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Hydrogen’s Mysterious Attraction

Posted in: Bond Breaker, Lesson Time! | May 16, 2016 | No Comments


One of the most common questions I’ve gotten about Bond Breaker is this: why in the world should Hydrogen attract protons? Something seems fishy, and I try to answer it once and for all in this video:

 

Shown_image

This is the second installment in my TestTubeGames Explains series (you learn about how magnetism is really just electricity here)

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Parallel Wires and Relativity

Posted in: Electric Shocktopus, Lesson Time!, TestTubeGames Explains, Velocity Raptor | January 14, 2016 | No Comments


The latest venture here at TestTubeGames?  Videos!

Teaching through games is great – and I love it (obviously) – but in addition to that, I thought I’d make some *videos* about the *science* behind the *games*.

…*…

And since everything I do needs a title (Feel Bad Friday, Gravity Simulator Image of the Day, etc), these videos have a name, too:

 

TestTubeGames Explains

In my first video, learn how Magnetism can be explained just by using Electricity and Relativity.

 

screenCap

 

What do you think?  Was it understandable?  Crazy confusing?  What other topics would you like to see in the future?

-Andy

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The Electric Field Mystery, part 1

Posted in: Electric Shocktopus, Lesson Time! | August 11, 2015 | No Comments


The forums were quiet.

…too quiet.

In this neck of the woods, Andy knew that meant trouble was brewing.  It was then that A Random Player walked into the room with a problem.  And that problem’s name was Shocktopus.

.     So I heard you made a game about electric fields.

.     That’s right, Andy said, looking up from the line of impeccably tilted test tubes that covered his desk.

.     And I heard the game is scientifically accurate.

.     Well, aside from the questionable marine bi   ology, I sure aim for it to be, Andy said cautiously, memories of Velocity Raptor’s bullet-conundrum swirling like a mist in the back of his head.

.     Then how do you explain *this*? Random said, throwing a picture down on the desk:

 

Screen Shot 2015-07-27 at 3.02.48 PM

Andy had seen volcanoes before… and they never looked like this.  There was nothing on the top of the screen, but there was an electric field emanating from it.

That was five nights ago, and Andy had been hitting the pavement ever since.  Looking for clues in a town full of shadows.  Where did he go wrong? Was this a bug?  Or worse: not a bug?  Either way, he knew that an understanding wouldn’t come easy.  Where did all this trouble begin?  Oh yes, with the Relaxation Method, which, as it turned out, was anything but.

The Case of the Relaxation Method

Back when he first started out as a gumshoe – programing the first prototypes of The Electric Shocktopus, the world was a simple place.  Electric fields were easy to come by, so long as you knew how solve a simple equation.  And with a computer at his fingertips, solving equations was one thing Andy could do.  Find all the electric charges in a level, find where the Shocktopus is, and you’ve found yourself the force.  Perfect forces.  Perfectly easy.

But, like all things in life, it was simple precisely up until the point when it was not.

And that point was when Andy took on the case of a grounded Conductor (who, incidentally, had been sent to the sleeper car without his supper).  As Andy pieced together the puzzle, he realized that you couldn’t just add conductors into The Electric Shocktopus.  There’s no simple rule for figuring out the forces, then.  No easy equation to solve.  For the fields affect the charges in the conductor, just as the charges in the conductor affect the fields.  You may as well try to track down a criminal who knows you’re on his tail.  Each move you make changes which way he’s running.

 

mystery_blog_1

 

But, just like with that criminal.  There was a solution.  You just needed boundaries.  Walls, gates… or in this case, an electrified fence would do the trick.

If you’ve got a conductor all around the outside of your level, you’ve got a way to hunt down the electric field.  It’s called the Relaxation Method.  And just like playing a game of cat-and-mouse, it involves a lot of guessing.  But with each guess, you get closer to the criminal.  Or closer to the final, true electric field.

 

mystery_blog_2

A Found Field

 

There’s only one problem.  By putting a ring of conductors up, you’ve changed things.

 

mystery_blog_3

It just wasn’t the same anymore.

 

So, how can you put up a ring of conductors around the level, while at the same time not having them affect the level?  In all his years watching crooks melt into the shadows, Andy knew that everything leaves a trace.  Footprints. Cigarette ash. Or warped electric fields.

But there had to be a way.  And Andy had to find it.

To be continued…

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How Repulsive…

Posted in: Bond Breaker, Lesson Time! | September 18, 2014 | No Comments


Bond Breaker is based on real physical chemistry, which means by playing around with it, you’re actually doing science experiments.  Last week, we looked at the Van der Waals forces, which pulls molecules together.  This week, we’re going to go a little more basic.

Like Charges Repel

Whoa, we’re going very basic, eh?  We’ve all heard the phrase ‘likes repel’ ever since we were in diapers.  Two protons, both having positive charges, will push away from one another due to the electric force.  The game includes this force, with each proton pushing on all the others:

 

"I just want personal space"

“I just want personal space”

 

As you can see, they all try to get as far away from one another as possible.  The calculations in the game are modeled completely after Coulomb’s Law, which tells us that the force between two charges is proportional to the inverse square of the distance between them.  To put that in terms that anyone who hasn’t taken a course about Electromagnetism can understand: if you double the distance between the charges, the force will drop to just one quarter of what it was.  It gets small fast.  And this is why, in Bond Breaker, you can only get so close to another proton… and no closer.

 

"Whoa, whoa, whoa: don't touch me."

“Whoa, whoa, whoa: don’t touch me.”

 

The connection between the force and distance can reveal itself in even more advanced ways, too.  Take, for instance, this level:

 

Think you can beat this level? Give it a try: http://bit.ly/1r2SGSD (It's the 'Bonus' Level)

Think you can beat this level? Give it a try: http://bit.ly/1r2SGSD (It’s the ‘Bonus’ Level)

 

It’s a new bonus level that I made just for this blog post, simply click the image above to play it in your browser.  Once you give it a try: what does this level (or should I say, experiment) have to teach us about Coulomb’s Law?  I’ll leave that to you, the player, to figure out.

And that’s one of my favorite things about making a game that stays true to the science: Each level is actually an experiment, which makes players scientists.

 

-Andy

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Van der Waals

Posted in: Bond Breaker, Lesson Time! | September 10, 2014 | 2 Comments


One of the best parts of making a game based on science is that while playing the game… you learn science.  Even if you don’t mean to!  Take, for instance, the Van der Waals force.

(If you haven’t played Bond Breaker yet, give it a go.  It’ll make this all go down a little easier)

 

Van der Waals

The Electric Force, at its core, is pretty basic.  You can sum it up with: “opposites attract, likes repel.”  If you put two positive charges together, they’ll push away from each other.  And if you put a positive near a negative, they’ll attract together.  A neutral object, with no positive or negative charges, will be unaffected by the Electric Force.

In Bond Breaker, you can make a lot of ‘neutral’ object.  A Hydrogen molecule, for instance, consists of two protons and two electrons.  (+2) + (-2) = 0.  Put two of them near each other, and the Electric Force shouldn’t do anything, right?  Well, in Bond Breaker you can try that out!  Below is a little level I made (just for you, blog-post-reader), to test out what happens when neutral molecules are near one another.  Click it in your browser, and go play with the level (it’ll be called the ‘Bonus Level’).

 

Click the image to go play this BONUS Bond Breaker level!

Click the image to go play this BONUS Bond Breaker level!

 

Okay, so the molecules attract.  But if they’re all neutral, why?

Van der Waals forces.

What, you need more information than that?  Well, then…

These forces are what make molecules attract to one another (and form into liquids, say).  The weakest form of VdW force is called the “London Dispersion Force,” and it’s what you encounter in the game.

 

London Dispersion Forces

‘Neutral’ molecules are not simply neutral.  The positive and negative charges aren’t sitting right on top of one another.  At any given moment, the molecule will have a dipole moment — meaning one side will be more positive, and one side will be more negative.  Kind of like a bar magnet with a North pole and South pole.  Imagine putting a bunch of magnets into a bag and shaking them.  It won’t take long until they’re all stuck together.

With a molecules like Hydrogen that are very symmetrical, the dipole is completely random.  Sometimes you’ll find the electrons more on the north side of the molecule, sometimes you’ll find them on the south side.  And this makes the force pulling the molecules together very weak.  But it’s still there.

Van der Waals forces, though weak, end up being important in everything from forming liquids to helping geckos stick to walls.  So the next time you’re sitting in the pool, watching your pet gecko play Bond Breaker, you know what force to thank.

 

-Andy

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Tour de Force – (part 1)

Posted in: Gravity Simulator, Lesson Time! | June 21, 2014 | 1 Comment


Another week — another Gravity Simulator Lesson!  This week: what happens when you change the laws of gravity?  In the past, I’ve touched on this topic a bit, but this time we’re really gonna dive in.

 

The Baseline: 1/r^2

Ah yes, Newtonian gravity.  The perfect place to start.  This is gravity as we know it in our universe (ignoring pesky things like general relativity for the moment).  With the “one-over-r-squared” law, gravity decreases inversely as the square of the distance between the two objects.

That’s a mouthful, but really it’s quite simple: If we get twice as far away from the center of the earth, it’ll pull on us with one quarter of the force it does now.  If we triple the distance, gravity will pull just one-ninth as hard.

And with this wonderful, happy force law, we get orbits that look like this:

 

Classic Inverse Square Law, what you'd find by default the Gravity Simulator

Classic Inverse Square Law, what you’d find by default the Gravity Simulator

(As usual, all the gifs you’ll see today are taken right from the Gravity Simulator.  Feel free to play around, yourself!)

 

Experiment 1: 1/r^2.1

In the Gravity Simulator, we can do something you can’t do in the real world: we can change the force of gravity and see what happens!  So let’s increase that exponent over ‘r’.  So now the force goes as “one-over-r-to-the-2.1.”  Doesn’t quite roll off the tongue as easy, eh?

Well, let’s see what happens:

 

The tiniest of changes -- a 1/r^2.1 force law.  Try it out in the Gravity Simulator with this code:  [ForceGr: r^(-2.1),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 62.5,y0: 17.5,vx: 0,vy: 2.7,t0: 0,who: 3,m: 0,c: 0], [x0: 0,y0: 17.5,vx: 0,vy: 0,t0: 0,who: 1,m: 1000,c: 1]

The tiniest of changes — a 1/r^2.1 force law. Try it out in the Gravity Simulator with this code: [ForceGr: r^(-2.1),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 62.5,y0: 17.5,vx: 0,vy: 2.7,t0: 0,who: 3,m: 0,c: 0], [x0: 0,y0: 17.5,vx: 0,vy: 0,t0: 0,who: 1,m: 1000,c: 1]

Huh, the orbits no longer match up.  That’s called precession — basically the planet is nearly making an ellipse as it goes around the star, but not quite.  The path keeps twisting around and around.  Turns out 1/r^2 is special — very few force laws will make orbits that don’t precess.

Why does making it 2.1 instead of 2 change things? Well, ‘r’ is raised to a slightly higher power, which means as two planets get further away, the force of gravity drops off faster than it would in our world.  And as they get closer, the force of gravity increases faster, too!

So when the planet comes in close, gravity pulls on it stronger (than in our universe at least), deflecting its path even more, which gives it a tighter curve.  That tighter curve when it’s close to the star means that the planet swings around faster than we’d expect.  So by the time it comes back out to its furthest distance again, it’s gone more than 360 degrees.  Matches with what we see above.  (Go, science!)

 

Experiment 2: 1/r^3

Let’s keep going!  Let’s raise ‘r’ to an even higher power!  We should expect even stronger precession, right?  Well, let’s see:

 

1/r^3 force law.  Try it in the Gravity Simulator with: Gravity Fun at TestTubeGames.com: [ForceGr: r^(-3),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 62.5,y0: 17.5,vx: 0,vy: 2.7,t0: 0,who: 3,m: 0,c: 0], [x0: 0,y0: 17.5,vx: 0,vy: 0,t0: 0,who: 1,m: 1000,c: 1]

1/r^3 force law. Try it in the Gravity Simulator with: Gravity Fun at TestTubeGames.com: [ForceGr: r^(-3),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 62.5,y0: 17.5,vx: 0,vy: 2.7,t0: 0,who: 3,m: 0,c: 0], [x0: 0,y0: 17.5,vx: 0,vy: 0,t0: 0,who: 1,m: 1000,c: 1]

Huh, now we’re getting something different.  When an object gets close, gravity starts pulling so strongly that the planet just spirals inward until it collides.  And if the object is too far away, it spirals outward… and since gravity gets a lot weaker the further out you go, it keeps spiraling out more and more, never to return.

 

Experiment 3: 1/r^10

What if we went really crazy with this?  (I always go really crazy with this.)  Let’s try a force law that decreases extremely fast.  1/r^10!!!! (Excitement, there, not factorials)  Now if two objects get twice as far apart, the force between them is about 1000 times smaller!  That makes for this weird world:

 

1/r^10 force law!  Play around with it in the Gravity Simulator with this code: Gravity Fun at TestTubeGames.com: [ForceGr: r^(-10),Qual: 1,Zoom: 1,xSet: 0,ySet: 0]

1/r^10 force law! Play around with it in the Gravity Simulator with this code: Gravity Fun at TestTubeGames.com: [ForceGr: r^(-10),Qual: 1,Zoom: 1,xSet: 0,ySet: 0]

Objects really don’t notice each other in that weird universe, they mostly travel in straight lines… at least until they get juuuust close enough to the star.  And once they do, they’re pulled quickly and without remorse into a collision.  Boy, I’m glad we don’t live there!  You can imagine how hard it would be to create galaxies in that universe, let along solar systems with stable orbits.

Tune in next week when we take things the other way… what happens for 1/r^1.9?  Or 1/r?  In the meantime, play around with your own force laws in the Simulator!

-Andy

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Black Hole or Bust

Posted in: Gravity Simulator, Lesson Time! | June 14, 2014 | No Comments


Last week I reported on General Relativity in the Gravity Simulator.  And now that relativity is in place — this means the sim can have BLACK HOLES!  YEAAAAA!

 

Created in the Gravity Simulator

Created in the Gravity Simulator

 

Black Holes

How hard could it be to add in black holes, right?

We all know that black holes are extremely massive, extremely dense objects.  (Mostly.)  Get close enough to them, and gravity pulls so strong, that not even light can escape.  Whoa!  So we just need to make a big star in the Gravity Simulator — and we get a black hole!

Well, no.  You’ll never get a black hole if you’re dealing with boring, old Newtonian gravity, though.  In the old simulator, say, you could make a star bigger and bigger and bigger, and all you’d get is a bigger star.  Any astroid or planet or star can escape its pull, so long as it’s moving fast enough.

 

Event Horizon

Around black holes, there’s a line of no return called the Event Horizon.  If you’re outside of this boundary — you could escape the black hole.  But the moment you cross it, you’re sunk.  You’ll get swept ever further into the black hole.

This happens because General Relativity contains our old friend Special Relativity.  And, if you’ll recall, a key part of Special Relativity is that nothing can travel faster than light.  The speed of light is the speed limit for everything.  The Event Horizon represents the line near enough to the black hole where, if you wanted to escape, you’d have to travel at light speed.  Fall in a bit closer, and gravity gets a bit stronger, and you’d need to go even faster than light to escape.  No can do.  You’re stuck.

 

Masses: 1000, 2000, 4000, 8000...

Masses: 1000, 2000, 4000, 8000…

In the pictures above, I draw where the Event Horizon would be on each the star.  The smaller the star is, the weaker the gravity it, and the closer you’d have to get to reach the Horizon.  In fact, most of the time, you’d actually have to go deep inside the star to find this line. Which means, it isn’t really an Event Horizon.  The calculations I used to draw these assumed that all the mass of the star is inside the Horizon.  As you can see above, that’s not the case.  The stars aren’t dense enough, which means: no Event Horizon and no black hole.

 

Masses: 8000, 10000, 14000, 16000...!

Masses: 8000, 10000, 14000, 16000…!

 

But if we get enough mass in place, the Horizon grows big enough that it swallows up the whole star — and we finally get our black holes!  Now let’s have some fun with them!

 

An orbit that precesses, much like Mercury. (Created in the Gravity Simulator)

An orbit that precesses, much like Mercury. (Created in the Gravity Simulator)

 

Black holes merging! (Created in the Gravity Simulator)

Black holes merging! (Created in the Gravity Simulator)

 

Black Holes have tons of neat properties, which you’ll all be able to check out in the next update to the Gravity Simulator.  Stay tuned for more General Relativistic fun!

 

-Andy

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Relativistic Gravity

Posted in: Gravity Simulator, Lesson Time! | June 4, 2014 | No Comments


Thought I’d keep you all apprised of the latest addition I’m working on:

RELATIVITY IN THE GRAVITY SIMULATOR.

At least, *mostly* relativity.  You see, we’d talked about adding in Black Holes… which would be awesome.  But not just *nom*nom*nom* generic sci-fi ‘sucks-stuff-in’ Black Holes.  This is TestTubeGames, after all.  So I wanted to at least get stuff approximately right.  Maybe so ‘orbits’ become something like this:

 

Light_deflection

Orbits around (into?) a black hole, taken from helpful, helpful wikipedia: http://bit.ly/1p7N7Qo

A while back, in an earlier chat in the forums, we found a General Relativistic formula for the attraction between two objects.  Seems reasonable that we could plug in that force law (after all, we’ve got change-able force laws already).  It won’t be precisely right (there won’t be gravity waves…), but it’ll get us close.

Simple, then, slap on a GR Force Law and call it a day!  Well, nope.  Because Black Holes have this neat feature where once something gets too close — ~~inside the event horizon ~~ — it’ll *never* come back out.  Mwahahaha.

Never, that is, unless it travels faster than light.  Which, in the real world (as far as we know) nothing does.  But in the Gravity Simulator, you can launch stuff at any speed!  Black Holes would lose all meaning, objects could escape at will.  They’d become just ‘really strong stars’ instead of ‘points of no return.’  Boo, hiss.

That means I need to add more relativity in the sim, to make objects obey the speed limit of light.  Now when something accelerates, it can get close to, but never reach the speed of light. And, lo and behold, we get neat orbits like this:

 

Relativistic Orbits

Look Familiar?  (Drawn with an unreleased version of Gravity Simulator)

 

Great.  Can we just paint that star black and stop there?

NO!

Because once you have objects traveling near the speed of light, well, then E=mc^2 becomes important.  Namely, mass is energy, energy is mass.  So what?  So EVERYTHING.  Imagine two stars colliding.  They rush inwards to meet one another, then *boom* they combine to form a single, stationary star.

 

Whaaa???

Whaaa??? (Drawn in an unreleased version of Gravity Simulator)

 

 

The total energy has to remain the same, which means that Kinetic Energy had to go somewhere.  In our sim, there’s only one place that energy can go: into rest mass.  Just as two subatomic particles can combine to form something massive (wee protons crashing into each other to make the Higgs boson, anyone?), two stars can combine to make one *huge* star.

What other parts of relativity will come into play down the line?  Well, the Schwartzschild radius is important. And relativistically slowed clocks are awesomely fun…

Where does this all end?  With a bang?  With a whimper?  Will the simulation collapse into a singularity under all the weight of the new code?  Stay tuned to find out!

-Andy

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Negative Mass

Posted in: Gravity Simulator, Lesson Time! | May 29, 2014 | No Comments


This week, I want to talk about negative mass.  Wait… negative mass?  Like antimatter?

No.

Antimatter, in spite of its ‘anti’ name, still has a positive mass.  An electron and an antielectron (positron) have opposite charge, but they have the same mass.  Charge flips to negative, mass stays the same.

If antimatter is out of the picture — what in the world does have negative mass?  Well, nothing really that we know of.  It would take a very strange, exotic form of matter to have negative mass.  The closest we come is with the complex Casimir Effect.  But let’s imagine that a chunk of negative mass does exist.  What do we know about it?

First off, negative masses would have… negative energies.  When an object is moving, it has a kinetic energy equal to 1/2*m*v^2.  If the mass is negative, the kinetic energy is negative.  Even when you look at E = mc^2, the rest energy of matter, you’d find that the negative mass would lead to negative energy here, too!

Strange stuff.  The faster it goes, the less energy it has.  The more you have of it, the less energy you’ve got.

Gravity Time!

What would a negative mass mean for gravity?  Well, we can check this out in the Gravity Simulator.  Two planets with positive mass will, of course, attract.

 

together

Load this Scene in the Simulator: [ForceGr: r^(-2),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: -125,y0: 0,vx: 0,vy: 0,t0: 0,who: 4,m: 1000,c: 1], [x0: 0,y0: 0,vx: 0,vy: 0,t0: 0,who: 4,m: 1000,c: 1]

This is because the force between them is:

 

NewtonsGravity

Where m and M represent their masses.  Positive masses mean a positive force, which in this case means the objects pull towards each other.

What about negative masses?  Suppose we had two planets with negative mass.

Looking at the equation above, if we change m to -m and M to -M… the equation doesn’t change at all!  (Negative*Negative = Positive!) So they should attract as before, right?

 

Apart

Load this Scene in the Simulator: [ForceGr: r^(-2),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 50,y0: 0,vx: 0,vy: 0,t0: 0,who: 4,m: -1000,c: 1], [x0: 75,y0: 0,vx: 0,vy: 0,t0: 0,who: 4,m: -1000,c: 1]

Wrong! 

There’s an extra bit of the puzzle we’ve left out.  The force hasn’t changed, but the acceleration has.  To figure out how something will move, we use F=ma.  The force is equal to the mass times acceleration.  Even if the force hasn’t changed in our example… if the sign of the mass has flipped, the sign of the acceleration has to flip.  So instead of the pulling force making the planets accelerate together… they accelerate apart.

With all that in mind, the final puzzle is: what happens when you have a positive mass and a negative mass.  Will they attract?  Will they repel?

The test is easy to do (Note, you can do all these tests and MORE in the Gravity Simulator):

Sideways

Load this Scene in the Simulator: [ForceGr: r^(-2),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 125,y0: 2.5,vx: 0,vy: 0,t0: 0,who: 4,m: -1000,c: 1], [x0: 90,y0: 2.5,vx: 0,vy: 0,t0: 0,who: 4,m: 1000,c: 1]

…whoa.  Why did that happen?  You’ve now got enough information to figure that one out.

As a final note, it would seem that having two planets suddenly zoom off the screen, ever faster, would violate Conservation of Momentum, or Conservation of Energy.  But in fact, it doesn’t.  As the positive mass gains positive kinetic energy, the negative mass gains negative kinetic energy.  They cancel out.

Same with the momentum.  Both planets may start moving in the same direction in that gif above, but the planet with negative mass has its momentum moving to the right.  (Momentum = m*v, so flipping the sign of the mass flips the direction of the momentum).  So even with strange negative masses around, conservation laws stay intact.

Will we ever really encounter a chunk of negative mass?  Who can say.  But at least we can make some good predictions about how we’d expect it to behave.

-Andy

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To Another Dimension!

Posted in: Gravity Simulator, Lesson Time! | May 21, 2014 | No Comments


One of the new visitors to the forums brought up a neat topic the other day: Dimensions. Namely, what would gravity look like, if instead of 3 spatial dimensions, we had 2? Or 4? Whoa.

I’ve talked about something similar before with the Electric Shocktopus, but this topic warrants more.  And it needs some swanky gifs from the new Gravity Simulator.

 

3D

Let’s start off simple.  The world we live in, for all practical purposes, is in three spatial dimensions.  You can go up-down, left-right, or forward-backward.  And in this world, Newton’s Law of Universal Gravitation tells us that two objects with mass will attract each other, according to:

NewtonsGravity

If you only care about the distance, the force goes as 1/R^2, which is why we call this an inverse-squared law.  Double the distance between the earth and the sun, and gravity will pull them together with 1/4 the strength.  Inverse square laws are beautiful, because they lead to cool things like closed and stable orbits.

 

1/R^2 Force law.  Ah, beautiful, closed orbits.  Home.

1/R^2 Force law. Ah, beautiful, closed orbits. Home.

 

All GREAT things for life!

Unfortunately, Newton’s Law of Gravity can’t help us when we’re in a world with a different number of dimensions.  The law only works in 3D.

 

Gauss’s Law

Instead, our starting point instead is Gauss’s Law.  Take a point mass, make an imaginary spherical surface around it.  Imagine that instead of gravity, the point mass is just shooting out 100 bullets in random directions.

pewpew

How many bullets pass through the sphere around the point mass?  No matter how big or small you make your sphere, the answer will always be 100.  That’s the core of Gauss’s Law.  The number of bullets passing through a sphere doesn’t depend on its radius.  When we’re talking about gravity, the bullets represent the Flux, which is simply the strength of gravity times the area of your sphere.

flux1

So then:

flux2

If the total flux is always the same, that means the Force of Gravity is proportional to 1/(Total Area of a Sphere).  In 3D, a sphere has a surface area of 4PI*R^2.  So the Force goes as the inverse of that, or 1/R^2.

 

2D

Suppose we were like the Flatlanders, and lived on a 2D plane.  We could move forward-backward and left-right, but not up-down.  How would that change gravity?

Well, Gauss’s Law still holds.  But this time our ‘sphere’ that we draw around the point mass is actually just a circle.  (Remember, we can’t leave our 2D surface!)  The ‘surface area’ of the circle is just its circumference: 2PI*R.  Which means that the Force of Gravity goes as 1/R.  And we get orbits that look like this:

1/R Force Law.  Notice that the orbits don't match up nicely anymore.

1/R Force Law. Notice that the orbits don’t match up nicely anymore.

 

 

4D

What if there were an extra dimension?  Suppose we lived in a world where we could move up-down, left-right, forward-backward, and… and… uh… 4Up-4Down.  We don’t have a word for those last two of course, since we never move through a fourth spatial dimension.  But nonetheless, we are armed with all the information we need to figure out how gravity would look.

This time the ‘sphere’ we make around a point charge is in 4D, which is hard to draw, but we can figure out what it must be like with a simple pattern.  In 2D, the ‘surface’ of a sphere is a line.  In 3D, the ‘surface’ of a sphere is an area.  So in 4D the ‘surface’ of a sphere should be a volume.  Its units should be length cubed.  No surprise, then, that the surface ‘area’ of the 4D sphere is 2*PI^2*R^3.  Which means that the Force of Gravity must go as 1/R^3.  So we get orbits like this:

 

A 1/r^3 Force Law.  Yikes, I'd hate to be on *any* of those planets.

A 1/r^3 Force Law. Yikes, I’d hate to be on *any* of those planets.

 

Beyond

You’ve now got everything you need to ponder the force of gravity in any number of dimensions.  And with the Gravity Simulator, you can check out what any of these worlds might look like.  Try to make a 5D solar system, I challenge you!

This was just a quick introduction to some interesting topics — so if you wanna know more, there are plenty of resources out there you can use.

-Andy

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