Using Reason as a Lever

I've developed a deliberate habit of teaching without notes.  To be sure, I've got a good idea of what I'd like to cover that day and a mental list of examples, derivations, and group-work problems to fall back on, but I love it when we get sidetracked for a while discussing something I hadn't anticipated.

For example, one of the most wonderful things about science is the way seemingly simple and irrelevant experiments can actually reveal an understanding about something much deeper.  In my freshman physics class, we were winding up a section on harmonic (spring/elastic) motion and I was suggesting that there were many different kinds of physical situations that could be (however approximately) described by the same equations.  This is the real reason for spending so much time on springs --- not that they'll spend any significant time in their careers using actual springs, but that knowing how to analyze a simple system means that they may be able to use it as an approximation for very much more complicated systems.

It turns out that the motion of a simple pendulum can be approximated by the same equations for elastic systems.  Suppose you have a string and a weight tied to one end:

A mass \(M\) on a string of length \(\ell\).  The mass feels a tension \(T\) pulling at an angle \(\theta\)
with respect to the vertical.  At this moment it's a distance \(x\) from the vertical.

Here, I suppose that the vertical motion is small compared to the horizontal motion (which is about right for small oscillations).  So then in the vertical direction, Newton's 2nd Law gives
\[T\cos{\theta} = Mg\]
and in the horizontal direction
\[T\sin{\theta} = Ma\]
Now, if the oscillations really are small, then \(\cos{\theta} \approx 1\).  And from the diagram, \(\displaystyle \sin{\theta}=\frac{x}{\ell}\), so substituting,
\[a=-\frac{gx}{\ell}\]
(The negative sign appears since the displacement is to the left).  The left-hand side (acceleration) is recognized as the second derivative of position, so the position of the mass at any time is given by
\[\ddot{x} = -\frac{g}{\ell}x\]
The cool thing is that this is exactly the same equation we would have typically just derived for a spring:
\[\ddot{x} = -\frac{k}{m}x\]
just with different letters.  The period of oscillations is
\[T=\frac{2\pi}{\omega}=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{\ell}{g}}\]
The time for each swing does not depend on the mass attached, something noticed first by Galileo.  It only depends on the length of the string and the planet you're on.

The last comment was intended to be a bit tongue-in-cheek, but we decided to explore it a bit.  In a previous class, we found out that the gravitational acceleration \(g\) close to the Earth's surface is
\[g=\frac{GM_e}{R_e^2}=9.81\;\text{m/s}^2\]
Now, plugging this in for \(g\), and also substituting \(M_e=\rho V_e\) where \(\rho\) is the average density for the Earth and \(\displaystyle V_e=\frac{4}{3}\pi R_e^3\), then a little algebra finally leaves us with
\[\rho = \frac{3\pi \ell}{T^2 RG}\]
For the Earth, for example, it's been known from ancient times (see Eratosthenes' Experiment) that \(R_e \approx 6.4\times 10^6\) meters, and Newton's Gravitational Constant \(G=6.67\times 10^{-11}\;\frac{\text{N-m}^2}{\text{kg}^2}\).  If you set up your little pendulum, say with \(\ell=1\) meter and observe that it swings back and forth with a period of \(T=2\) seconds per swing, you've now just found out that the average density for the entire Earth is
\[\rho \approx 5,500\;\text{kg/m}^3\]
That's kind of a remarkable number if you think about it; you've now measured the density of everything on and in the Earth with a rock and a piece of string.  Further, it turns out that the density of water is \(1,000\;\text{kg/m}^3\), and rocks have a (very roughly) average density of something like \(3,000\;\text{kg/m}^3\).  But when walking around the surface of the Earth, you pretty much only encounter rock and water, so if the Earth was only composed of that stuff you'd expect the average density to be somewhere between those two numbers.  The fact that the measured number is quite a bit higher than that is very interesting.  A reasonable conclusion would be that the center of the Earth is made up of something far denser than ordinary rock (we now think, indeed, that the center of the Earth is largely iron/nickel, which has a density of something like \(9,000\;\text{kg/m}^3\)).

But how amazing!  Suppose you land on some planet for which you know the size (which doesn't sound too unreasonable, and we've already seen a direct way to measure it).  If you just pull out a rock and some string you can rightly say that, by measuring how long it takes the rock to swing back and forth, you're actually sampling the core of the planet!!  This sort of indirect experiment and reasoning is done in science all the time --- a common objection from someone unfamiliar with the equations might be "how can you know what the core is made of if you can't go down there and sample it?"  Of course, we don't know what the core is made of, but we can predict that it's likely something dense like iron, and predictions like these have consequences.  A spinning iron core is likely to produce an associated magnetic field, which we do more directly observe, and iron is predicted to be a very common element created by stars out of which to build rocky planets, so this (along with many other lines of seismic evidence) seems a very likely conclusion.  Illustrating this sort of relationship between the equations derived in class and some potentially wonderful application is, I think, terribly important not only for students to retain the formulas and techniques, but most importantly to build a real appreciation and respect for the process and chains of reasoning so important to the fundamental process of science.

Experiments of a similar spirit are done often.  One of the most profound is the Large Hadron Collider in Europe.  The idea is that by colliding small particles together at tremendous energies on small scales, we're building tiny laboratories at extremely high temperatures similar to the temperatures and energies existing at the earliest moments after the Big Bang.  Think of it --- on this tiny speck of a planet and only 500 generations removed from first figuring out how to plant crops and write, we can actually figure out how the Universe evolved from a trillionth of a second after it began to its present state.  It is an astonishing trophy for the process of reason.

TAU Chapter 3 --- Seasons and Phases

Seasons

Before investigating the consequences and orbits of our Sun-centered (heliocentric) system of major planets and minor objects, I'll describe a couple of very important local phenomena --- seasons and phases.

If you asked a large number of people why it was that we had seasons, and why (in the Northern Hemisphere) it was hot in July, the most common answer would probably be that we were closer to the Sun in July.  It's an answer that makes common sense; the Sun really does appear to be like a hot fire in the sky, and we all know from experience that we're warmer when we're closer to the fire.  This, however, is a case when a common-sense answer is the wrong one (understanding a flaw in a bad argument often makes a correct argument easier to remember).  If you were to assert "it's hot in July because we're close to the Sun then" to someone from Australia (or anywhere else in the Southern Hemisphere), you'd get a strange look --- July is their winter.  How could it be that it's hot in the Northern Hemisphere because it's close to the Sun, and cold in the Southern presumably because it's farther away?  We're all on the same planet, so the explanation makes no sense.  It is especially implausible if we appreciate the true scale of the size of the Earth and its real distance from the Sun.

True scaled sizes of the Sun and Earth (though not representative of our distance
from the Sun).

The Sun is 100 times larger in diameter than the Earth, so the true scale of the two objects looks like the image above.  That image, by the way, is only meant to represent the difference in sizes --- it does not indicate how far we are away from the Sun.  For that, look at the image below.

The scaled distance from the Sun to the Earth

Note that, if you look at the true distance scale, you can hardly even see the Earth unless you look at the full-resolution image--- it's barely a pixel!   Now it really doesn't make any sense to say that, at the same moment, one part of the tiny Earth is hot because it is near the Sun and the other half of the speck is cold because it is farther.  In fact, it turns out that our path around the Sun is an ellipse (not quite a circle), and the closest approach to the Sun actually occurs in January.

The actual "reason for the season" turns out to be our orbital tilt.  If you imagine the Earth turning around an axis running from the South to the North poles, that axis doesn't point straight up and down (relative to our orbit around the Sun) but rather is tilted by about \(23.5^{\circ}\).

The Earth's \(23.5^{\circ}\) tilt, bringing summer to the Northern Hemisphere.

As you can see from the image, this has a number of interesting effects.  While the Earth is in this position as it turns about its axis, notice that someone standing at the North Pole is always illuminated by the Sun during the day, but someone standing at the South Pole never sees it rise.  This happens as long as the person is inside the Arctic Circle (or Antarctic Circle in the south).  This particular orientation is called summer in the Northern Hemisphere.  The crucial point is that the direction of the tilt does not change as we go around the Sun.  Six months later, the Earth is halfway around the Sun, and so the axis, still oriented in the same direction, now points away from the sun, causing winter.  (Asterisk!  Actually, the direction of the axis does change a tiny bit.  The Earth wobbles a little, like a spinning top that's slowing down, so seen over thousands of years the axial arrow will make a little circle.  But it takes something like 26,000 years to wobble around once, so it's pretty accurate to say that during one year there's not much change in its direction.)

Northern Hemisphere in winter (left) and summer (right)  [not to scale].

Remember the above image does not remotely describe the relative sizes of the Earth and Sun, nor their distance from each other; it just shows the relative orientation of the Earth as it goes around the Sun.  Another interesting effect is that the red arrows show the direction someone would look if it were midnight and they looked straight up --- there is a completely different set of stars visible in that direction as opposed to what they would see 6 months later.  That's why you can only see some constellations in the winter in the Northern Hemisphere but not in the summer.  If the red arrow at the left side of the image is pointing towards the constellation of Orion, for example (which can be seen easily in the winter), then you should be able to understand why it's not visible in the summer.  It's still there, of course, but to see it you'd have to look towards the Sun (in the daytime).  

The path of sunlight in summer.

The path of sunlight in winter.

Now it's easy to see why it should be warmer in the summer --- if the pole is tilted towards the Sun, there are 2 effects:  (a) the day is longer; in fact, looking back at the earlier image, you can imagine what happens as you go farther towards the pole.  Summer days get longer and longer, and when you cross the Arctic Circle the Sun never sets; and (b), the sunlight hitting any particular place in the summertime is more direct and concentrated (as you can see in the images), impacting a smaller area than in winter.  In contrast, in winter, sunlight is spread over a larger area, and therefore weaker and less efficient at heating.  So in winter, not only is weaker sunlight falling on your area, but the days are shorter too, so there isn't as much time during the day for heating.

It's good to reflect on what the process of science really is --- too often it's taught as a series of facts to be memorized.  It's true that you certainly can just memorize that what causes the seasons is the Earth's orbital tilt.  Perhaps you can pass some simple quiz or test with such knowledge, but it's not the kind of thing that stays with you the rest of your life.  Instead, the valuable thing is to use and appreciate the power of arguments; by this I don't mean unpleasant verbal sparring, but rather the chain of reasoning that leads you to a reasonable conclusion.  Knowing why the tilt explains the seasons is much more wonderful (and easy to remember) that just recalling the fact.  Knowing why the first idea (it's summer because we're closer to the Sun) is a bad argument is almost equally as valuable; otherwise one may fall back on fuzzy thinking and lazy reasoning just out of convenience.  Having the ability to throw out a previously held idea because of the weight of new evidence is the mark of an educated, mature individual (and a scientist!)  It happens all the time -- I think one should always be prepared to throw out ideas and beliefs if later evidence shows them to be suspect.  

Moon Phases

Now let's try to understand why the Moon exhibits different phases.  Just like with the explanation of the seasons above, the key is appreciating the geometry and alignment between the Earth, Sun, and Moon.  That the Moon cycles through a repeating pattern of phases is a consequence of the following 3 simple ideas:

• The Moon does not emit its own light.  It does not glow --- it's essentially a giant rock in space. The only reason we see it at all is because sunlight bounces off of it and is reflected to us.
• The Moon orbits the Earth.  It takes about 27 days for the Moon to make one trip around the Earth (which is the origin of the word month).
• The Sun is farther away than the Moon.  This means that the Moon *always* comes between the Earth and Sun, and that we're often seeing the unlit "dark side" from behind during crescent phases.

It might be easiest to see the consequences of the geometry by looking at an illustration.  The image below does not represent the true sizes and distances of the objects, but is meant to show what happens when you look at a lit sphere (the Moon) that is lit by a distant source (the Sun)

The Earth-Moon-Sun system (not to scale).

Just to make the point, here's an illustration of the relative sizes and distance between the Earth and Moon (below).  The Moon is only about a quarter of the diameter of the Earth and about 30 Earths away.  Also, notice in the image that the orbit of the Moon is somewhat inclined (about 5 degrees) relative to the direction of the Sun.  This means that only very rarely are we in the Moon's shadow (a solar eclipse).  When the Moon is behind the Earth, it is possible for it to be in the Earth's shadow, which is a lunar eclipse.  The lunar eclipse is somewhat more likely, mostly because since the Earth is bigger than the Moon, it casts a bigger shadow.

Relative sizes and distance between Earth and Moon.

Now let's look at this system in motion.  I'll suppress the motion of the Earth around the Sun as well as our 24-hour spin as that can get a little confusing.  We'll see a top-down view running simultaneously with a changing aspect that will give you an idea what the phase looks like when the Moon is seen from the Earth.

You might have to play the movie a few times to get the hang of it (and pause it often while it's playing!) but you can start to see what's happening:  from above, you can see that half of the Moon's surface is always lit by the Sun as it goes around the Earth.  The different phases happen when we look at the half-illuminated Moon from different points of view.  For the new and crescent phases, we're looking at the Moon from behind, and we're between the Moon and Sun during the gibbous and full phases.  That's really about all there is to it.

Diagram showing the Moon phases (and timings) along with its position.

The above diagram shows the animations in static form --- the inner set of circles shows the half-lit surface of the Moon as it orbits the Earth, and the outer set shows what the Moon at that time would look like from the Earth.  In addition, I've indicated the time of day for someone standing on Earth's surface (noon if the Sun would appear directly overhead, midnight if the Sun is on the other side of the Earth).  Now you can predict when a certain phase would rise and set!  For example, according to the diagram, the 1st quarter phase should be high in the sky at sunset.  In that case, the moon would then rise at noon and set at midnight (subtracting and adding 6 hours, respectively -- see illustration below).  The full moon should always rise at sunset, be high in the sky at midnight, and set at sunrise.  

The third-quarter moon at sunrise

What you actually see from the Earth should be something like the image above.  At sunrise, the third-quarter moon should be high in the sky.  The Sun is about 400 times further away than the Moon so the position of the Sun should be taken as just the direction of the sunlight.  The waxing crescent moon (below) at noon should be visible as shown, although it might be tough to see if the atmosphere is bright.  

Waxing crescent moon at noon