Using Reason as a Lever for Understanding
Weighing the Earth with a Piece of String and a Rock!
Over the years I've developed a deliberate habit of teaching without notes. To be sure, I've taught this stuff for a long time and 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 the real win for me (and maybe the students) is when we get sidetracked for a while discussing something I hadn't anticipated. And, frankly, I like to imagine science teaching is in some sense less about the literal content and more about the process of doing science, asking questions, and trying different approaches to figuring things out.
For example, one of the most wonderful things about science is the way simple and seemingly irrelevant experiments can actually reveal an understanding about something much deeper. It’s mind-boggling, really, that tying a rock to a piece of string and then timing how long it takes to swing back and forth is a way to weigh the Earth and sample the core! Read on…
Anecdote time! In my freshman physics class, we were finishing 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, and then you push it to the side a little:
Here, I’m supposing that the vertical motion is small compared to the horizontal motion (which is about right for small oscillations). For those of you who’ve had a bit of physics, the standard recipe is to use Newton’s 2nd Law: add up all the forces acting on the mass (in each direction, horizontal and vertical) and then set it equal to its mass time acceleration. In the vertical direction,
where g is the gravitational acceleration on the Earth’s surface (9.81 m/s per second). In the horizontal direction
Now, if the oscillations really are small, then we play one of our favorite tricks:
And from the diagram,
so substituting,
— the negative sign appears since we moved the mass to the left.
The cool thing is that this is exactly the same equation we would have typically just derived for a spring:
just with different letters. For a spring, “k” is the stiffness of the spring and “m” is the mass we attach to the spring. Anyway, this tells us that the motion of a mass on a spring and a mass at the end of a pendulum is described by the same equations!
Ok, so in class we then figure out that the time for one swing (the period) is just
I said something like “cool, the time for each swing does not depend on the mass attached, something noticed first by Galileo. Every pendulum with a 1-meter long string has a period of almost exactly 2 seconds, no matter how massive the thing is at the end of the string! 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 actually depends on the mass of the Earth
and its size, which has been known for 2200 years thanks to an ingenious experiment done by Eratosthenes. So if
then:
where G is just the number
But this is why physics is fun! Look, we have that same g in the equation for the period of our little pendulum, so that suggests a nifty little experiment: what if we didn’t know the mass of the Earth ahead of time (and why would you, really)? You could cut a 1-meter string, tie a rock at the end, and let it swing a little. If we observe the period to be, say, 2 seconds, we can weigh the Earth! With a little algebra,
That still blows me away — you can sit and watch a pendulum swing back and forth and calculate the mass of the entire Earth. Not only that, but knowing that density is mass/volume,
and using the same pendulum, we can calculate the average density of the Earth (… algebra break…)
It turns out that the density of water is 1,000 (in the same units), and rocks have a (very roughly) average density of something like 3,000. 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. And as far down as you can realistically dig, you only encounter more of the same stuff — rock and water. 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). You can do the same thing for the Moon, and what you’d find out is that the Moon has an average density much closer to rock, which would make sense if the Moon is pretty much rock all the way down to the core. But then you might go down the rabbit-hole of asking why the Moon and Earth should be so different — maybe they didn’t form in the same way out of the same stuff? It’s now thought that they probably didn't! So much interesting stuff and cool questions from watching a simple pendulum.
But how amazing! Suppose you land on some planet for which you know the size (which doesn't sound too unreasonable). 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.
On Deck:
The next article I’m working on looks at the phenomenon of “streaks” for random events (like hits in baseball, or the “hot hand” in basketball), and why it’s so seductively easy to build misleading narratives around that idea.
If you’re a student/teacher and want to see lots of worked examples that I like to include in my classes when I teach the “standard” University Physics 1 and 2 courses, feel free to browse the (growing) collection of 150+ videos at
And if something is especially cool and you’re inclined to leave a “tip” I’m not above coffee or pizza:
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Wow I did not know that we still don’t know what’s at the center of the earth. That’s so crazy!