Earth Size

Earth Size
Relative Size of the Sun and Earth

Sunday, June 22, 2014

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,
(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
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.