Chapter 4: Conservation of Energy

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Welcome to the Deep Dive.

Today we're tackling a really fundamental concept from Richard Feynman's Six Easy Pieces.

We're focusing just on his chapter about the conservation of energy.

It's arguably the most central law in all of physics.

Everything hinges on it in a way.

Right.

But it's also famously abstract, isn't it?

It's not like conserving, I don't know, water in a tank.

Exactly.

That's the fascinating part.

Feynman points out right away that this law, this conservation principle, it has no known exceptions.

It works everywhere for everything we've ever observed.

From like stars exploding down to a ball rolling down a hill.

But the thing being conserved energy well, it isn't a physical thing.

You can't hold it.

It's a number,

a mathematical quantity.

So the idea is you calculate this number before something happens, say, before two cars collide, and then you calculate it again afterwards considering all the pieces and heat and sound and the numbers the same.

Precisely.

The total amount stays constant, even though it might change form dramatically.

Feynman uses that analogy of a bishop on a chess board if it starts on a red square, no matter how complex the game gets.

It always lands on a red square.

Okay.

But what is this number?

How do we even start to grasp it if it's not, you know, tangible?

Well, that's where Feynman's genius comes in.

He uses this brilliant analogy to help us build up the idea.

It involves Dennis the Menace and his toy blocks.

Ah, the blocks.

I remember this.

Okay, lay it out for us.

So imagine Dennis has 28 blocks.

They're indestructible.

His mother counts them every day.

The rule, the law in this little world is there are always 28 blocks.

That number, 28, is conserved.

Just pull enough.

But then she counts one day and finds only 27 or maybe 26.

The law seems broken.

Okay.

So what does she do?

Does she just give up on the law?

No, she investigates.

She looks under the rug, finds a block, checks outside the window.

Maybe Dennis threw one out.

She accounts for blocks coming in or out.

Maybe his friend Bruce brought some over or took some home.

She adjusts the count based on these physical movements.

Right.

That makes you track the physical items.

But then comes the twist.

Dennis locks some blocks in his toy box and he won't let his mother open it.

Now she can't see all the blocks.

Ah, so how can she possibly know if the total is still 28?

The law seems unverifiable now.

This is the crucial step towards abstraction.

She doesn't give up on the law.

Instead, she finds a way to calculate the number of blocks inside the box without seeing them.

Oh.

She discovers that each block weighs, say, three ounces.

And the empty box weighs 16 ounces.

So she weighs the lock box.

Let's say it weighs 25 ounces.

Okay.

She subtracts the weight of the empty box that's 16 ounces gone, leaving nine ounces.

Then she divides that by the weight of one block, three ounces.

Nine divided by three is.

Three.

So she calculates there must be three blocks in the box.

Exactly.

She adds that calculated number, three, to the number of blocks you can physically see outside the box.

And lo and behold, the total always comes back to 28.

She's presuming the law by adding an abstract term based on weight.

That's clever.

But what if things get even more complicated,

like Dennis throws blocks into a bucket of dirty water?

Feynman pushes the analogy further.

Yeah.

Now she can't just weigh them easily.

So she develops another abstract term.

Maybe she notices the water level was originally six inches high and each block submerged raises it by a quarter inch.

So she measures the new water subtracts the original six inches, divides the difference by a quarter inch.

And that gives her another calculated number of water blocks.

Wow.

So her formula gets quite complex blocks seen plus blocks calculated from weight, plus blocks calculated from water height.

Right.

And the amazing thing is the sum still consistently equals 28.

She maintains the conservation law through calculation through abstract terms representing unseen or transformed blocks.

And the punch line

The punch line is in real physics, we eventually find we don't have any block seen term at all.

Energy isn't little particles we count directly, not in the way we usually think.

It's all calculation.

We have formulas for gravitational energy, kinetic energy, heat energy, electrical energy.

And we add up all these calculated values.

And the total number remains constant.

We don't actually know what energy is in some fundamental block -like sense.

It's defined by its conservation,

by the fact this calculated sum never changes.

It's a purely abstract mathematical idea.

Okay, that really clarifies the block analogy.

It's about how we account for something conserved, even when we can't directly see it.

So if energy is this abstract calculation, how did physicists even figure out this conservation law holds true?

Did they just guess?

Not exactly guess, but it starts from a very powerful theoretical assumption, a principle about what cannot happen.

It's the idea of no perpetual motion.

Right, the idea that you can't build a machine that runs forever and does work for free.

You can't get something for nothing.

Precisely.

Feynman shows how you can use just this one assumption, no free lunch, to logically derive the formula for one form of energy,

gravitational potential energy.

How does that work?

He uses a thought experiment with weightlifting machines.

Imagine you have two machines.

Machine A is perfect, totally reversible, no friction.

Machine B could be any machine, maybe inefficient.

Okay.

Both machines do the same basic job.

They lower a one -unit weight by a distance of one unit, and in doing so, they both lift a heavier three -unit weight.

Machine A lifts at a distance x.

Machine B lifts at a distance y.

Got it.

One standard weight down, one heavier weight up.

Different machines, maybe different heights lifted.

Now the key question is, could machine B lift the weight higher than machine A?

Could y be greater than x?

What if it could?

If y was greater than x, you could do this.

Run machine B forward, lifting the three -unit weight up to height y.

Then, since y is higher than x, you can just let that weight fall from y down to x.

That falling weight could do some extra work for you, free energy.

Okay, I see that.

You gained something from that extra height difference.

Then you take the reversible machine A, you run it backwards.

Since it's reversible, running it backwards means it takes the three -unit weight at height x and lowers it back down while lifting the original one -unit weight back up to its starting position.

So let's trace that.

The one -unit weight is back where it started.

The three -unit weight is back at height x, but we got free work when it dropped from y to x.

Exactly.

The whole system is almost back to its initial state, except you extracted some network.

That's perpetual motion, getting something for nothing, since we assume that's impossible.

Then y cannot be greater than x.

Machine B can't lift it higher than the perfect reversible machine A.

Right.

And if machine B was also reversible, you could run the whole argument the other way around and prove x can't be greater than y either.

Which means for any reversible machine doing this job, the height must be exactly the same.

X must equal y.

It's a universal height determined purely by the weights involved and the starting drop.

Yes.

The principle of no perpetual motion forces this conclusion.

And we could even figure out what that height x is.

Feynman describes a simple setup, like weights on racks, where you can see by balancing forces.

That if a one -pound weight drops one foot and lifts a three -pound weight a distance x, then three times x must equal one foot.

Correct.

So three dollars x equal one foot, which means six tackles equals thirteen a three foot.

And the quantity that's being balanced here, the thing that's conserved in this reversible process, is weight multiplied by height.

That's it.

We've just derived the formula for gravitational potential energy.

Energy equals weight times height.

Just from logical argument based on no free lunch.

And of course, experiments confirm this relationship holds.

So potential energy is energy something has because of its position, like being high up.

Generally, yes.

Its position relative to something else, like the Earth's gravitational field.

Or it could be electrical potential energy due to position in an electric field.

The change in potential energy is always related to a force acting over a distance.

This principle seems really powerful.

Can we use it to understand actual machines without getting lost in gears and levers?

Absolutely.

That's one of the main points Feynman makes.

The conservation law lets you analyze complex devices, sometimes instantly, just by looking at the overall energy balance.

Forget the internal details.

Give us an example.

Okay, think of an inclined plane, like a ramp.

Feynman uses a three -four -five triangle setup.

Imagine lifting a one -pound weight straight up by three feet.

Okay, the potential energy gain is one pound times three feet.

Right.

Now, instead of lifting directly, you use a ramp that's five feet long along the slope to reach that same three -foot height.

You pull the one -pound weight up the ramp using a counterweight, W, that falls down along the ramp's five -foot length.

So the work done by the weight, W, is W times five feet.

The work done on the one -pound weight is one pound times three feet.

And if the system moves smoothly, without gaining speed, energy conservation says the work done must balance the energy gained.

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

W times five equals three.

So W must be 35 of a pound.

We figured out the force needed just by looking at the distances, not the angle or friction or anything.

Exactly.

It's incredibly efficient.

Feynman even mentions Stavinas's cool epitaph design, a chain draped over a triangle, which shows the same principle through static balance.

What about something like a screw jack?

Those things let you lift a car with minimal effort.

Same principle.

Imagine a jack with a long handle, say 20 inches, and the screw has 10 threads per inch.

This means for every full turn of the handle, the jack lifts the weight by 110th of an inch.

Okay.

Now, suppose you want to lift one ton that's 2 ,000 pounds by just one inch.

To lift it one inch, how many times do you have to turn the handle?

If one turn is 110th inch, you need 10 turns.

Right.

And in 10 turns, how far does your hand move at the end of that 20 inch handle?

Well, one turn is as local with circumference, two pi radius.

So two pi 20 inches, about 126 inches.

10 turns is about 1260 inches.

So my hand moves 1260 inches applying some small force W.

The car moves up one inch, pushed by 2 ,000 pounds.

Energy conservation.

Work in equals work out.

W times 1260 inches must equal 2 ,000 pounds times one inch.

So W is 2 ,000 divided by 1260, which is, wow, only about 1 .6 pounds.

See, the conservation law tells you the force advantage immediately, just based on the distances involved.

It bypasses all the complex mechanics of the screw itself.

This even works for that aren't moving.

Feynman talks about the principle of virtual work.

Yes, this is really neat.

You can figure out forces in a static system like a balanced lever or rod by imagining a tiny hypothetical movement of virtual displacement.

How does imagining something help?

Let's say you have an eight foot rod balanced at one end.

It has a hundred pound weight, two feet from the pivot, a 60 pound weight at the center, four feet.

And you want to know what weight W you need at the far end, eight feet to balance it all.

Let's ignore the rod's own weight for simplicity.

Okay.

A balancing act.

Now imagine the end of the rod with weight W moves down by a tiny amount, say four inches.

Because of the pivot, the center point with 60 pounds must rise half that distance, two inches.

And the point two feet out with a hundred pounds must rise a quarter of that, just one inch.

Right.

Simple geometry of the lever.

Now calculate the change in potential energy for this imagined movement.

W goes down four inches, loses energy.

60 pounds goes up two inches, gains energy.

A hundred pounds goes up one inch, gains energy.

So energy change, energy gained, energy lost, 62 plus 101W4.

And if the rod was initially perfectly balanced, then for this tiny virtual movement, the net change in potential energy must be zero.

The system doesn't spontaneously gain or lose energy just by wiggling slightly around its balance point.

So 62 plus 101W4 equals zero.

That's 120 plus 100 plus 104W.

So 220 equals 4W.

Which means W equals 55 pounds.

You found the balancing weight just by considering the energy changes in a hypothetical motion.

That's the principle of virtual work.

Amazing.

Okay.

So we have potential energy due to position.

What happens when things actually start moving?

Yeah.

Like a pendulum swinging.

Ah, the pendulum.

Perfect example.

When the pendulum bob is at the top of its swing, it's momentarily stopped.

All its energy is potential energy, weight height.

Then it swings down.

As it falls, it loses height.

So its potential energy decreases.

But it picks up speed.

Since energy must be conserved, that lost potential energy has to transform into something else.

The energy of motion.

Exactly.

Kinetic energy.

At the very bottom of the swing, its height is minimum.

So potential energy is minimum, maybe zero, if we set the baseline there.

But its speed is maximum.

All the initial potential energy has become kinetic energy.

And that kinetic energy is what allows it to swing back up the other side, right?

Converting back into potential energy as it slows down.

Precisely.

The kinetic energy, kE, must be equivalent to the work it could do if it were stopped.

Which is equal to the potential energy it came from.

So kE equals weight times the height, h, it fell from.

Or could reach again.

Is there a formula relating kinetic energy directly to speed?

Yes.

By using the physics of falling objects, we find that the height h is related to the velocity v by h equals v for

where g is the acceleration due to gravity.

Substituting that into kE, weight h gives us the standard formula.

Kinetic energy, weight v for you, it's proportional to the square of the velocity.

Okay.

So we have potential energy and kinetic energy.

Are those the only forms we need to worry about to keep the total conserved?

Oh, not by a long shot.

The conservation law only holds if we keep discovering and adding new forms of energy to our calculation as we explore more phenomena.

Like what?

Well, consider friction.

When the pendulum eventually slows down and stops, where did the potential and kinetic energy go?

Ah, it just disappeared.

According to the conservation law, it can't just disappear.

It must have changed form again.

It turns into heat energy.

Yes.

Friction causes the atoms in the pendulum string, the pivot, and the surrounding air to jiggle around more randomly and vigorously.

This disorganized microscopic kinetic energy of is what we perceive and measure as heat.

So the large -scale organized energy of the swing becomes small -scale disorganized energy.

That's a huge idea.

Heat isn't some weird fluid.

It's just atomic motion.

That was a major breakthrough.

We also need elastic energy for things like stretched springs.

They store potential energy that can become kinetic.

There's electrical energy related to forces between charges.

Chemical energy, like in batteries or fuel.

Right.

Feynman describes chemical energy as basically a combination of the kinetic energy of electrons within atoms and the electrical potential energy due to their interactions with atomic nuclei.

Yeah.

Rearranging atoms in chemical reactions changes this energy, releasing or storing it.

What about light?

That's radiant energy.

It's energy carried by electromagnetic waves, wiggling in the electromagnetic field, as Feynman puts it.

And then there's nuclear energy associated with the forces inside the atomic nucleus.

It's not fully understood in the same way as electrical forces, but it's definitely a distinct form that has to be included in our energy accounting, especially in nuclear reactions.

And the really big one, mass itself, EMC only.

Yes, mass energy, Einstein's famous equation.

It tells us that mass is a form of energy.

If you have mass, you have energy simply because you exist.

The C in that equation is the speed of light, which is a huge number.

So C squared is enormous.

It means a tiny amount of mass corresponds to a vast amount of energy.

Is there experimental proof for this?

Absolutely.

If Feynman mentions particle physics examples, like when an electron meets its antiparticle apothetron, they annihilate each other, poof, they disappear completely.

Your mass vanishes.

Their mass vanishes, but in its place, pure radiant energy, like gamma rays appears.

The amount of radiant energy produced exactly matches the energy equivalent of the vanished mass calculated by EMC Wood.

It's a direct conversion, proving mass is part of the total conserved energy.

So the list of energy forms just keeps growing as we learn more.

It seems like whenever the conservation law looks like it's failing, we just invent a new form of energy to make it work.

That's a slightly cynical way to put it, but there's truth there.

However, these new forms aren't just arbitrary inventions.

They have specific formulas, they can be measured independently, and they allow us to make accurate predictions.

The law's power lies in its universality and predictability across all these forms.

Is energy conservation related to other conservation laws in physics?

Yes, it's part of a deeper structure.

Feynman groups the conservation laws.

There are three subtle ones, conservation of energy, conservation of linear momentum, related to motion in a straight line, and conservation of angular momentum, related to rotational motion.

Way subtle.

In quantum mechanics and advanced theory, these three are deeply connected to fundamental symmetries of time.

Energy conservation is linked to the fact that the laws of physics don't change over time.

Linear momentum conservation is linked to the laws being the same everywhere in space.

Angular momentum is linked to the laws being the same no matter how you're oriented in space.

Wow, okay.

That's deep.

What are the other types of conservation laws?

Then there are three simple counting laws, which feel more like Dennis's blocks.

There's conservation of electric charge.

The total net charge, positive minus negative, never changes in any interaction.

Okay.

Conservation of number baryons are particles like protons and neutrons.

The total count of baryons minus anti -baryons is constant.

Right.

And conservation of lepton number leptons include electrons, muons, and neutrinos.

Again, the total number of leptons minus anti -leptons stays the same.

These are like fundamental accounting rules for particles.

So we've come full circle.

Back to counting things, but also understanding this incredibly abstract yet absolutely fundamental mathematical quantity called energy that underpins everything.

It really is the bedrock.

But there's one final crucial distinction Feynman implicitly touches upon,

which becomes vital later in physics, especially thermodynamics.

What's that?

It's the difference between energy being conserved and energy being available or useful.

The total amount of energy in the universe is constant, yes, but its form matters.

How so?

Think about the heat energy in the ocean.

There's a tremendous amount of energy in the random motion of all those water molecules, but it's very difficult to extract that disorganized energy and make it do useful work, like powering a city.

It's low quality disorganized energy.

Whereas the chemical energy in oil or the potential energy of water behind a dam, that's more organized, easier to use.

Exactly.

While the total energy is always conserved when you burn fuel or let water The conversion process often turns some high quality useful energy into low quality disordered heat, like waste heat from an engine.

This tendency for energy to become less useful, more disordered, is related to the concept of entropy and the second law of thermodynamics.

So just because energy is conserved doesn't mean we don't have an energy crisis in terms of finding useful accessible forms.

Precisely.

And that brings us to a really thought -provoking point Feynman makes right the end of the chapter, looking towards the future.

What does he say?

He talks about the sheer amount of energy available, particularly locked up in mass via EMCA SOF.

He gives this stunning example.

If we could master controlled thermonuclear reactions, like fusion.

The process that powers the sun.

Yes.

If we could make that work efficiently here on Earth, using deuterium from ordinary water, he calculates that processing just 10 quarts of water per second.

10 quarts, like a couple of gallons.

10 quarts per second could release enough energy to equal the entire electrical power generation of the United States at the time he was writing.

The energy is there, locked in the mass of water.

Wow.

So the fundamental challenge isn't that energy disappears because the conservation law guarantees it doesn't.

Right.

The challenge for humanity, for our future, isn't the existence of energy.

It's the technology, the understanding, the ingenuity required to convert the abundant forms of energy around us, like mass or fusion fuel, into forms we can actually use effectively and safely.

That's a powerful perspective to end on.

The physics tells us the energy is conserved.

The engineering challenge is utilization.

It really underscores how profound and practical this abstract law truly is.

Well, that's certainly given us a lot to think about.

A fantastic deep dive into just one chapter of Feynman, showing the power of starting with basic principles.

Thank you for joining us.