Chapter 4: Conservation of Energy

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Welcome to the Deep Dive.

We're here to tackle challenging ideas and, hopefully, make them clearer and accessible for you.

Today, we're jumping into a really fundamental concept

the conservation of energy.

We're using Chapter 4 of the Brilliant Feynman Lectures on Physics as our guide.

Now, this law, it's quite something.

It applies to absolutely everything we know in nature.

No exceptions found, ever.

It's total.

But here's the tricky part, Feynman points out right away.

It's not about a thing.

It's not a mechanism.

It's this abstract number, a quantity, that just stays the same, no matter what happens.

Right, and that's the big mental shift you need to make.

It's tough at first.

You have to stop asking, well, what is energy?

Because honestly,

physics doesn't have that answer.

Instead, the focus becomes entirely different.

It's, okay, how do we calculate this number?

How do we track it as it seems to change form?

It's all about the calculation, the accounting.

So our mission today is to follow Feynman's lead,

track this constant number through all sorts of changes, things moving, electricity, even nuclear stuff.

We'll use those really vivid thought experiments he was famous for.

Let's start with how he introduces the basic idea, this abstract conservation.

He uses that wonderful analogy, Dennis the Menace and his blocks.

Ah, yes, the blocks, classic.

So Dennis has 28 blocks.

They're indestructible.

You can't break them.

And his mother, she's playing the role of the physicist here.

She insists the total count must always be 28.

And at first, it's simple, right?

She just counts them, 28 today, 28 tomorrow, easy.

But Dennis is, well, Dennis.

So things get complicated fast.

One day, maybe she only counts 27.

Panic.

Where's the missing block?

Exactly.

So she searches under the rug, maybe.

Ah, there it is.

Counts again.

Back to 28.

She's just tracking the location, really.

But the real challenge, the part that gets to the heart of energy is when she can't see the blocks.

Say three are missing and she knows they're in a lockly toy box.

Right.

She can't open the box, but she needs that number 28 to hold true.

So what does she do?

She needs a formula.

She has to figure out how many are in the box without looking.

Precisely.

She's clever.

She knows each block weighs, say, three ounces, and she knows the empty box weighs 16 ounces.

So she weighs the whole box.

If it weighs, um, 25 ounces total, she can calculate 25 ounces, 16 ounces is nine ounces divided by three ounces per block.

That's three blocks.

She's accounted for them numerically, even though they're hidden.

Spot on.

And it gets more complex.

Maybe one block is in bath, raising the water level.

She needs a water energy term based on that.

Another goes over the fence.

Another term.

So she ends up with this potentially huge formula, adding up calculated numbers for blocks under the rug, blocks in the box, blocks in the water.

And the sum of all those numbers must always equal 28.

Unless, you know, Bruce brings over some extra blocks, adding energy to the system.

Okay.

Here's the mind bending part.

Feynman says in physics, there are no blocks.

Energy is the formula.

It's that final number, the sum of all these different calculated terms, gravitational energy, kinetic energy, heat, electrical, chemical, nuclear mass energy.

That sum is what's conserved.

That analogy just perfectly captures the process.

We observe that something is conserved, this number.

Then we use logic, pure reason to figure out the formulas for each piece of the sum, which brings us nicely to section four two, where Feynman actually derives the formula for one of those terms, gravitational potential energy, the energy something has just because of its height.

And he derives it using just one single hypothesis.

Just one.

The idea that perpetual motion is impossible.

You can't get something for nothing.

No free lunch.

It's a powerful way to reason, similar to how Carnot thought about heat engines.

Basically, if you have a machine, say one that lifts weights, and you run it through a cycle, so it ends up exactly where it started.

Then it can't have produced a net lifting of weight overall, unless you put energy in from somewhere else.

Okay, so the thought experiment.

We imagine perfect idealized machines, reversible ones.

They can run forwards or backwards without losing any energy.

Let's call one machine A.

Right.

Machine A lowers a one unit weight by one foot, and it uses that work to lift a three unit weight up some distance.

Let's call that height X.

Now imagine another reversible machine, machine B, that does the same thing, lowers one unit by one foot, lifts three units, but maybe it lifts them to a different height Y.

Okay, now the crucial step is proving that X must equal Y.

Y.

Let's think.

Suppose Y was actually higher than X.

Could we exploit that?

We could.

You'd use machine B to lift the three units up to height Y.

Then you let that weight fall back down, maybe driving something else, until it only reaches height X.

So you've extracted some work from that extra height difference, Y minus X.

Exactly.

And then you use machine A running backwards, which costs nothing, because it's reversible to lift the original one unit weight back up to its starting point using the three unit weight now at height X.

And the net result.

Everything's back where it started, but we got some useful work out of that YX difference.

Which is perpetual motion, free energy.

And since we assume that's impossible.

Y cannot be greater than X.

And you can run the same logic the other way around, assuming A lifts higher.

So X can't be greater than Y either.

They must be equal.

X equals Y.

It doesn't matter what kind of machine it is.

Levers, gears, hydraulics, if it's reversible and does this weight exchange, the height ratio is fixed.

It's a universal constant for that weight ratio.

Okay, so we know this universal height X exists, but what is it?

What's the number?

How do we find the value of X?

Another brilliant thought experiment.

The three ball rack.

Picture this.

You have a reversible machine.

When you lower one ball by one foot, it lifts a rack containing three identical balls up by that unknown distance X we just talked about.

Okay, one ball down, one foot.

Rack with three balls up, distance X.

Now the clever setup.

Imagine the three balls start on a little platform.

Above that platform are shelves spaced exactly X apart vertically.

Got it.

Shelves are X high.

So we run the machine.

One ball goes down one foot.

The rack with the three balls goes up by X.

Now because the shelves are spaced X apart, the three balls on the rack are now perfectly level with the next shelf up.

So we can just gently roll them off onto that shelf.

Okay, so they're now sitting X higher than where they started.

Right.

Now think about the net change.

We lowered one ball by one foot.

We ended up with three balls, each X higher.

But we can simplify.

Imagine we lifted and used the machine in reverse to lift the original ball back up one foot.

That should balance out, right?

Let me think.

Feynman's actual argument is slightly different, I think.

He considers the whole cycle.

Roll one ball down one foot.

Lift the rack of three balls up X.

Roll all three balls horizontally onto the next level shelf, which is X higher.

Then lower the empty rack back down X, which gives back some potential.

Then somehow reset.

Wait, let's look at figure 4 -2, the simpler way he puts it later.

The net result of lowering one ball by one foot is that one of the three balls could be considered lifted by three X.

Is that it?

Yes, that's a clear way to summarize the outcome of the intricate steps involving the rack.

Through those clever maneuvers, the overall effect, when you balance the books, is equivalent to taking the energy from lowering one ball by one foot and using it to lift one ball by a total distance of three X dollars.

Okay, that makes sense.

The machine itself is just a means to an end.

The net energy change is what matters.

Lowering one ball one foot gives some energy.

Lifting one ball three to six dollars costs some energy.

And to avoid perpetual motion getting free energy out of nowhere, the energy gained must equal the energy spent.

The work done must balance.

So the work done by the ball going down, weight by one foot, must equal the work done on the ball going up.

Weight X three X dollars.

The weights cancel out.

Which means one foot must equal three X dollars.

So X has to be exactly one third of a foot.

Exactly.

And that gives us the general rule for gravitational potential energy.

It's the weight times the height.

W times H.

That's the number we track for energy due to positioning gravity.

It's really elegant how it comes just from no perpetual motion.

And this principle, conservation of energy, it suddenly makes sense of so many things, doesn't it?

Like levers and pulleys.

Absolutely.

All those complicated rules you learn about mechanical advantage, they just collapse into this one idea.

Work in equals work out, ideally.

Or energy is conserved.

Take the inclined plane example.

That classic three four five triangle.

You put a one pound weight on the slope.

How much weight W do you need hanging straight down to balance it?

Well, if the balancing weight W moves down the full five foot slant distance, the one pound weight only moves up the vertical three foot height.

So conservation demands W times five feet must equal one pound times three feet.

Which means W is instantly three eight or five fives of a pound.

You need to worry about angles or forces directly.

Just track the energy via W times H.

It's even more impressive with something complex like a screw jack, you know, for lifting a car.

Right.

Huge weight.

Let's say we want to lift one ton 2000 pounds.

The handle might be what 20 inches long and maybe 10 threads per inch on the screw.

Okay.

So to lift that ton, just one single inch, you have to turn the screw 10 full rotations.

And each rotation, the end of that 20 inch handle moves in a circle.

Circumferences two pi are roughly two, three, 20, about 120 something inches per turn.

Let's call it 126 inches roughly like fine and does.

So for 10 turns, the force on the handle moves 1260 inches.

Whoa, okay.

So the work done by you is force W times 1260 inches.

The work done on the car is 2000 pounds times one inch.

Set them equal.

W times 1260 inches equals 2000 pound inches.

Solve for W.

It's only about 1 .6 pounds.

You can lift a ton with less than two pounds of force just by trading force for a huge distance.

Amazing.

This whole approach gets formalized in something called the principle of virtual work.

It's a neat trick.

How does that work?

It sounds like work that doesn't actually happen.

Sort of.

You use it when things are perfectly balanced.

In equilibrium, nothing is actually moving.

But to find an unknown force, you imagine a tiny hypothetical movement of virtual displacement.

And then you demand that the total work done during this imaginary tiny movement must add up to zero because the system is balanced.

I see.

Let's try that bar example from Fast Four Six.

An eight foot bar pivoted somewhere.

There's 100 pounds hanging two feet from one end, 60 pounds hanging four feet from that same end.

And we're lifting the very end eight feet away.

What force W do we need?

Right.

It's balanced.

Now imagine we push the lifting in W down by a tiny amount, say four inches, just hypothetically.

Okay.

Virtual displacement, four inches down at the end.

Because of the lever geometry, the 60 pound weight, which is halfway to the pivot from the end, must go up half that amount.

So two inches up.

And the 100 pound weight, which is only a quarter of the way out, must go up a quarter of the amount, one inch up.

Got it.

So W goes down four inches, 60 pounds goes up two inches.

A hundred pounds goes up one inch.

Now apply the principle.

Total virtual work must be zero.

The work done by W force times distance must equal the work done on the other way.

It's force times distance.

So W times four inches down must equal 60 pounds times two inches up plus a hundred pounds times one inch up.

Calculate that out.

62 equals 120.

101 equals 100.

Total upward work is 220 pound inches.

So W four equals 220 divide by four.

W is 55 pounds.

Found it without calculating torques.

That's really powerful.

It really is a unifying idea.

Okay.

So far we've mostly dealt with statics or potential energy.

What about things that are moving?

The pendulum example is perfect.

At the top, it has gravitational potential energy, WXH.

At the bottom H is zero.

So the potential energy is gone.

Where did it go?

It had to turn into something else, right?

To keep the total number constant, it turned into the energy of motion, kinetic energy.

So kinetic energy, KE, is basically the energy an object has because it's moving.

And it's defined as the amount needed to conserve the total when particular energy changes.

And through relating the speed at the bottom to the height it fell from, you can derive the formula.

Which comes out as KE for one two meet V two two.

One half times mass times velocity squared.

That's the one.

And again, like WXH, it's technically an approximation.

It works brilliantly for everyday speeds and masses.

But if things get really fast, close to the speed of light, you need relativity.

And if heights get enormous, gravity isn't quite constant, so WXH needs adjusting.

Exactly.

The formulas might get more complex in extreme cases, but the underlying principle that the total energy is conserved holds true always.

So we keep adding terms to our energy spreadsheet.

We have gravitational potential energy, WXH, kinetic energy, 12 mV2.

What else?

What about a stretched spring?

Good one.

A stretched spring can definitely do work when you release it, maybe launch something.

So it must store energy.

We call that elastic energy.

Okay, elastic energy.

That gets converted into kinetic energy when the spring unwinds.

But why do things eventually stop?

Even a pendulum in a vacuum eventually slows down due to friction in the pivot.

Where does that energy go?

Does it seem to be potential or kinetic or elastic anymore?

Ah, the energy leak.

It's not really lost, though.

It turns into heat energy.

Heat.

So what is heat in this energy accounting picture?

It's actually just kinetic energy again.

But instead of the whole object moving together, it's the tiny atoms inside the object jiggling around randomly, all that bumping and vibrating at the atomic level.

So the organized motion of the pendulum gets converted into disorganized random motion of atoms in the pivot and the air.

Precisely.

And that random jiggling is what we measure with a thermometer as temperature.

Heat is disordered energy.

And that explains why real machines aren't perfectly reversible.

Some energy always gets lost, as heat due to friction is still conserved, but it's now in a less useful disorganized form.

Exactly.

That's a key idea leading towards thermodynamics.

Okay, so the list of energy forms keeps growing.

We need to account for everything.

What about light?

Radiant energy.

Light carries energy, obviously.

We now understand light fundamentally as oscillations or wiggles in the electromagnetic field.

So you could say it's a form of electrical energy propagating through space.

And chemical energy, like in batteries or food or fuel.

That's about how atoms interact and bond together.

It involves the kinetic energy of the electrons whizzing around inside the atoms and the electrical potential energy due to the

repulsions between electrons and nuclei.

Right.

Rearranging atoms can release or store this energy.

And then we get to the really big stuff.

Nuclear energy.

Right.

That's the energy associated with how the particles inside the nucleus, the protons and neutrons, are arranged.

Changing that arrangement, like in fission or fusion, can release enormous amounts of energy.

And finally, the most fundamental connection.

Mass itself as energy.

Mass energy.

Einstein's famous Msc 22 Toms.

It means that mass itself is a form of energy.

Just by existing, an object has an inherent energy content equal to its mass times the speed of light squared.

And this isn't just theoretical, right?

They're a real example.

Oh, absolutely.

Particle physics sees it all the time.

Take an electron and its antiparticle, a positron.

If they meet, they can annihilate each other.

They completely disappear.

Oof.

Gone.

And what comes out?

Pure radiant energy.

Gamma rays, usually.

And the amount of energy in those gamma rays perfectly matches the total mass of the electron and positron, converted using etemp C22s.

Mass was converted directly into energy, and the total conserved value includes that Mamma T22 term.

It's an incredible accounting system that connects motion, position, heat, light, chemistry, nuclear forces, and even existence itself through this one conserved number.

It's one of the pillars of physics.

But it's important to remember, it's not the only conservation law.

Feynman points out there are actually six main ones we know of.

Six, okay.

How does he categorize them?

He makes a really insightful split into two groups of three.

There are three subtle laws, which are deeply connected to the symmetries of spacetime.

Symmetries, like how physics doesn't depend on where or when you do an experiment.

Exactly.

The first subtle law is conservation of energy itself.

It turns out this is mathematically linked to the fact that the laws of physics don't change over time.

An experiment works the same way today as it did yesterday or well tomorrow.

Time invariance implies energy conservation.

Wow.

Okay, what are the other two subtle ones?

Conservation of linear momentum.

That's linked to the fact that physics is the same everywhere in space.

Location independence and conservation of angular momentum is linked to the fact that physics doesn't depend on orientation in space.

Rotational invariance.

So energy, linear momentum, angular momentum, conservation, all stem from fundamental symmetries of the universe regarding time, location, and direction.

That's profound.

It really is.

Then you have the other three laws, which Feynman calls the simple laws.

These feel much more like Dennis counting his blocks.

Simple counting.

How so?

Well, there's conservation of charge.

You just count the total electric charge, positive charges, minus negative charges.

That number always stays the same in any interaction.

Okay, like adding up ones and negative ones.

Precisely.

Then there's conservation of baryon number.

Baryons are heavy particles like protons and neutrons.

You assign them a value of plus one.

Antibaryons get maggots one.

The total sum is conserved.

Another counting rule.

And the third is conservation of lepton number.

Leptons are light particles like electrons and neutrinos.

Again, plus one for particles, minus one for antiparticles.

Count them up.

The total is conserved.

So charge, baryons, leptons are conserved by simple addition and subtraction, like counting objects.

But energy is different.

Energy conservation feels different, doesn't it?

It's not counting discrete blobs.

It's this continuous numerical quantity that comes from complex formulas.

One -cline -one times H2, $12 -millibit, two -suzzi, two, et cetera, all added together.

Its conservation is tied to that deep time symmetry, not just simple counting.

That brings up a really practical point near the end of the chapter.

If energy is always conserved, how can we have an energy crisis?

Why do we worry about running out of fuel?

Ah, the crucial difference between conserved energy and useful or available energy.

The total amount never changes, but its quality, its ability to do useful work, degrades.

Like the heat energy example.

The random jiggling of atoms in the ocean contains a vast amount of energy, but...

But it's incredibly disorganized.

It's very hard to get that random motion to push a piston in a coordinated way.

You need energy in an organized form, chemical potential energy and fuel, gravitational potential energy in a reservoir to easily convert it into useful work.

So energy gets conserved, but it tends to spread out and become more disorganized over time.

That's related to entropy, right?

Exactly.

The laws of thermodynamics, particularly the second law dealing with entropy, govern the availability of energy.

The total number stays the same, but its usefulness decreases as entropy increases.

That makes sense.

We don't destroy energy, we just convert it into less useful forms, mostly waste heat.

Which leads to the final point about our energy sources, sun, coal, oil, nuclear.

They're all about tapping into concentrated, organized forms of energy.

And Feynman ends with that mind -boggling potential of fusion.

Right, the controlled thermonuclear reaction.

He estimated that harnessing the mass energy in just 10 quarts of water per second via fusion.

Could supply all the electrical power needed for the entire United States at the time.

The sheer density of energy locked up in mass, according to UMC 2022, is just astronomical.

So wrapping this up, energy conservation is this incredibly powerful, abstract mathematical principle.

It's a number calculated from various formulas that mysteriously stays constant through every known physical process.

It connects mechanics, heat, electricity, light, chemistry,

nuclear physics, even mass itself.

It's the universal currency of physics, the ultimate accounting principle.

And yet, as Feynman subtly reminds us, we don't really know why it works on a fundamental mechanism level.

Especially since energy doesn't seem to come in countable units like charge or particles.

Its deepest connection seems to be to the symmetry of time itself.

A number that links everything.

Born from the very indifference of the universe to when things happen.

That's definitely something to think about.

Thank you for joining us this deep dive into the conservation of energy.

We hope following Feynman's logic helps you track that elusive number no matter how it tries to hide.

We'll see you next time.