruples it. If we compare the first and third
lines, however, we find that doubling the mass only doubles
the energy. This suggests that kinetic energy is proportional
to mass times the square of velocity, $mv^2$, and further
experiments of this type would indeed establish such a
general rule. The proportionality factor equals 0.5 because
of the design of the metric system, so the kinetic energy of
a moving object is given by
\begin{equation*}
                KE    =    \frac{1}{2}mv^2   \qquad   .
\end{equation*}

<%
  fig(
    'can-imploding',
    %q{%
      A vivid demonstration that heat is a form of
      motion. A small amount of boiling water is poured
      into the empty can, which rapidly fills up with
      hot steam. The can is then sealed tightly, and
      soon crumples. This can be explained as
      follows. The high temperature of the steam is
      interpreted as a high average speed of random
      motions of its molecules. Before the lid was put
      on the can, the rapidly moving steam molecules
      pushed their way out of the can, forcing the
      slower air molecules out of the way. As the steam
      inside the can thinned out, a stable situation was
      soon achieved, in which the force from the less
      dense steam molecules moving at high speed
      balanced against the force from the more dense
      but slower air molecules outside. The cap was
      put on, and after a while the steam inside the
      can reached the same temperature as the air outside.
      The force from the cool,
      thin steam no longer matched the force from the
      cool, dense air outside, and the imbalance of
      forces crushed the can.
    },
    {
      'width'=>'wide',
    }
  )
%>

<% end_sec() %>
<% begin_sec("Energy in general") %>

<% marg(0) %>
  % not sure why the special label is necessary in the following, but it is; if I just do "coin," I get a different ref
<%
  fig(
    'coin',
    %q{%
      The spinning coin slows down. It looks like conservation of energy is violated, but it isn't.\label{fig:coin-duh}
    }
  )
%>
<% end_marg %>

By this point, I've casually mentioned several forms of energy: kinetic, gravitational, heat, and
sound. This might be disconcerting, since we can get throughly messed up if don't realize that
a certain form of energy is important in a particular situation. For instance, the spinning coin in figure
\figref{coin-duh} gradually loses its kinetic energy, and we might think that conservation of energy was
therefore being violated. However, whenever two surfaces rub together, friction acts to create heat.
The correct analysis is that the coin's kinetic energy is gradually converted into heat.\index{energy!heat}\index{energy!sound}\index{heat}\index{sound!energy}


One way of
making the proliferation of forms of energy seem less scary is to realize that many forms of energy
that seem different on the surface are in fact the same. One important example is that heat is actually
the kinetic energy of molecules in random motion, so where we thought we had two forms of energy, in fact
there is only one.
Sound is also a form of kinetic energy: it's the vibration of air molecules.

 This kind of unification of different types of energy has been a process that has
been going on in physics for a long time, and at this point we've gotten it down the point where there
really only appear to be four forms of energy:
\begin{enumerate}
\item kinetic energy
\item gravitational energy
\item electrical energy
\item nuclear energy
\end{enumerate}
We don't even encounter  nuclear energy in everyday life (except in the sense that sunlight
originates as nuclear energy), so really for most purposes the list only has three items on it. Of these
three, electrical energy is the only form that we haven't talked about yet. The interactions
between atoms are all electrical, so this form of energy is what's responsible for all of chemistry. The
energy in the food you eat, or in a tank of gasoline, are forms of electrical energy.\index{energy!electrical}\index{energy!nuclear}\index{electrical energy}\index{nuclear energy}

\begin{eg}{You take the high road and I'll take the low road.}\label{eg:high-road-low-road}
\egquestion Figure \figref{high-road-low-road} shows two ramps which two balls will
roll down. Compare their final speeds, when they reach point
B. Assume friction is negligible.

\eganswer Each ball loses some gravitational energy because of its
decreasing height above the earth, and conservation of
energy says that it must gain an equal amount of kinetic energy
(minus a little heat created by friction). The balls
lose the same amount of height, so their final speeds must be equal.
\end{eg}

\vspace{8mm}

<% marg(20) %>
<%
  fig(
    'high-road-low-road',
    %q{Example \ref{eg:high-road-low-road}.}
  )
%>
\spacebetweenfigs
<%
  fig(
    'orionnebula',
    %q{Example \ref{eg:birthofstars}.}
  )
%>
<% end_marg %>


\begin{eg}{The birth of stars}\label{eg:birthofstars}\index{Orion Nebula}
Orion is the easiest constellation to find. You can see it in the winter, even if you live
under the light-polluted skies of a big city. Figure \figref{orionnebula} shows an interesting
feature of this part of the sky that you can easily pick out with an ordinary camera (that's how
I took the picture) or a pair of binoculars. The three stars at the top are Orion's belt, and the
stuff near the lower left corner of the picture is known as his sword --- to the naked eye, it
just looks like three more stars that aren't as bright as the stars in the belt. The middle ``star''
of the sword, however, isn't a star at all. It's a cloud of gas, known as the Orion Nebula,
that's in the process of collapsing
due to gravity. Like the pool skater on his way down, the gas is losing gravitational energy.
The results are very different, however. The skateboard is designed to be a low-friction device,
so nearly all  of the lost gravitational energy is converted to kinetic energy, and very little
to heat. The gases in the nebula flow and rub against each other, however, so most of the gravitational
energy is converted to heat. This is the process by which stars are born: eventually the core of
the gas cloud gets hot enough to ignite nuclear reactions.
\end{eg}

\vspace{8mm}

\begin{eg}{Lifting a weight}
\egquestion At the gym, you lift a mass of 40 kg through a height of 0.5 m. How much gravitational
energy is required? Where does this energy come from?

\eganswer 
The strength of the gravitational field is 10 joules per kilogram per meter, so after you lift the weight,
its gravitational energy will be greater by $10\times40\times0.5=200$ joules.

Energy is conserved, so
if the weight gains gravitational energy, something else somewhere in the universe must have lost some.
The energy that was used up was the energy in your body, which came from the food you'd eaten. This is what we refer to
as ``burning calories,'' since calories are the units normally used to describe the energy in food,
rather than metric units of joules.

In fact, your body uses up even more than 200 J of food energy, because it's not very efficient. The rest
of the energy goes into heat, which is why you'll need a shower after you work out. We can summarize this
as
\begin{equation*}
\text{food energy} \rightarrow \text{gravitational energy} + \text{heat} \qquad .
\end{equation*}
\end{eg}

\begin{eg}{Lowering a weight}
\egquestion After lifting the weight, you need to lower it again. What's happening in terms of energy?

\eganswer 
Your body isn't capable of accepting the energy and putting it back into storage. The gravitational
energy all goes into heat. (There's nothing
fundamental in the laws of physics that forbids this.
Electric cars can do it --- when you stop at a stop sign, the car's kinetic energy is absorbed
back into the battery, through a generator.) 
\end{eg}

<% marg(0) %>
<%
  fig(
    'welding',
    %q{%
      Example \ref{eg:absorb-and-emit-light}.
    }
  )
%>
\spacebetweenfigs
<%
  fig(
    'irbike',
    %q{%
      Example \ref{eg:absorb-and-emit-light}.
    }
  )
%>
<% end_marg %>

\begin{eg}{Absorption and emission of light}\label{eg:absorb-and-emit-light}
  Light has energy. Light can be absorbed by matter and transformed into heat, but the
  reverse is also possible: an object can glow, transforming some of its heat energy into
  light. Very hot objects, like a candle flame or a welding torch, will glow in the visible
  part of the spectrum, as in figure \figref{welding}.

  Objects at lower temperatures
  will also emit light, but in the infrared part of the spectrum, i.e., the part of the
  rainbow lying beyond the red end, which humans can't see.
  The photos in figure \figref{irbike} were taken using a camera that is sensitive to infrared light.
  The cyclist locked his rear brakes suddenly, and skidded to a stop. The kinetic energy of the bike
  and his body are rapidly transformed into heat by the friction between the tire and the floor.
  In the first panel, you can see the glow of the heated strip on the floor, and in the second panel,
  the heated part of the tire.
\end{eg}

\begin{eg}{Heavy objects don't fall faster}\index{free fall}
Stand up now, take off your shoe, and drop it alongside a much less massive object
such as a coin or the cap from your pen.

Did that surprise you? You found that they both hit the ground at the same time.
Aristotle wrote that heavier objects fall faster than
lighter\index{Aristotle}\index{Catholic Church}\index{Church!Catholic}
ones. He was wrong, but Europeans believed him for thousands of years, partly because
experiments weren't an accepted way of learning the truth, and partly because the
Catholic Church gave him its posthumous seal of approval as its official
philosopher.

Heavy objects and light objects have to fall the same way, because conservation
laws are additive --- we find the total energy of an object by adding up the energies
of all its atoms. If a single atom falls through a height of one meter, it loses a certain
amount of gravitational energy and gains a corresponding amount of kinetic energy.
Kinetic energy relates to speed, so that determines how fast it's moving at the end
of its one-meter drop. (The same reasoning could be applied to any point along the
way between zero meters and one.)

Now what if we stick two atoms together? The pair has double the mass, so the amount
of gravitational energy transformed into kinetic energy is twice as much. But twice
as much kinetic energy is exactly what we need if the pair of atoms is to have the same
speed as the single atom did. Continuing this train of thought, it doesn't matter how many
atoms an object contains; it will have the same speed as any other object after dropping
through the same height.
\end{eg}

<% end_sec() %>
<% end_sec() %>
<% begin_sec("Newton's Law of Gravity",0) %>\index{gravity!Newton's law of}
Why does the gravitational field on our planet have the particular value it does? 
For insight, let's compare with the strength of gravity elsewhere in the
universe:\index{Newton, Isaac!law of gravity}\index{gravitational field!Newton's law of gravity}
\index{field!gravitational!Newton's law of gravity}

\begin{tabular}{|p{50mm}|p{45mm}|}
\hline
location      & $g$ (joules per kg per m) \\
\hline
asteroid Vesta (surface) & 0.3 \\
earth's moon (surface)   & 1.6 \\
Mars (surface)           & 3.7 \\
earth (surface)          & 9.8 \\
Jupiter (cloud-tops)     & 26 \\
sun (visible surface)    & 270 \\
typical neutron star (surface) & $10^{12}$ \\
black hole (center)      & infinite according to some theories, on the
   order of $10^{52}$ according to others \\
\hline
\end{tabular}

A good comparison is Vesta versus a neutron star. They're roughly the same size, but they have
vastly different masses --- a teaspoonful of neutron star matter would weigh a million tons!
The different mass must be the reason for the vastly different gravitational fields. (The notation
$10^{12}$ means 1 followed by 12 zeroes.)
This makes sense, because gravity is an attraction between things that have mass.

The mass of an object, however, isn't the only thing that determines the strength of its
gravitational field, as demonstrated by the difference between the fields of the
sun and a neutron star, despite their similar masses.  The other variable that matters is
distance. Because a neutron star's mass is compressed into such a small space (comparable
to the size of a city), a point on its surface is within a fairly short distance from every
part of the star. If you visited the surface of the sun, however, you'd be millions of miles
away from most of its atoms.

As a less exotic example, if you travel from the seaport of
Gua\-ya\-quil, Ecuador, to the top of nearby Mt. Cotopaxi, you'll experience
a slight reduction in gravity, from 9.7806 to 9.7624 J/kg/m. This is because
you've gotten a little farther from the planet's mass. Such differences in the
strength of gravity between one location and another on the earth's surface were
first discovered because pendulum clocks that were correctly calibrated in one country
were found to run too fast or too slow when they were shipped to another location.

The general equation for an object's gravitational field was discovered by
Isaac Newton, by working backwards
from the observed motion of the planets:\footnote{Example \ref{eg:moonorbit} on page \pageref{eg:moonorbit} shows
the type of reasoning that Newton had to go through.}
\begin{equation*}
        g = \frac{GM}{d^2} \qquad ,
\end{equation*}
where $M$ is the mass of the object, $d$ is the distance from the object, and $G$ is
a constant that is the same everywhere in the universe. This is known as Newton's
law of gravity.\footnote{This is not the form
in which Newton originally wrote the equation.}
This type of relationship, in which an effect is inversely proportional to the
square of the distance from the object creating the effect, is known as an
inverse square law.\index{inverse-square law} For example, the intensity of the
light from a candle obeys an inverse square law, as discussed in subsection
\ref{subsec:inverse-square} on page \pageref{subsec:inverse-square}.

<% marg(0) %>
<%
  fig(
    'newton',
    'Isaac Newton (1642-1727)'
  )
%>
<% end_marg %>

<% self_check('mars-and-venus',<<-'SELF_CHECK'
Mars is about twice as far from the sun as Venus. Compare the strength of the sun's
gravitational field as experienced by Mars with the strength of the field
felt by Venus.
  SELF_CHECK
  ) %>

Newton's law of gravity really gives the field of an individual atom, and
the field of a many-atom object is the sum of the fields of the atoms.
Newton was able to prove mathematically that this scary sum has an unexpectedly
simple result in the case of a spherical object such as a planet: the result is
the same as if all the object's mass had been concentrated at its center.\index{gravitational constant, $G$}

Newton showed that his theory of gravity could explain the orbits of the planets, and
also finished the project begun by Galileo of driving a stake through the heart of
Aristotelian physics. His book on the motion of material objects, the \emph{Mathematical
Principles of Natural Philosophy},\index{Mathematical Principles of Natural Philosophy}\index{Principia Mathematica}
was uncontradicted by experiment for 200 years,
but his other main work, \emph{Optics},\index{Optics}
 was on the wrong track due to his conviction
that light was composed of particles rather than waves. He was an avid alchemist,
an embarrassing fact that modern scientists would like to forget. Newton was
on the winning side of the revolution that replaced King James II with William and Mary
of Orange, which led to a lucrative post running the English royal mint; he worked hard at
what could have been a sinecure, and took great
satisfaction from catching and executing counterfeiters. Newton's personal life was less
happy, as we'll see in chapter \ref{ch:electricity}.

\begin{eg}{Newton's apple}\label{eg:newtonsapple}\index{moon!gravitational field experienced by}\index{Newton, Isaac!apple myth}
A charming legend attested to by Newton's niece is that he first conceived of
gravity as a universal attraction after seeing an apple fall from a tree. He
wondered whether the force that made the apple fall was the same one that made the
moon circle the earth rather than flying off straight. Newton had astronomical data
that allowed him to calculate
that the gravitational field the moon experienced
from the earth was 1/3600 as strong as the field on the surface of the earth.\footnote{See example
\ref{eg:moonorbit} on page \pageref{eg:moonorbit}.}
(The moon has its own gravitational field, but that's not what we're talking about.)
The moon's distance from the earth is 60 times greater than the earth's radius,
so this fit perfectly with an inverse-square law: $60\times60=3600$.
\end{eg}

<% marg(40) %>
<%
  fig(
    'earth-moon-apple',
    'Example \ref{eg:newtonsapple}.'
  )
%>
<% end_marg %>

<% end_sec() %>
<% begin_sec("Noether's Theorem for Energy",0,'noether-energy') %>
\index{Noether's theorem!for energy}\index{energy!Noether's theorem}
Now we're ready for our first full-fledged example of Noether's theorem.
Conservation of energy is a law of physics, and Noether's theorem
says that the laws of physics come from symmetry. Specifically, Noether's
theorem says that every symmetry implies a conservation law. Conservation
of energy comes from a symmetry that we haven't even discussed yet, but one that
is simple and intuitively appealing: as time goes by, the universe doesn't
change the way it works. We'll call this time symmetry.\index{symmetry!time}\index{time symmetry}

We have strong evidence for time symmetry, because when
we see a distant galaxy through a telescope, we're seeing light that has taken
billions of years to get here. A telescope, then, is like a time machine. For all
we know, alien astronomers with advanced technology may be observing our planet
right now,\footnote{Our present technology isn't good enough to let us pick
the planets of other solar systems out from the glare of their suns, except in a few
exceptional cases.}
but if so, they're seeing it not as it is now but as it
was in the distant past, perhaps in the age of the dinosaurs, or before life
even evolved here. As we observe a particularly distant, and therefore ancient,
supernova, we see that its explosion plays out in exactly the same way as
those that are closer, and therefore more recent.

Now suppose physics really does change from year to year, like politics, pop music,
and hemlines. Imagine, for example, that the ``constant'' $G$ in Newton's
law of gravity isn't quite so constant. One day you might wake up and find that you've
lost a lot of weight without dieting or exercise, simply because gravity has gotten
weaker since the day before. 

If you know about such changes in $G$ over time, it's the ultimate insider
information. You can use it to get as rich as Croesus, or even Bill Gates.\index{Gates, Bill}
On a day when $G$ is low, you pay for the energy needed to lift
a large mass up high. Then, on a day when gravity is stronger,
you lower the mass back down, extracting its gravitational energy.
The key is that the energy you get back out is greater than what you
originally had to put in. You can run the cycle over and over again, always
raising the weight when gravity is weak, and lowering it when gravity is strong.
Each time, you make a profit in energy. Everyone else thinks energy is conserved,
but your secret technique allows you to keep on increasing and increasing the amount
of energy in the universe (and the amount of money in your bank account).

The scheme can be made to work if anything about physics changes over time, not
just gravity. For instance, suppose that the mass of an electron had one value
today, and a slightly different value tomorrow. Electrons are one of the basic
particles from which atoms are built, so on a day when the mass of electrons is low, every
physical object has a slightly lower mass. In problem \ref{hw:changing-electron-mass}
on page \pageref{hw:changing-electron-mass}, you'll work out a way that this could
be used to manufacture energy out of nowhere.\label{text:changing-electron-mass}

Sorry, but it won't work. Experiments show that $G$ doesn't change measurably over
time, nor does there seem to be any time variation in any of the other rules by which the universe
works.\footnote{In 2002, there have been some reports that the properties of atoms
as observed in distant galaxies are slightly different than those of atoms here and
now. If so, then time symmetry is weakly violated, and so is conservation
of energy. However, this is a revolutionary claim, and it needs to be examined carefully.
The change being claimed is large enough that, if it's real, it should be detectable
from one year to the next in ultra-high-precision laboratory experiments here on earth.}
If archaeologists find a
copy of this book thousands of years from now, they'll be able to reproduce all the
experiments you're doing in this course.

I've probably convinced you that if time symmetry was violated, then
conservation of energy wouldn't hold. But does it work the other way around? If time
symmetry is valid, must there be a law of conservation of energy? 
Logically, that's a different question. We may be able to
prove that if A is false, then B must be false, but that doesn't mean that if A is
true, B must be true as well.
For instance, if you're not a criminal, then you're presumably not in jail, but just
because someone is a criminal, that doesn't mean he is in jail
--- some criminals never get caught.

Noether's theorem does work the other way around as well: if physics has a certain
symmetry, then there must be a certain corresponding conservation law. This is
a stronger statement. The full-strength version of Noether's theorem can't
be proved without a  model of light and matter more detailed than the one
currently at our disposal.

<% end_sec() %>
<% begin_sec("Equivalence of Mass and Energy",0,'massenergy') %>

<% begin_sec("Mass-energy") %>\index{mass-energy}\index{mass!equivalence to energy}\index{energy!equivalence to mass}
You've encountered two conservation laws so far: conservation of mass and conservation
of energy. If conservation of energy is a consequence of symmetry, is there a
deeper reason for conservation of mass?

Actually they're not even separate conservation laws. Albert Einstein found,\index{Einstein, Albert}
as a consequence of his theory of relativity, that mass and energy are equivalent, and
are not separately conserved --- one can be converted into the other. Imagine that
a magician waves his wand, and changes a bowl of dirt into a bowl of lettuce. You'd be
impressed, because you were expecting that both dirt and lettuce would be conserved
quantities. Neither one can be made to vanish, or to appear out of thin air. However,
there are processes that can change one into the other. A farmer changes dirt into
lettuce, and a compost heap changes lettuce into dirt. At the most fundamental
level, lettuce and dirt aren't really different things at all; they're just collections
of the same kinds of atoms --- carbon, hydrogen, and so on.

We won't examine relativity in detail in this book, but mass-energy
equivalence is an inevitable implication of the theory, and it's the only part of the
theory that most people have heard of, via the famous equation $E=mc^2$. This equation
tells us how much energy is equivalent to how much mass: the conversion factor is the square
of the speed of light, $c$. Since $c$ a big number, you get a really really big number
when you multiply it by itself to get $c^2$. This means that even a small amount of mass
is equivalent to a very large amount of energy. 

<%
  fig(
    'eclipse',
    'Example \ref{eg:eclipse}.',
    {
      'width'=>'wide'
    }
  )
%>
\label{fig:eclipse}

\pagebreak

\begin{eg}{Gravity bending light}\label{eg:eclipse}
Gravity is a universal attraction between things that have mass, and since the energy
in a beam of light is equivalent to some very small amount of mass, we expect that
light will be affected by gravity, although the effect should be very small.
The first experimental confirmation of relativity
came in 1919 when stars next to the sun during a solar eclipse were
observed to have shifted a little from their ordinary
position. (If there was no eclipse, the glare of the sun
would prevent the stars from being observed.) Starlight had
been deflected by the sun's gravity. Figure \figref{eclipse} is a
photographic negative, so the circle that appears bright is actually the
dark face of the moon, and the dark area is really the bright corona of
the sun. The stars, marked by lines above and below them, appeared at
positions slightly different than their normal ones.
\end{eg}

<% marg(0) %>
<%
  fig(
    'newspaper-eclipse',
    'A New York Times headline from November 10, 1919, describing the observations discussed in example \ref{eg:eclipse}.'
  )
%>
\label{fig:newspaper-eclipse}
<% end_marg %>

\begin{eg}{Black holes}\index{black hole}
A star with sufficiently strong gravity can prevent light
from leaving. Quite a few black holes have been detected via
their gravitational forces on neighboring stars or clouds of gas and dust.
\end{eg}


Because mass and energy are like two different sides of the same coin, we may speak of
mass-energy, a single conserved quantity, found by adding up all the mass and energy,
with the appropriate conversion factor: $E+mc^2$.

\pagebreak

\begin{eg}{A rusting nail}\label{eg:rustingnail}
\egquestion
An iron nail is left in a cup of water
until it turns entirely to rust. The energy released is
about 500,000 joules. In theory, would a sufficiently
precise scale register a change in mass? If so, how much?

\eganswer
 The energy will appear as heat, which will be lost
to the environment. The total mass-energy of the cup,
water, and iron will indeed be lessened by 500,000 joules. (If it
had been perfectly insulated, there would have been no
change, since the heat energy would have been trapped in the
cup.) The speed of light in metric units is
$c=3\times10^8$ meters per second (scientific notation for
3 followed by 8 zeroes), so converting to mass units, we have
\begin{align*}
                m         &=    \frac{E}{c^2}  \\
                        &= \frac{500,000}{\left(3\times10^8\right)^2} \\
                         &=    0.000000000006\  \text{kilograms}   \qquad   .
\end{align*}
(The design of the metric system is based on the meter, the kilogram, and the
second. The joule is designed to fit into this system, so the result comes
out in units of kilograms.)
The change in mass is too small to measure with any
practical technique. This is because the square of the speed
of light is such a large number in metric units.
\end{eg}
%
<% end_sec() %>
<% begin_sec("The correspondence principle",nil,'correspondence-principle') %>\index{correspondence principle!defined}
The realization that mass and energy are not separately conserved is our first example
of a general idea called the correspondence principle. When Einstein came up with
relativity, conservation of energy had been accepted by physicists for decades, and
conservation of mass for
over a hundred years.\index{correspondence principle!for mass-energy equivalence}

Does an example like this mean that physicists don't know what they're talking about?
There is a recent tendency among social scientists to 
deny that the scientific method even exists, claiming that
science is no more than a social system that
determines what ideas to accept based on an in-group's
criteria. If science is an
arbitrary social ritual, it would seem difficult to explain
its effectiveness in building such useful items as
airplanes, CD players and sewers. If voodoo
and astrology were no less scientific in
their methods than chemistry and physics, what was it that
kept them from producing anything useful?
This silly attitude was effectively skewered in a famous hoax
carried out in 1996 by New York University physicist Alan Sokal.\index{Sokal, Alan} Sokal wrote
an article titled ``Transgressing the Boundaries: Toward a Transformative 
Hermeneutics of Quantum Gravity,'' and got it accepted by a cultural studies
journal called \emph{Social Text}.\footnote{The paper
appeared in \emph{Social Text} \#46/47 (1996) pp. 217-252. The full text
is available on Professor Sokal's web page at www.physics.nyu.edu/faculty/sokal/.}
The scientific content of the paper is a carefully constructed soup of
mumbo jumbo, using technical terms to create maximum confusion; I can't make
heads or tails of it, and I assume the editors and peer reviewers at
\emph{Social Text} understood even less. The physics, however, is mixed
in with cultural relativist statements designed to appeal to them ---
``\ldots the truth claims of science are inherently theory-laden and self-referential'' ---
and footnoted references to academic articles such as 
``Irigaray's and Hayles' exegeses of gender encoding in fluid mechanics \ldots
and \ldots Harding's comprehensive critique of the gender ideology underlying
the natural sciences in general and physics in particular\ldots''
On the day the article came out, Sokal published a letter explaining that
the whole thing had been a parody --- one that apparently went over the heads
of the editors of \emph{Social Text}.

What keeps physics from being merely a matter of fashion is that it has to agree
with experiments and observations. If a theory such as conservation of mass or
conservation of energy became accepted in physics, it was because it was supported
by a vast number of experiments. It's just that experiments never have perfect
accuracy, so a discrepancy such as the tiny change in the mass of the rusting nail
in example \ref{eg:rustingnail} was undetectable. The old experiments weren't all
wrong. They were right, within their limitations. If someone comes along with a
new theory he claims is better, it must still be consistent with all the same
experiments. In computer jargon, it must be backward-compatible. This is called
the correspondence principle: new theories must be compatible with old ones in
situations where they are both applicable. The correspondence principle tells us
that we can still use an old theory within the realm where it works, so for instance
I'll typically refer to conservation of mass and conservation of energy in this
book rather than conservation of mass-energy, except in cases where the new theory
is actually necessary.

Ironically, the extreme cultural relativists want to attack what they see
as physical scientists' arrogant claims to absolute truth, but what they
fail to understand is that science only claims to be able to find partial, provisional truth.
The correspondence principle tells us that each of today's scientific truths can be superseded
tomorrow by another truth that is more accurate and more broadly applicable. It also
tells us that today's truth will not lose any value when that happens.
<% end_sec() %>
<% end_sec() %>
 %===============================================================================
 %===============================================================================

<% begin_hw_sec %>

<% begin_hw('mg-to-kg') %>__incl(hw/mg-to-kg)<% end_hw() %>

<% begin_hw('units') %>__incl(hw/units)<% end_hw() %>

<% begin_hw('backyard') %>__incl(hw/backyard)<% end_hw() %>

<% begin_hw('furlongs') %> The speed of light is $3.0\times10^8$  m/s. Convert
this to furlongs per fortnight.  A furlong is 220 yards, and
a fortnight is 14 days.  An inch is 2.54 cm.\answercheck
<% end_hw() %>

<% begin_hw('micrograms') %> Express each of the following quantities in micrograms:\\
(a) 10 mg, (b) $10^4$ g, (c) 10 kg, (d) $100\times10^3$ g, (e) 1000 ng. \answercheck
<% end_hw() %>

<% begin_hw('estrogen') %>__incl(hw/estrogen)<% end_hw() %>

<% begin_hw('jumpkeandpe') %>__incl(hw/jumpkeandpe)<% end_hw() %>

<% begin_hw('throw-down-and-up') %>__incl(hw/throw-down-and-up)<% end_hw() %>

<% begin_hw('pulley') %>
(a) If weight B moves down by a certain amount, how much does weight A
move up or down?\hwendpart
(b) What should the ratio of the two weights be if they are to balance?
Explain in terms of conservation of energy.
<% end_hw() %>
<% marg(50) %>
<% fig('hw-pulley','Problem \ref{hw:pulley}.') %>
<% end_marg %>


<% begin_hw('slidingmagnets') %>
        (a) You release a magnet on a tabletop near a big piece
        of iron, and the magnet leaps across the table to the iron.
        Does the magnetic energy increase, or decrease?
        Explain.\hwendpart
        (b) Suppose instead that you have two repelling
        magnets. You give them an initial push towards each other,
        so they decelerate while approaching each other. Does the
        magnetic energy increase, or decrease? Explain.
<% end_hw() %>

<% begin_hw('spacesuit') %>__incl(hw/spacesuit)<% end_hw() %>


<%
  fig('hw-colliding-balls','Problem \ref{hw:colliding-balls}.',
    {
      'width'=>'wide',
    }
  )
%>
<% begin_hw('colliding-balls') %>
        The multiflash photograph below shows a collision
        between two pool balls. The ball that was initially at rest
        shows up as a dark image in its initial position, because
        its image was exposed several times before it was struck and
        began moving. By making measurements on the figure,
        determine whether or not energy appears to have been
        conserved in the collision. What systematic effects would
        limit the accuracy of your test? (From an example in PSSC
        Physics.)
<% end_hw() %>

<% begin_hw('rocketweight') %>
How high above the surface of the earth should a rocket be in order to
have 1/100 of its normal weight? Express your answer in units of earth radii.\answercheck
<% end_hw() %>

<% begin_hw('changing-electron-mass') %>__incl(hw/changing-electron-mass)<% end_hw() %>

<% begin_hw('rustingnailsmallc') %>__incl(hw/rustingnailsmallc)<% end_hw() %>

<% begin_hw('freeneutron') %>__incl(hw/freeneutron)<% end_hw() %>

<% begin_hw('drop-rock') %>__incl(hw/drop-rock)<% end_hw() %>

<% end_hw_sec %>
<% end_chapter() %>