Q: How was the number π first discovered? How did we first figure out it was 3.14…?

Physicist: Unfortunately, the use of the number π is older than recorded history so… who knows?  But the use of π at the beginning of recorded history wasn’t too sophisticated, so we can make guesses.

π is the ratio of a circle’s circumference to its diameter.  Everything about π boils down to that definition.

Measure the distance around and across any circle, then divide, and you’ve got π.

That the distance across a fixed shape is proportional to the distance around it is nothing special.  The same is true for absolutely every shape.  If you double the size of any shape, both the distance across and around with double as well, so their ratio stays the same.

Measure the distance around and across any square, then divide, and you’ve got 4.

Knowledge of the fact that the ratio of the circumference to the diameter of circles is some particular fixed number is older than recorded history.  The fact that the ratio isn’t exactly three takes a little care and precision.  Figuring out that the ratio has an infinite and non-repeating decimal expansion (that starts with 3.14159265358979323846264…) takes a little math and time.

A Babylonian tablet from around 4,000 years ago that declares the value of π to be 3.  Could be worse.

If you can nail one end of a piece of string down and tie a quill or piece of charcoal to the other, then you can draw a damn-near-perfect circle.  With a longer piece of string (at least 2π times as long, in fact) and a ruler, you can measure the circumference and as long as you’re careful, it’s pretty obvious that π≠3.  As long as the error in your measurements is below 4%, you can tell the difference.  The Babylonians created the Code of Hammurabi and some pretty impressive buildings, so presumably they could create a meter stick with centimeter marks (or rather, a kuš stick with šu-si marks).  It turns out the tablet in the picture above is really a “cheat sheet” for roughly approximating the area of circles.  They were known to have used $\frac{25}{8}=3.125$, an approximation within half a percent of the correct value.  Which, for the bronze age, is fine.

Since π shows up practically anytime you’re doing math involving circles, there were a heck of a lot of opportunities for people in the ancient world to notice it and we can’t be sure which first caught their eye (this is the drawback of discoveries that are older than history).  For example, the volume, v, of a barrel with height h and diameter d is $v=\frac{\pi}{4}hd^2$.  If you want to predict how much water a cylindrical barrel can hold to within a couple percent, then you need to know π to at least a couple digits.

All of this is to say that π, while full of mathematical mysteries or whatever, is not some abstract idea.  It’s something you can physically measure.  Well… you can tell it’s not three anyway.  The larger the circle and the better your measurements, the more digits of π you can discover, but the law of diminishing returns kicks in faster than an all-you-can-eat-week-old-sushi buffet.

By the time you know π out to N digits, you know the ratio of the circumference to the diameter of a circle to on the order of 1 part in 10N.  For example, if you know that π3.14, then you can fit a bike tire onto a rim to within a cm.  If you know that π3.1415, you can build a fence around a circular acre to within a cm.  And if you know that π3.1415926535, you can wrap a cable around the Earth with less than a cm of wasted cable.  Arguably, knowing π out to ten or more digits is aggressively pointless, but that has never stopped mathematicians from practicing their cruel craft.  Not once.

The definition of π not only gives us a recipe for physically measuring it, but also hundreds of ways to mathematically derive it, and that’s where the real precision comes from.  Mathematicians like Archimedes and Liu Hui, and some nameless Egyptian a couple millennia before them, were able to approximate π using polygons.  Liu Hui calculated π accurately out to three digits, a record slightly better than Archimedes’, that held for about a thousand years.  Which is strange.

During the Roman siege of Syracuse, general Marcellus ordered that Archimedes should be captured alive because clever knows no nation.  Unfortunately, geometry-ing to the very end, Archimedes was a jerk to the very first Roman he met.

Either Archimedes and/or historians really dropped the ball, or people in ancient Greece were more level-headed about knowing lots of digits of π than we are today.  For a given circle, the points on an “inscribed polygon” touch the circle (so it’s inside) and the middle of the edges of a “circumscribed polygon” touch the circle as well (so it’s outside).  Archie approximated π by inscribing and circumscribing 96-gons in and around the circle and calculating their perimeters.  An inscribed polygon gives a lower bound, and a circumscribed polygon gives an upper bound, for the value of π.  But here’s the thing: Archie didn’t just find the perimeter of 96-gons, he invented an iterative algorithm to calculate the perimeter of 2n-gons given n-gons.  That is, he started with hexagons (6-gons) and bootstrapped up to 12-gons, 24-gons, 48-gons, and then inexplicably stopped at 96-gons.  Evidently he had better things to do than calculate more digits of π.  Which, to be fair, is not a high bar.  He may have just declared the problem solved, since anyone following his procedure could find as many digits as they’d want, and moved on to heat rays or something (seriously, dude tried to build a solar heat ray to defend Syracuse).

Every time Archimedes’ iterative algorithm is used the estimate for π gets about 4 times better (its rate of convergence is 1/4).  Which is less impressive than it sounds; that’s about 3 decimal digits every 5 iterations.  Archie used it 4 times to go from hexagons to enneacontahexagons for an accuracy of about 3 decimal places.  If he had bothered to repeat the process, say, ten more times, he would have known the first 9 decimal digits.  Definitely not useful, but potentially brag-worthy.

Modern algorithms put those old approximations to shame instantly.  Archimedes’ technique converges linearly to the true value of π (you gain about the same number of digits every time you use the algorithm).  Things didn’t really start to pick up until we invented quadratically converging algorithms, which double the number of known digits with each iteration.  That is: if you know ten, then after the next iteration you’ll know twenty.  The fastest algorithms today converge nonically (the accurate decimal expansion gets nine times longer with each step).

The definition of π, the ratio of the circumference to the diameter of a circle, allows us to measure it directly, but inaccurately, or calculate it precisely, but pointlessly.  The more abstract properties of π, like going on forever without repeating (which it does) or containing every possible pattern (which it might), require more than brute force discovery of digits.  These more abstract properties are based on the definition of π, not its value, and can typically be proven or disproven without knowing even a single digit.  Math is useful in the physical world, but it doesn’t “live here”.  π has physical significance, but we use its mathematical properties to learn about it.

Answer Gravy: Ancient people were very clever and when they lived long enough they even got to prove it.  This gravy is the math behind Archimedes’ algorithm.  This isn’t exactly the way he wrote it.  Ancient Greek mathematicians suffered from the demonstrably false belief that a minimum of words is the secret to a maximum of understanding.  So, even translated, their proofs still read like Greek.

Mr. Medes’ method was this.  If In is the perimeter of the inscribed polygon and Cn is the perimeter of the outer n-gon, then:

$C_{2n}=\frac{2C_nI_n}{C_n+I_n}\quad\quad I_{2n}=\sqrt{I_n C_{2n}}$

You could rigorously prove that as you increase the number of sides the perimeters will get closer to π (the circumference of the circle) using lots of equations or something, or you can just draw a picture and say “look… they clearly do”.

A circle (dashed line) with its inscribed and circumscribed n-gons (blue) and 2n-gons (red).  The length of each segment is the total perimeter divided by the number of segments (hence all the dividing by n’s).

If you stick six triangles together, you get a hexagon and with a little trigonometry you find that if your circle has a diameter of 1, then the inscribed hexagon’s perimeter is I6 = 3 and the circumscribed hexagon’s perimeter is C6 = 2√3 3.46.

To find the perimeters for dodecagons, plug C6 and I6 into the iterative equations:

$C_{12}=\frac{2(3.46)(3)}{3.46+3}\approx3.215$ $I_{12}=\sqrt{(3)(3.215)}\approx3.106$

These perimeters are both closer to π than those before the iteration and since $I_n<\pi for all n, this gives us an ever-diminishing range where π can be found.  Here’s why it works:

In/circum-scribing n-gons on a circle produces some useful symmetries.  In particular, it allows us to draw some triangles and rapidly figure out their angles.

One of the sides from an inscribed n-gon and a corner from a circumscribed n-gon.  The lines inside are from the new 2n-gons.  The red shaded triangles are similar to each other (have the same angles) and the blue outlined triangles are similar.

A full circle is 360 degrees, so each side of an n-gon spans 360/n degrees.  In this case, a is half of one of those angles, so a=180/n.

Both b’s are complimentary to a (they sum to 90), since the sum of angles in a triangle is 180 and the third angle is 90 (a symmetry we get from the fact that radial lines always hit a circle at a right angle).  So, b = 90-180/n.

c and b are complimentary, so c = a = 180/n.

c+d = 180, so d = 180-c = 180-180/n.

The sum of angles in a triangle is 180, so d+e+e = 180 and e = 90-d/2 = 90/n.

Finally, b+e+f = 90, so f = 90-b-e = 90/n.

Since f=e, the two red triangles have the same angles: they’re “similar triangles“.  Similarly, since c=a the two blue triangles have the same angles and are also similar.  When two triangles are similar, the ratios of their sides are the same.

The lengths involved.

Using the similarities on the blue triangles we can say:

$\begin{array}{rcl} \frac{\left(\frac{C_n}{2n}\right)}{\left(\frac{I_n}{2n}\right)}&=&\frac{\left(\frac{C_n}{2n}-\frac{C_{2n}}{4n}\right)}{\left(\frac{C_{2n}}{4n}\right)} \\[4mm] \frac{C_n}{I_n}&=&\frac{2C_n-C_{2n}}{C_{2n}} \\[2mm] \frac{C_n}{I_n}&=&\frac{2C_n}{C_{2n}}-1 \\[2mm] \frac{C_n}{I_n}+1&=&\frac{2C_n}{C_{2n}} \\[2mm] C_n+I_n&=&\frac{2C_nI_n}{C_{2n}} \\[2mm] C_{2n}\left(C_n+I_n\right)&=&2C_nI_n \\[2mm] C_{2n}&=&\frac{2C_nI_n}{C_n+I_n} \\[2mm] \end{array}$

and of course, using the red triangles:

$\begin{array}{rcl} \frac{\left(\frac{I_{2n}}{2n}\right)}{\left(\frac{C_{2n}}{4n}\right)}&=&\frac{\left(\frac{I_n}{n}\right)}{\left(\frac{I_{2n}}{2n}\right)} \\[4mm] \frac{I_{2n}}{C_{2n}}&=&\frac{I_n}{I_{2n}} \\[2mm] \left(I_{2n}\right)^2&=&I_nC_{2n} \\[2mm] I_{2n}&=&\sqrt{I_nC_{2n}} \\[2mm] \end{array}$

So, starting with a little geometry and the definition of π we can construct a bootstrapping method for approximating it better the longer we bother to work at it.

Multiplication and long division can be done by hand pretty easily.  The hardest part of this iterative algorithm facing any ancient person is the square root.  Also making the scratch paper.  Fortunately, there are tricks for that too.  For example, if you want to take the square root of S, just make a guess, x, then calculate $\frac{1}{2}\left(x+\frac{S}{x}\right)$ and what you get will be a number closer to $\sqrt{S}$ than your original guess, x.  This method was known to the Babylonians and Archimedes (it is, in fact, “the Babylonian method”) and it converges quadratically, so it achieves whatever (reasonable) accuracy you’re hoping for almost instantly.

The point here is that you can, through the injudicious use of reason, time, and scratch paper alone, find π to as many digits as you would ever need.

You can read a little more about the Babylonian tablet here.

Posted in -- By the Physicist, Engineering, Geometry, Math | 6 Comments

Q: Given two points on the globe, how do you figure out the direction and distance to each other?

Physicist: The very short answer is: use the spherical law of cosines so you can do trigonometry on a sphere.

This is a seriously old problem that needed to be solved before we became a routinely globe-trotting species.  If you have the latitude and longitude of your location and a destination, you could just move east/west until you match the destination’s longitude then move north/south until you match the latitude.  But that’s a big waste of time, as the so-intuitively-obvious-it-barely-deserves-a-name “triangle inequality” will tell you.

The “triangle inequality” says that the sum of any two sides of a triangle are longer than any one side.  So if you’re traveling from Nicaragua to Norway, don’t go via Nigeria.

As you may have heard, the shortest distance between two points is a straight line.  For curved surfaces the closest you can get to a straight line is a “geodesic”, which is a straight line on small scales, but it you step back may be going all over the place.  If you wrap a ribbon around a gift (or any manner of festive package) and manage to keep it flat, you’ve found a geodesic, because if the path of the ribbon changes direction, the ribbon kinks.  On a sphere like Earth, where you’re stuck moving along the surface (instead of drilling straight through like some kind of compulsive, fire-proof mole), the geodesics are “great circles”.  If you walk in a straight line, you’re walking on a great circle.

A great circle is the path you’ll follow if you walk in a straight line on a sphere.  Smaller circles result from not-straight-lines (continuously turning a little to the side).

Doing geometry on a sphere is more difficult than geometry on a flat plane, but not by a hell of a lot.  Geometry, despite what some pudunk Greek philosophers may say, is basically just playing with triangles.  A nice, universal rule for any triangle is the Law of Cosines which relates the lengths of the three sides, a, b, and c, with the angle opposite one of those sides, C: $c^2=a^2+b^2-2ab\cos(C)$.

Left: A triangle made from great arcs. Right: A triangle made from straight lines.

Slap that triangle on the side of a sphere and that law becomes the Spherical Law of Cosines:

$\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)$

Here the meaning of a, b, and c has changed a little.  Lengths on a sphere can be described by angles (drawn from the center to the surface).  If you’re using radians (which you always should), those angles are $\frac{\textrm{the overland distance}}{\textrm{Earth's radius}}$.  This is generally easier to use than the actual distance.  For example, the angle from the north pole to the equator is 90° (obviously) while the physical distance is about 6,200 miles (different on every planet).

Lucky for us, positions on Earth are usually described in terms of angles (latitude and longitude) and not distances, so the spherical law of cosines is ready to go.  Just put the corners at the north pole, where you’re at, and where you’re planning to go.

If you put the corners of the triangle at a pole, your location, and your destination, then you can read off the lengths of two sides (90-latitude) and the angle at the pole (the difference of the longitudes).  From these there’s enough information to find the distance and bearing to the destination.

The law of cosines (either of them) relates four things: all three sides and one of the angles.  If you have any three of those pieces of information you can solve for the forth.  With the latitudes we have two sides and with the difference of the longitudes we got an angle.

First (because you always should) convert your angles from degrees to radians by multiplying by $\frac{\pi}{180}$: $\theta_1=\frac{\pi}{180}(90-latitude1)$, $\theta_2=\frac{\pi}{180}(90-latitude2)$, and $\phi=\frac{\pi}{180}|longitude1-longitude2|$.

$\cos\left(\frac{\textrm{distance}}{\textrm{Earth's radius}}\right)=\cos\left(\theta_1\right)\cos(\theta_2)+\sin\left(\theta_1\right)\sin(\theta_2)\cos(\phi)$

and therefore

$\textrm{distance}=(\textrm{Earth's radius})\arccos\left[\cos\left(\theta_1\right)\cos(\theta_2)+\sin\left(\theta_1\right)\sin(\theta_2)\cos(\phi)\right]$

Boom!  There’s the distance.  And with three sides in hand (the two given by the latitudes and the distance from, like, two seconds ago) you can find your “bearing”, the angle between north and your destination.

$\cos\left(\theta_2\right)=\cos\left(\theta_1\right)\cos\left(\frac{\textrm{distance}}{\textrm{Earth's radius}}\right)+\sin\left(\theta_1\right)\sin\left(\frac{\textrm{distance}}{\textrm{Earth's radius}}\right)\cos\left(\textrm{bearing}\right)$

and so

$\textrm{bearing}=\arccos\left[\frac{\cos\left(\theta_2\right)-\cos\left(\theta_1\right)\cos\left(\frac{\textrm{distance}}{\textrm{Earth's radius}}\right)}{\sin\left(\theta_1\right)\sin\left(\frac{\textrm{distance}}{\textrm{Earth's radius}}\right)}\right]$

For example, if you happen to be traveling from Nueva Guinea, Nicaragua (11.6932° N, 84.4540° W) to Nesflaten, Norway (59.6466° N, 6.7997° E), the angles involved are $\theta_1=\frac{\pi}{180}(90-11.6932)\approx1.367$, $\theta_2=\frac{\pi}{180}(90-59.6466)\approx0.530$, and $\phi=\frac{\pi}{180}\left|6.7997-(-84.4540)\right|\approx1.593$.  So the distance is

$\textrm{distance}=(3959miles)\arccos\left[\cos\left(1.367\right)\cos(0.530)+\sin\left(1.367\right)\sin(0.530)\cos(1.593)\right]=(3959miles)(1.406)=5566miles$

and the bearing is

$\textrm{bearing}=\arccos\left[\frac{\cos\left(0.530\right)-\cos\left(1.367\right)\cos\left(\frac{5566}{3959}\right)}{\sin\left(1.367\right)\sin\left(\frac{5566}{3959}\right)}\right]=0.538rad=30.1^o$

If you want to walk in a straight line from downtown Nueva Guinea to uptown Nesflaten, you’d face due north, turn about 30 degrees to the right, and walk in a straight line for about two and a half months, while ignoring the Atlantic Ocean to the best of your ability.

So how do you calculate distance and bearing?  A small bucket of math.  Today if you want to calculate sines and cosines you break out a computer.  So you may be wondering how they did this math back in the days of sailing ships (without GPS).  They didn’t.  Calculating was done once, then written down, then looked up many times.  Once upon a time books were useful!

A very boring book from the 17th century wherein you can look up the values of trig functions.

The people who actually did the calculating were “computers”; pitiable folk who sat in dark rooms and slowly went mad and rarely got to sail around the world.

Posted in -- By the Physicist, Geometry, Math | 8 Comments

Q: Is it possible to eat all of the ice cream in a bowl?

Physicist: If you’ve ever sat next to someone as they rang an almost-empty ice cream bowl like church bells on Sunday, then you’ve probably asked yourself some variation of this question.

Still plenty left.

Clearly, the issue is that a spoon and a bowl aren’t osculating curves.  The radius of a spoon’s curvature is typically less than an inch while a bowl’s is several inches.  With different curvatures, when you drag a spoon across the bottom of an ice-cream-laden bowl, they only contact each other at a tiny point and you can only remove ice cream along a thin strip (Matching the curvatures is why you instinctively start using the spoon sideways when the ice cream is low).  So if you wanna get your bowl perfectly clean, you gotta figure out how wide that strip is.

Around the point of contact there’s a gap between the spoon and bowl that gets bigger the farther away you get.  But so long as the ice cream molecules are bigger than that gap, they’ll be caught.  Water, at about a quarter of a nanometer across, is the smallest molecule in ice cream’s molecular menagerie.

To figure out the width of the strip (red bar) removed by the pass of a spoon, you have to find how far you can get from the point of contact before the gap is big enough for the smallest molecules to pass through.

The bottom of a circle with a radius R can be closely approximated by a parabola of the form $y=\frac{1}{2R}x^2$ (There’s nothing too special about parabolas; every curve that doesn’t have sharp corners can be closely approximated by circles and vice versa, it’s just that parabola math is easy).  Spoons have a radius of around 1 cm.  Bowls have a radius of around 7 cm.  The distance from the center (the point of contact) to the edge of the strip, x, such that water molecules won’t be able to slip through the gap between the bowl and the passing spoon is given by $\frac{1}{2(0.01m)}x^2-\frac{1}{2(0.07m)}x^2=0.25\times10^{-9}m$.  Solve for x and you get $x=2.4\times10^{-6}m$, so the strip should be about 5 micrometers across; on the order of a tenth of a hair’s width.

The wide ice-cream-free swaths you see in practice aren’t regions where all of the ice cream has been removed, but regions where it’s been left thin enough to see though.  So, if you really wanted to scrape a bowl clean of every molecule of ice cream, you’d need to carefully “scan” one strip after another, about four million and change times.  Just to make sure you don’t miss any (else you and your IC OCD would be forced to start over), it couldn’t hurt for those strips to overlap.  Call it an even ten million passes.  At, say, two passes per second (the approximate speed of a kid charging headlong into a brain freeze), it would take you almost two months to scour your bowl clean.  About a week into that, you’ll get the overwhelming urge to ask for seconds.  Or at least do something else.

So far, this is all a bit idealized.  The Platonic perfection of parabolas doesn’t apply to actual spoons and bowls.  If you zoom in close enough, ceramics are like moonscapes with plenty of room for ice cream to hide and the tip of a spoon looks more like a stainless steel mountain range.  Truly, a rocky road.

The surface of a ceramic with the (highly ideal) 5 micrometer wide ice-cream-molecule-stripping path of a spoon.

As undeniably clever as the approach above is, that “10,000,000 passes with a spoon” estimate is built on unrealistic premises.  Sadly, you can’t eat all the ice cream in a conventional bowl using compulsive spooning alone.

But that doesn’t mean it’s impossible eat all of the ice cream, you just have to stretch the rules a little.  For example, you could eat as much as you would like, then put the bowl into a kiln.  By and large, organic molecules (sugar, fat, cellulose, etc.) burn at temperatures below 500°F and turn into new compounds, and a good kiln heats things up to 1700°F.  A ceramic bowl would survive the heat intact, but the chemicals that make ice cream ice cream (as opposed to ash) wouldn’t.

No ice cream may escape.

So if you define “eating all the ice cream in the bowl” as “there was ice cream, I ate ice cream, now it’s all gone”, a kiln is a good way to do it.  But if you think destroying the evidence is somehow cheating, there is one more, perhaps not as delicious, alternative guaranteed to work.

Eat the bowl.

Willy Wonka demonstrating the simplest solution for consuming every atom of ice cream (or whatever else).

The empty ice cream bowl picture is from here.

The extreme close-up of a ceramic is from here.

The kiln picture is from here.

The Willy Wonka picture is from Willy Freaking Wonka.

Posted in -- By the Physicist, Geometry, Philosophical | 8 Comments

Q: Could the “proton torpedoes” in Star Wars be a thing?

Physicist: If you’re a fan of Star Wars you may remember proton torpedoes from Episode IV as the only weapon that, when fired down an exposed thermal exhaust port on the Death Star, would “…start a chain reaction which should destroy the station.“, because obviously “The shaft is ray-shielded, so you’ll have to use proton torpedoes.“.

Fans of Star Trek, on the other hand, think they just read “photon torpedoes” twice.

According to the perfectly named Wookieepedia, proton torpedoes release clouds of high energy protons on impact.  Now technically, that’s what all explosives do.  By mass, atoms are almost entirely made of protons and neutrons, each of which are around 2000 times more massive than electrons, and (for most elements) show up in roughly equal numbers.  So about half of the mass of practically anything is protons (the glaring exception is hydrogen, which is 1 proton and 1 electron).

Incidentally releasing clouds of protons, because that’s what’s in clouds of anything, is clearly not the idea behind proton torpedoes nor the spirit of this question.  Neutron bombs (which are real things) release a heck of a lot of neutrons and not too much else.  Proton torpedoes should be like neutron bombs, just with protons.

Once again physics, the queen of sciences and mother of buzzkills, has things to say.  The thing about protons is that long before you notice them doing damage by physically impacting things, you notice their electrical charge.  A proton is like a mosquito when you’re trying to sleep; the mass is not what’s important.

There are incomprehensible electrical charges buried inside of everything, we just don’t notice them because they’re normally almost perfectly balanced.  One gram of protons, without accompanying electrons to bring their net charge to zero, would create an electric field strong enough strip electrons from their host atoms and ionize air more than 15 km away.  It wouldn’t be too healthy for any other materials either; a field this large and intense could pull lightning bolts through granite at a range of 10 km.

In the immediate vicinity of this Tesla coil the electric field exceeds the 3kV/mm needed for air to spontaneously ionize (hence the faint glow around the rod).

So a proton torpedo with a one gram payload does a heck of a lot of damage just by existing.  Luke’s aim didn’t really need to be that good.

Ironically, the big effect of detonating the torpedo and releasing the protons is that the electric field actually drops.  As the last post talked about too much, you can understand a lot about electric and gravitational fields by “drawing bubbles” around them.  The total field flowing out of a bubble is proportional to the amount of charge inside of that bubble.  So if you’re standing far enough away, the field you feel stays about the same whether or not the torpedo has exploded.  As the shell of protons expands and repels itself, the field outside of it stays about the same while the field inside of it disappears.  If you’re up close, when the expanding shell of protons passes you (assuming you’re alive) you’d find yourself in a region with little or no electric field.  The most destructive thing you can do with a pile of protons is not blow it up, but keep it together in as small a place as possible.

Left: A torpedo (or anything else) with an excess of positively-charged protons produces an electric field.  Right: As the shock front of proton shrapnel expands, the field outside stays about the same while the field inside drops to around zero.

The only way to block the electric field created by a bucket of charge is to surround it with the same amount of the opposite charge.  Both positive and negative charges create electric fields, but those fields counteract each other.  That’s why, despite containing many kg of protons, you doesn’t destroy everything around you.

Unfortunately for BlasTech Industries (famed maker of clumsy, random blasters), counteracting electric fields is exactly what charges always do, given the chance.

Negative charges (blue) flow toward positive charges (red), because that’s what charges do, until they balance.  Once they do, the net electric field outside the torpedo is zero.

So even if you intentionally store protons in you proton torpedo with the hope of weaponizing its electric field, the first thing that will happen is any nearby matter will ionize.  The loosed electrons will fly toward and coat the torpedo while the stripped nuclei (which are full of protons) beat a hasty retreat.  Assuming the casing is a perfect insulator, in the end you’d have a torpedo with a gram of extra protons inside, about half a milligram of electrons outside (because electrons are about 2000 times less massive than protons), and no devastating electric field.

You could keep the protons and electrons together until the torpedo hits its target, then release the protons but keep the electrons in place (I mean… somehow).  Then you could transport your munitions without preemptively killing everything around them.  But that would be like firing bullets tied to (very strong) rubber bands; the electrons and protons would rather fly back together than do anything else.

So “proton torpedo” may just be a code name like “tomahawk missile” (noteworthy for its lack of tomahawks).  That, or George Lucas is playing a little fast and loose with physics.

Posted in -- By the Physicist, Particle Physics, Physics | 9 Comments

Q: Why does gravity pull things toward the center of mass? What’s so special about the center of mass?

The original question was: How do you measure d in the equation for universal gravitation ($f=\frac{GMm}{d^2}$)?  It can’t be the distance between the surfaces of the objects, or when they touched f would be “infinite.”  It can’t be the distance between their centers of masses because if I shot a bullet into a rubber ball their centers of mass might coincide, and again d would be zero.  It can’t be the distance between the center of mass of one object and the surface of the other object, because there would be two possible d’s.

Physicist: Nature abhors a singularity.  So when you see a nice glaring one (like this case, where d=0 makes the force jump to infinity) it’s worth pausing to consider how the universe finagles its way out.  When you apply Newton’s Law of Universal Gravitation, $f=\frac{GMm}{d^2}$, to planets and stars, d is indeed the distance between the centers of mass.  But when to use that standard and why it works is a little subtle.

One of the most important ideas in physics is that the exact same laws apply to everything, everywhere, always.  Nothing and nowhere gets special treatment, and that includes the center of mass.

Nothing is sacred.

The stuff at the center of the Earth does pull on you, but that’s nothing special; so does literally everything else.  Phone books, your worst enemy, that thing you saw that one time, everything on Earth (and otherwise) pulls on you according to Newton’s law.  Every atom pulls on every other atom.

All mass creates gravity and pulls things toward it.  Here on Earth, we always feel a pull toward the center of the Earth (top), so it might seem as though there’s something special down there.  In reality, we’re experiencing the collective pull of everything in and on the Earth (bottom).

The reason you don’t notice a pull toward each individual thing around you is that gravity is weak.  The physical (as opposed to emotional) attraction between you and someone you’re walking next to is a force on the order of the weight of a mote of dust (tens of nanograms).  The air moving in and out of your lungs pushes you around far, far more.

The reason you do notice a pull toward the Earth is that there’s a lot of Earth to do the pulling; around 100,000,000,000,000,000,000,000 times the mass of the person next to you.  Sure it’s farther away (around 4,000 miles on average), but there’s a lot of it.  Heck, it blocks half the sky (the lower half).

Finally, the reason everyone feels a pull toward the center of the Earth is that, for most intents and purposes, the Earth is a perfect sphere.  No matter where you are and no matter how you look at it, you can draw a line through the middle of the Earth and the two sides will be mirror opposites of each other.  As much as one half pulls you to the right, the other pulls you to the left.  The collective pull of both halves is right down the middle; toward the line between them.

The matter on the left side of the (red) line pulls you down and to the left, the matter on the right pulls you down and to the right.  Because the two sides are essentially the same, the “left-and-right-ness” cancel out, leaving a net down.  For a sphere, you can repeat the same argument for any (yellow) line through the center.

The same argument applies to any line that separates the two halves, and all of those lines pass through the center of mass.  Technically, that’s how the center of mass is defined.  So, because the mass of the Earth is spherically symmetric, gravity around here points toward the center of mass.

However!  Gravity does not, in general, point toward the center of mass.  Take for example the Earth-Moon system.  Despite its non-contiguous-ness, it’s still a perfectly legit collection of mass.  The center of mass of the Earth and Moon combined is a point just a little below the Earth’s surface.  That fact is important when you consider how the two orbit each other (the Moon circles that point while the Earth kinda wobbles around it), but it is almost entirely unimportant when you consider how gravity pulls on things presently sitting on Earth’s surface.

The Earth-Moon system to scale.  Notice that gravity does not pull you toward this system’s center of mass (the red dot) it pulls you toward the largest nearby thing (the pale blue dot).

That all said, Earth is only almost perfectly spherical.  You may have noticed that the world is not a perfectly polished ball; there are mountains, oceans, gigatons of missing single socks, and that non-uniformity penetrates well below the surface.  Those variations in the distribution and density of mass means that Earth’s gravity changes (very, very slightly) from place to place, both in magnitude and direction.

Deviations in the strength of Earth’s gravity.  The average surface gravity of Earth is around 980 Gal (1 Gal = 1 cm/s2).  Surface gravity points away from Earth’s center the farthest where the colors are changing the fastest.

By flying a pair of satellites in formation over the Earth and measuring the distance between them we can detect them speeding up and slowing down with respect to each other as one, then the other, passes over differently-gravity’d regions.

The GRACE (Gravity Recovery And Climate Experiment) satellites, cleverly called “GRACE-1” and “GRACE-2”, shot lasers at each other from 2002 to 2017 to measure Earth’s gravity field over that time.  They were so stupefyingly sensitive that they could detect changes in the location of water and ice through the gravity they generate (hence the “climate” part).

Finally, to more directly address the original question, since gravity is a collective pull toward each individual piece of mass and not necessarily a pull toward the center of mass, Newton’s universal law of gravitation has to be applied with a little finesse.  $f=\frac{GMm}{d^2}$ works fine (with d the distance to the center) as long as you’re outside of a sphere of mass, it stops working as soon as you’re inside because the mass above you counteracts the pull of the mass below you.  In the case of a bullet shot through a rubber ball, this is why the gravitational force never jumps to infinity; the equation that says it should ceases to apply.  The universe always has an obnoxious way to get out of having to deal with singularities.

Because gravity is an “inverse square force” (the “$d^2$” in “$f=\frac{GMm}{d^2}$” ) it has the remarkable property that the total gravitational field pointing into any closed “bubble” is proportional to the amount of mass contained within that bubble.  Inverse square laws also describe the way light gets dimmer with distance: so if you build a glass bubble around a light source, no matter how big or what shape the bubble is, the same total amount of light flows out of it.

Left: The force through any particular part of the bubble can change, but the total depends only on the amount of mass inside.  Right: If the mass is distributed spherically, then the force is the same everywhere on a spherical bubble (what would make it different?) and we can see why the outer layers of a planet have no impact on the gravity inside of a planet.

The beauty of this is that it allows us to do what should be an arduous calculation (adding up the contribution from every tiny chunk of matter) in a manner that appeals to physicists: stupid easy and deceptively impressive.  One of the immediate conclusions of this is that, as long as you’re outside of the sphere of mass in question, there’s no difference between the gravity of a sphere of mass and the same amount of mass concentrated at a point.  So if the Sun collapsed into a black hole, everything in the solar system would continue to orbit it as though nothing had happened, because the gravity would only have changed below where the surface of the Sun used to be.

The second cute conclusion of this is that if you’re inside of a sphere of mass, only the layers below you count toward the gravity you feel.  So the closer you are to the center, the less mass is below you, and the less gravity there is.  If you were in an elevator that passed through the Earth, you’d know you were near the center because there’d be no gravity.  With the same amount of mass in every direction, every atom in your body would be pulled in every direction equally, and so not at all.  A sort of gravitational tug of war stalemate.

As you approach a sphere of mass gravity increases by the inverse square law, but drops linearly inside the sphere.  If the density varies (which it usually does), that straight line will bow up a bit.

Newton’s Universal Law of Gravitation applies to everything (hence the “universal”).  But instead of applying to the centers of mass of pairs of big objects, it applies to every possible pair of pieces of matter.  In the all-to-common-in-space case of spheres, we can pretend that entire planets and stars are points of mass and apply Newton’s laws to those points, but only as long as we don’t need perfection (which is usually).  As soon as things start knocking into each other, or the subtle effects of their not-sphere-ness become important, you’re back to carefully tallying up the contribution from every chunk of mass.

The pedestal with nothing on it used to have a guy stabbing a bird.  The picture is from here.

The GRACE stuff is from NASA.

Posted in -- By the Physicist, Equations, Philosophical, Physics | 4 Comments

Q: In relativity, how do you define “the observer”?

Physicist: Whenever you listen to a physicist drone on about relativity (and thank you for your time), you’ll often hear them say things like “…from the perspective of a moving observer…” or “…the observer sees…”.  That’s all very fine and good, but how do you actually define the perspective of that observer?

When you describe something from your own perspective you say things like “it’s ten feet in front of me” or “it’s to my left” or “it passed me by at 30 mph”.  You personally define a set of directions (forward, left, etc.) and distances (however far away something is) relative to yourself.  Far more subtly, your perspective also includes time.  The “future direction” is just as personal and subjective as all of the other direction and distance stuff.  The subjectiveness of time was one of the great insights of Einstein’s relativity.

A short, more pedantic answer is that the observer determines the coordinate system.  That’s just the mathematical way of saying that when you talk about the location and time that things happen, you just describe them in terms of your location and what your watch says.

You are at the origin (center) of a very particular, personal coordinate system.

We’re so accustomed to sharing a coordinate system, like (so called) universal time or latitude and longitude, that it’s easy to fall under the assumption that there really is such a thing as “correct coordinates”.  In reality, such “universal coordinates” are useful only because they’re something a lot of people have bothered to agree on.  And to be fair, it is a lot more useful to tell someone where you are in terms of street corners or latitude and longitude than it is to use your own personal coordinate system, which would amount to saying “I am where I am”.  Philosophically interesting, but not useful.

So while you could describe an observer’s trajectory through a city using some kind of standard coordinate system, that’s not the observer’s perspective.  That’s a description of their location using someone/thing else’s perspective.  Form their own perspective, an observer doesn’t observe their position changing so much as they observe the city moving around them.

As you read this, you would be well within your rights to say “the screen is two feet in front of me” (or however far it is) or more succinctly “the location of the screen is x=2” (where the x direction is forward, the y direction is left, and the z direction is up).  From anyone else’s perspective, the location of the screen is different.  For the person sitting next to you on the subway (if you’re reading this on a subway), the location of the screen might be “x=2 and y=3” or even more succinctly “(2,3,0)”.

If you’ve ever taken an intro physics course, you’ve been subjected to the parable of the guy kicking a ball off a cliff.  The problem is something like “there’s this guy at the top of a 20 m tall cliff who, in a sudden pique of ludophobia, kicks a ball off of the cliff so that it is initially traveling sideways at 3 m/s.  Where does it land?”.  The answer, you will be thrilled to learn, is about 6 m from the base of the cliff.

When you’re setting up this problem you’re faced with an ancient conundrum that has been haunting physicists since long before the age of cartography: what coordinates should I use?  The most correct answer is: whatever lets you be as lazy as possible.

The yellow coordinates are the point of view of the kicker, the red coordinates are the most convenient, and the blue coordinates work fine, but they’re awful.

If you put the center of your coordinates at the base of the cliff, with the axes aligned with the ground and the cliff (red axes in picture above), then one axis gives you height off the ground while the other tells you distance from the cliff.  Very convenient.  In this case the base of the cliff is at (0,0), the ball is kicked at (0,20), and lands in the water at (6,0).  You could use the kicker’s point of view (yellow axes), in which case you would describe the base of the cliff as 20 m below them, and the ball lands 20 m down and 6 m out, (6,-20).

The beauty of physics is that it’s capable of imitating the universe’s supreme deference.  You can write physical laws without ever specifying coordinates, allowing us to define the stage (make up coordinates) and let the universe play out however it likes.  So if you really, really wanted to, you could use completely arbitrary coordinates (blue axes).  But don’t.  The level of the water is no longer just “z=0” or “z=-20”, it’s an equation.  Gravity no longer points in the “negative z direction”, but off to the side.  You can still do the problem and by applying the exact same physical laws, but you have to do a lot more work and that’s categorically unacceptable for a physicist.

In normal space, direction is part of the coordinate system (forward, left/right).  We’re already used to that notion when we talk to each other (as in “You’ve got something on the left side of your face.  No my left, your right.  Well it’s all over the place now.”).  And you already know how to change your coordinate directions.  If you want your “left direction” to become your “forward direction” you just rotate 90° to the left.  If you want your “up” to become your “forward”, just rotate backward 90° (lie down).  Easy.

Observers oriented differently relative to each other will disagree on what directions “forward” (solid lines) and “sideways” (dashed lines) are.

In relativity, the future is just another direction.  Obviously time is a little special.  You can measure distances in any spacial direction using a ruler or string, but to measure “distance” in the time direction you need a clock.  So to define an observer in relativity, you define a coordinate system with them at the center and a time defined by what their clock says.

In relativity, an observer’s personal coordinate system includes their personal time.

It’s hard to picture the “time direction”.  Firstly because it requires thinking in four-dimensional terms and secondly because clocks and rulers are super different.  That’s why physicists futz about with math and coordinates all the time; they allow us to extend our intuition from what we can picture to what we can’t.

It turns out that you can do something similar to changing direction with time.  Exchanging one direction in space for another is called “rotation” (no big deal).  Exchanging one direction in space for the time direction is called a “Lorentz Boost” or, if you don’t speak physicist, it’s called “start moving in that direction”.  Someone facing a different direction relative to you will have a different perspective on what the “forward direction” is and someone moving relative to you will have a different perspective on what the “future direction” is.

Observers moving differently relative to each other will disagree on what directions “future” (solid lines) and “now” (dashed lines) are.

So when you hear things like “when something moves very fast it experiences less time” it’s important to parse out whose time you’re talking about and even what moving means.  This is actually a statement about how things behave in your perspective.  As the observer, time and space are defined relative to your position and clock.  When someone is moving, they’re moving relative to you and according to your coordinate system.  When someone is moving through time slower, it’s according to your clock.  The real headaches kick in when you realize that from the perspective of the other person, you’re the one who’s moving and experiencing time slower.  This feels like a contradiction, but keep in mind that different observers all have their own clocks and coordinates.  There’s a lot more detail on that here.

Here on Earth we’re all moving at about the same speed.  The fastest someone else is likely to be moving relative to you is at most 2000 mph (if you happen to be both on opposite sides of the Earth and on the equator).  Relativistic effects, including disagreeing about time, are only worth talking about when your relative velocities are an appreciable fraction of light’s modest 670,000,000 mph.  The most accurate clocks in the world are capable of detecting disagreements induced by walking slowly (relative velocities of less than 1 meter per second).  But we humans don’t care about differences of a few parts in a quintillion, so while the differences in our clocks are detectable (when we work really hard to detect them), they aren’t worth worrying about.

Posted in -- By the Physicist, Physics, Relativity | 10 Comments