Chapter 5: The Theory of Gravitation Simplified

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Welcome back to the Deep Dive.

Today we're tackling, well, really, one of the absolute giants of physics, the law of gravitation.

Absolutely, and we're using Richard Feynman's insights from Six Easy Pieces as our guide.

Exactly.

We want to trace how this idea came about from just watching the stars to predicting unseen planets.

The goal here is to give you a really solid grasp of how

this deceptively simple rule basically runs the cosmos.

It really is one of the most far -reaching ideas the human mind has come up with.

It grew out of centuries of just looking up and wondering.

So let's start right there.

What's the core principle?

Okay, the basic idea is this.

Every object attracts every other object.

Simple enough, right?

Right.

But the how is specific.

The force of attraction is proportional to how massive each object is.

So heavier things pull harder.

Makes sense.

And, crucially, it gets weaker the farther apart they are.

Specifically, it's inversely proportional to the square of the distance between them.

That inverse square part is key, isn't it?

Yeah.

Double the distance and the force drops to a quarter of what it was.

Precisely.

That mathematical relationship, botto -fracking, looks neat, yeah.

But the journey to figuring that out and what it unlocks, that's what we're digging into.

It wasn't just plucked out of thin air.

Not at all.

It starts way back with people just trying to make sense of how the planets move across the sky.

Okay, so Copernicus had already suggested planets orbit the sun, not the earth.

But what path do they take?

That was the big question.

And the person who really shifted the approach was Tycho Brahe.

His tremendous idea was basically enough philosophy, let's get really good data.

Accurate measurements.

Extremely accurate.

For the time.

He spent years meticulously recording the positions of planets.

Those tables, that raw data, were the foundation.

And then came Kepler, his student, right, who actually crunched those numbers.

Yes.

Kepler took Brahe's mountain of data and, after incredible effort, found these three, well, Feynman calls them, very beautiful and remarkable but simple laws that described exactly how planets move.

Let's quickly go through them.

Law number one.

The shape of the orbit.

Everyone assumed circles.

Perfect circles.

But Kepler found they move in ellipses.

An ellipse isn't just any oval shape, right?

It's specific, like a squashed circle.

You can think of it that way.

Technically, it's a shape where, for any point on the path, the sum of the distances to two fixed points inside, the foci is always constant.

And crucially, the sun isn't at the center.

It's at one of those foci.

Which means the planet is sometimes closer to the sun, sometimes farther away.

Exactly.

And that leads right into law number two.

Speed.

They speed up when they get closer to the sun and slow down when they're farther out.

Right.

But Kepler quantified it beautifully.

He showed that if you draw a line from the sun to the planet,

the radius vector, that line sweeps out equal areas in equal times.

So imagine the area the planet traces out in, say, a week when it's far away and moving slowly.

That long, skinny slice of area.

It's exactly the same size as the short, fat slice of area it traces in a week when it's close to the sun and zipping along.

Perfect.

That's the equal areas law.

And the third law.

This one connects different planets, doesn't it?

Yes.

It relates how long a planet takes to orbit its period, let's call it T dollar, to how large its orbit is.

Specifically, the semi -major axis.

Okay.

Kepler found that the square of the period, TDI2, is proportional to the cube of the semi -major axis, three through dollars.

Or, as Feynman puts it, T dollars is proportional to a raise to the power of three halves.

So the farther out a planet is, the disproportionately longer its year is.

Exactly.

These three laws were a huge breakthrough.

They described how things moved, but they didn't explain why.

Right.

That's the next step.

And for that, we needed Newton, but Newton first needed Galileo.

Absolutely critical.

Galileo's principle of inertia.

Before him, the thinking was you needed a continuous force just to keep something moving.

Like angels pushing the planets along their paths.

Kind of, yeah.

But Galileo said, no, an object in motion stays in motion, in a straight line, at a constant speed, unless a force acts on it.

So force isn't needed to keep moving, only to change the motion, speed it up, slow it down, or change direction.

That's the key insight.

And Newton applied this to planets.

A planet wants to coast in a straight line, tangentially away from the sun.

But it doesn't.

It curves.

It curves.

So there must be a force acting on it.

And since it's constantly curving towards the sun, the force can't be pushing it along its path.

It has to be pulling it inward towards the sun.

Precisely.

The force is directed towards the sun.

That debunked the whole angels pushing sideways idea.

Okay.

So Newton knows there's an inward force.

How does he use Kepler's descriptive laws to figure out the nature of that force, the inverse square part?

He works backward from Kepler's laws.

First, Kepler's second law, the equal areas law.

Newton showed mathematically that the only way a planet could sweep out equal areas in equal times is if the force acting on it is purely radial.

Meaning it points directly towards the central point, the sun.

No sideways component at all.

Exactly.

Any sideways force would mess up that equal area relationship.

So step one, force is radial.

Okay.

Now how strong is that force?

How does it change with distance?

That's where Kepler's third law comes in, the 2D dollar propto 833 relationship.

Newton plugged this relationship into his laws of motion, specifically five a dollar combined with the geometry of orbits.

And the math just worked out.

The math demanded that for this relationship between period and orbit size to hold true,

the force pulling the planet toward the sun must decrease in proportion to the square of the distance from the sun.

It had to be a hundred to a two.

Wow.

So it wasn't a guess.

It was a mathematical consequence of Kepler's observations.

It was a derivation, a stunning piece of logic.

And then comes the really audacious leap.

Right.

Newton says, okay, this force explains the planets.

Maybe it's the same force that makes an apple fall from a tree, the same force holding the moon in orbit around earth.

That's the universality concept.

It's not some special celestial force.

It's everyday gravity, just acting over vast distances.

So how do you test that?

You need to compare how things fall near earth versus how the moon falls.

Exactly.

We know, or Newton knew, roughly how fast things accelerate here.

An object falls about 16 feet in the first second.

The moon is much farther away, about 60 times the earth's radius.

So if the force follows the inverse square law, the pull in the moon should be 1 ,602 times weaker.

Which is $1 ,300 and $600.

So in one second, the moon shouldn't fall 16 feet, but $3 ,600 of 16 feet.

And does it?

When you calculate how far the mu deviates from a straight line path in one second, how far it falls towards earth, it comes out to about one twentieth of an inch.

And is that $3 ,300, $3 ,600 of 16 feet?

It matches perfectly.

Well, after Newton got the correct value for the earth's radius, apparently his first calculation was off, because the distance measurement wasn't quite right then.

But once corrected, it was spot on.

That's incredible confirmation.

But I'm still stuck on this idea of the moon falling, it stays up there.

How is falling compatible with orbiting?

Ah, yes.

That's a brilliant conceptual point Feynman explains with the cannonball analogy.

Imagine firing a bullet horizontally from a very high mountain.

Okay, gravity pulls it down, it falls 16 feet in the first second vertically while also moving horizontally.

Right.

Now fire it faster.

It goes farther before hitting the ground, but still falls 16 feet vertically in that first second.

Now imagine firing it really fast,

about five miles per second horizontally in the time it takes the bullet to fall 16 feet downwards.

The earth, because it's curved, has curved away beneath it by 16 feet.

Exactly.

So the bullet is 16 feet lower, but the ground is also 16 feet lower.

It never actually gets any closer to the surface.

It's continuously falling, but it's moving sideways so fast that it just falls around the earth.

That's orbit.

That's what the moon is doing.

That's what satellites do.

Wow.

Okay.

That's the genius of it.

It elegantly connects falling apples and orbiting moons.

So Newton derived Kepler's laws, tested it with the moon.

What else did this single law suddenly explain?

You mentioned tides early.

Yes, tides.

Why too high tides a day?

It's because the moon's gravitational pull isn't uniform across the earth.

It follows the inverse square law, remember?

So it pulls harder on the side of earth closer to it.

Right.

It pulls the water on the near side most strongly, creating a bulge towards the moon.

Okay.

That's one bulge.

Where does the second one come from?

Well, it pulls the solid earth less strongly than the near side water, and it pulls the water on the far side even less strongly.

Ah, so the far side water gets kind of left behind as the earth is pulled towards the moon more strongly than it is.

That's a good way to put it.

It creates a second bulge on the opposite side.

Yeah.

And as the earth spins underneath these two bulges, we pass through a high tide, then a low tide, then another high tide, then another low tide, roughly every 24 hours.

Exactly.

All from differential gravitational force.

Amazing.

What about the speed of light discovery?

How did gravity play into that?

That was Olly Roemer observing Jupiter's moons.

He noticed that the timings of when Jupiter's moons passed behind the planet weren't quite regular.

They seemed faster sometimes?

Slower other times?

Yes.

They appeared to be ahead of schedule when Earth was closest to Jupiter in its orbit and behind schedule when Earth was farthest away.

Why would that be?

Roemer realized it wasn't the moon's changing speed.

It was the time it took the light from Earth.

When Earth was farther away, the light simply took longer to travel the extra distance.

So the predictable gravitational clockwork of Jupiter's moons acted as a standard, and the variations revealed the travel time of light.

Precisely.

It gave the first real estimate for the speed of light way back in 1656.

Just incredible consequences flowing from understanding gravity.

But maybe the most jaw -dropping prediction was finding a whole new planet, right?

Neptune?

Oh,

absolutely.

Astronomers noticed Uranus wasn't quite following its predicted path, even after accounting for the poles of Jupiter and Saturn.

There were these tiny persistent irregularities.

Wobbles.

Wobbles, exactly.

Two mathematicians, Adams and Le Verrier, independently thought, what if there's another planet out there beyond Uranus pulling on it?

Using only Newton's law of gravitation.

Using only the law and the observed wobbles in Uranus's orbit.

They calculated where this unseen planet must be to cause those

perturbations.

And they told astronomers.

Point your telescopes here.

And bam, there is Neptune, found purely through the predictive power of the 1 or Q2 law.

You can't get much better validation than that.

It really shows the power.

And this law isn't just local, is it?

It stretches across the cosmos.

We see it everywhere.

Double stars orbiting each other in ellipses, just like Kepler described for planets.

Huge globular clusters with thousands of stars held together.

Entire galaxies spinning in spirals held by gravity.

Even clusters of galaxies,

vast agglomerations, millions of light years across, all interacting gravitationally.

The 1 or Q2 law seems to work on the grandest scales imaginable.

But it also works on the small scale, right?

How did they actually measure this force between, say, two objects in a lab?

It must be incredibly weak.

It is unbelievably weak.

That was the challenge for Henry Cavendish around 1798.

His experiment is famously known as weighing the Earth.

How did he possibly measure the attraction between, like, two bowling balls?

He used a very delicate setup called a torsion balance.

Imagine a thin wire or fiber hanging vertically, holding a horizontal rod with two small lead balls at the ends.

Okay, like a tiny dumbbell hanging from a thread.

Exactly.

Then he brought two much larger lead balls near the small ones, on opposite sides.

And the tiny gravitational attraction between the big balls and the small balls would twist the fiber.

A minuscule amount, yes.

But by measuring that tiny angle of twist, Cavendish could calculate the force between the balls.

This directly verified the inverse square law for everyday objects.

And because he measured the force between known masses at a known distance, he could finally calculate that constant of proportionality in Newton's equation, the gravitational constant g.

Precisely.

He determined Dailer's.

And once you know day dollars and you know how strongly the Earth pulls on you, and you know the radius of the Earth, you can rearrange the formula and calculate the mass of the entire Earth for the very first time.

That's why it's called weighing the Earth.

All thanks to measuring that incredibly tiny twist.

So we know how it works with incredible precision.

$1 G frac, our crew too.

But Feynman points out Newton didn't explain the mechanism.

What is gravity?

Right.

Newton described it perfectly, but offered no explanation for how the force acts across empty space.

What carries the pull?

Did anyone figure that out later?

Well, people tried.

There was an idea about tiny particles like gravitons being absorbed, creating a shadow effect.

But that theory predicted that moving objects, like planets, should feel a kind of drag or resistance from this particle flow.

Which would make them spiral into the sun eventually, but they don't.

Exactly.

So that mechanism didn't work.

And honestly, we still don't have a fully accepted mechanical explanation for gravity at the quantum level.

It remains a deep puzzle.

And tied into that puzzle is its incredible weakness, right?

Compared to other forces.

Yes, particularly electromagnetism.

Both gravity and electricity follow that same beautiful inverse square law for their forces.

But the strengths are just wildly different.

Remind us of the scale for two electrons.

Okay, take two electrons.

They repel each other electrically because they have the same charge.

They also attract each other gravitationally because they have mass.

How much weaker is the gravity?

The electrical repulsion is stronger than the gravitational attraction by a factor of, well, it's followed by 42 zeros.

$4 .17 times 10 is 42.

10 to the power of 42.

That number doesn't even make sense.

It's so huge.

It's staggering.

Gravity is absurdly weak at the particle level.

Why this enormous disparity exists between forces that otherwise look similar mathematically?

That's one of the biggest unanswered questions in fundamental physics.

Yet this incredibly feeble force shapes the entire universe.

Mind boggling.

Another key point Feynman emphasizes is the equivalence of gravitational mass and inertial mass.

What's the difference again?

Gravitational mass is what determines the gravitational force on an object, basically.

It's weight.

Inertial mass is its resistance to acceleration, how hard it is to push, Esma.

And experiments show these two types of mass the same or proportional.

They appear to be perfectly proportional to an astonishing degree of accuracy, like one part in $109 or even better, thanks to experiments by Utviz and later Dick.

And this equivalence is why astronauts feel weightless.

Precisely.

Because the spaceship and the astronaut inside are both falling around the Earth at the exact same rate.

If their gravitational mass, how hard Earth pulls them, wasn't perfectly proportional to their inertial mass, how much they resist that pull, they wouldn't fall together.

One would lag behind the other inside the cabin.

But they don't.

The chalk floats right next to them because everything falls at the same rate, thanks to that equivalence.

Exactly.

It's a profound principle.

Finally, we have to mention Einstein.

Newton's law isn't the absolute final word, is it?

No.

The big issue is that Newton's law implies gravity acts instantaneously across space.

If the sun vanished, Earth would instantly fly off its orbit, according to Newton.

But Einstein's relativity says nothing.

Not even gravity's influence can travel faster than light.

Correct.

So Einstein had to modify the law.

His theory of general relativity describes gravity not as a force, but as a curvature of spacetime caused by mass and energy.

This effect propagates at the speed of light.

And this new theory made slightly different predictions.

Yes, especially in strong gravity fields or for very precise measurements.

One famous prediction was that because energy has equivalent mass, light itself, which carries energy, should be bent by gravity.

And they observed this, starlight bending as it passed near the sun during an eclipse.

They did.

It was a major confirmation of Einstein's theory.

So while Newton's law is incredibly accurate for almost everything we experience, Einstein's provides a deeper, more correct picture.

So to wrap up, we started with Kepler describing how planets move, then Newton deriving the universal 100 -triangle -2 law to explain why.

A law that then explained falling apples, the moon's orbit, tides, helped measure the speed of light, predicted Neptune, described galaxies.

And was measurable in the lab, allowing us to weigh the Earth.

All from that simple inverse -square relationship.

It's an absolutely stunning intellectual achievement.

The power flowing from that simple mathematical form is just immense.

Yet, as Feynman highlights, the deepest questions remain.

We still don't really know the machinery behind it, why it's so incredibly weak compared to other forces.

And how it ultimately fits together with quantum mechanics.

That's the frontier.

So we leave you with this thought.

Gravity, this whisper of a force, dictates the structure of everything from planets to galaxy clusters.

If such a simple law, $5 propto fract in, could unlock so much understanding, what other simple elegant principle might still be out there, waiting to be discovered, perhaps finally explaining that baffling ten -pol factor and uniting gravity with the rest of physics?

Something to ponder.