# Lecture 1: Introduction to Energy

Don't Panic

Sections

Albert Einstein (1879--1955) circa 1905.

This was when he was a patent office clerk and discovering special relativity which includes E=mc**2.

``When I was young I despised all authority---and I have been punished for it by having been made into an authority myself.''---from memory. This is my favorite Einstein quote.

Credit: unknown.

This picture was obviously taken in the first decade of the 20th century and by U.S. copyright law is now out of copyright. According to the informative, but not authoritative, source WebMuseum, Paris copyright in all other jurisdictions would have expired if the holder died more than 70 years ago.

# What is Energy?

``Energy is everything.''---anonymous
The
epigraph is an aphorism---and aphorisms are as true as they are false---which is also an aphorism.

So energy isn't truly everything.

But it does turn up everywhere in the physical world---and everywhere it is important---but it remains a bit mysterious.

We use the term energy qualitatively in everyday life---and I think correctly qualitatively---when we discuss our amount of get-up-and-go.

We use the term energy exactly when we discuss the food energy---which we measure with that ridiculous unit the food calorie---electrical energy which we measure with that ridiculous unit the kilowatt-hour).

But what is a general definition?

Actually, energy defies simple definition---most who try to admit as much---and so does your truly.

There is probably no one-sentence adequate definition.

But after a lot of mulling I offer a one-sentence definition plus some multi-sentence explication. At the moment---and maybe only for the moment---I find it satisfying.

Energy is the quantified and conserved element of structure and capacity for change.

1. It's like a quantified stuff in a way.

You can make something with energy: i.e., structure.

You can always change that structure.

Not necessarily easily.

But how much change in structure is possible is limited by how much energy you have: you can't create it or destroy it---this is the famous conservation of energy that we discuss below in section Energy in different forms of structure has different forms.

Concretely, this means that different forms of energy have different formulae for their calculation.

But all those forms are energy because any form can be turned into any other form.

Not necessarily easily.

2. Now what does force have to do with energy?

Forces are physical relationships between things that can cause change or can balance other forces.

Forces transform energy between its different forms.

3. Why is energy so hard to define?

Well above we said it's like a stuff.

In way yes, in a way no.

What what we ordinary think of stuff is some substance.

And one can have pure substances.

But there is NO such thing as pure energy.

That is to say energy separated from the other properties of the physical system you find the energy in.

If there was a pure energy, we could just say well here we have a lump of it and here are it's prices properties.

But we can't do that.

4. Why is energy so abstract?

It's seems always rather remote from direct observables: like length, velocity, and mass.

So usually we have to calculate it from formulae in terms of those more direct observables.

Actually, as we'll discuss in section E=mc**2, energy IS mass.

So if we could measure mass exactly in all cases, we would be measuring energy exactly.

Practically though we can't usually measure mass exactly enough, to determine energy that way for many purposes.

So we have to measure other observables and calculate energy.

The above definition and elaboration was intended to be general---and so is necessarily abstract.

Let's look at two concrete cases of energy: kinetic energy and gravitational potential energy.

1. Kinetic Energy: This is the energy of motion.

In this case, the structure is the velocity of an object relative to some inertial frame of reference (which we will describe in the section Inertial Frames.

For example, the ground is sufficiently an inertial frame for the present discussion.

Kinetic energy is precisely defined because there is a formula for calculating it from observables.

The formula for calculating it for a particle is

```
KE = (1/2)*m*v**2

where

KE is the usual symbol for kinetic energy in physics,

m is the particle's mass,

v is the particle's speed,

and

**2 is an old fortran way of writing to the 2nd power.
```
A particle in this context is anything small enough that one can ignore it's size and internal structure.

If an object can't be regarded as a particle, one should use the speed of the center of mass which is the mass-weighted average position of the object---but that's a refinement we'll skirt.

The metric system unit of energy can be read derived from the kinetic energy formula: one gets kg*(m/s)**2

which has the special name of joule (J)

and that rhymes with drool

and that honors James Joule (1818--1889)

which rhymes with bowel

and that starts with b ...

(Apologies to the The Music Man.)

All energy forms are measured in joules or units that can be converted into joules.

But where exactly is the kinetic energy?

It's reasonable to say that it is in a moving object and spread distributed throughout it.

The kinetic energy of each little part is given by (1/2)*m*v**2, where m is the mass of the part and v is the speed of the part.

But there is a subtlety.

Velocity depends on the frame of reference.

For example, an object in a moving car has a velocity relative to the ground, but not relative to the car's own frame of reference.

So kinetic energy is dependent on the frame of reference.

In going from one frame of reference to another, one has to do a transformation of kinetic energy.

Which leads to another unsightly fact.

The conservation of energy principle dictates that energy is neither created nor destroyed, but how much of it you have depends on your frame of reference.

Is there frame of reference or set of frames of reference in which true absolute amounts of energy are measured.

I don't really know. Probably some does.

But it might be the set of frame of reference that participate in the mean expansion of the universe.

We'll discuss this briefly in the section Inertial Frames.

2. Gravitational Potential Energy is the energy of position in a gravitational field.

Any object with mass establishes a gravitational field around it.

The gravitational field is just a thing at every point in space that causes the gravitational force at that point.

There is an energy associated with locating an object in the the gravitational field of another object.

By locating the first object relative to another you have established a structure.

Often, but not always, this energy can be called gravitational potential energy.

The general formula for gravitational potential energy is a bit beyond our scope.

But the formula for near the Earth's surface is simple enough.

```              PE = mgy
```

It is never created or destroyed as far as we know---an observational fact and theoretical justified as we'll discuss below in section Conservation of Energy.

The physicsy way of expressing this is to say that there is conservation of energy or that ENERGY IS CONSERVED.

For example, in any closed system, the total amount energy stays constant.

A system is any set of objects that you wish to consider together.

Outside the system is the environment.

When the system and environment do not interact, the system is closed.

Division of the world into system and environment is an idealization that helps further the analysis.

Usually both system and environment are studied through ideal models and one increases the realism as one's understanding improves.

Since the system is of main interest, one usually only models the environment insofar as it affects the system.

To go further energy seems to be conserved in any little interaction---which is why it is converved overall in any isolated system.

The last statement takes some qualification in that it is sometimes tricky to define what constitutes a single interaction apart from everything else.

But in the idealization that it can be done, conservation of energy should hold since then the interaction could be regarded as a closed system.

But conservation of energy MAY be GLOBAL too: i.e., conservation holds for the universe as whole taken as an isolated system---but we're not quite sure.

But though conserved, energy is protean. When almost anything changes, energy is transformed from one form of energy to another form.

In fact, the whole discussion of changes in a system can often be discussed in terms of transformations of energy.

And so you might say energy is like a substance---except all substances actually can be created and destroyed: even those made of a pure element as we now know---so, in fact, energy is NOT like a substance in some ways.

Also a substance is can be refined to give a pure sample.

But energy is never seen ``pure''---it is always in some form.

The Earth at night circa 2000oct23.

Electromagnetic radiation or light just has an associated energy (or speaking loosely is a form of energy) despite a mild temptation to think of it as the purest form of energy.

The image is a collage, since it's not night everywhere on Earth at once---though Dracula would wish it otherwise.

Credit: NASA: Visible Earth. NASA allows free use of this image, but apparently has not declared it public domain which is NASA's usual practice. Image by Craig Mayhew and Robert Simmon, NASA GSFC.

And even then we don't see the form---at least not as directly as we see length or motion.

We see more direct observables and we calculate the energy by measurements of the more direct observables.

For example, motion has an associated energy: kinetic energy.

The formula for calculating it for a particle is

```
KE = (1/2)*m*v**2

where

KE is the usual symbol for kinetic energy in physics,

m is the particle's mass,

v is the particle's speed,

and

**2 is an old fortran way of writing to the 2nd power.
```
A particle in this context is anything small enough that one can ignore it's size and internal structure.

If an object can't be regarded as a particle, one should use the speed of the center of mass which is the mass-weighted average position of the object---but that's a refinement we'll skirt.

The metric system unit of energy can be read derived from the kinetic energy formula: one gets kg*(m/s)**2

which has the special name of joule (J)

and that rhymes with drool

and that honors James Joule (1818--1889)

which rhymes with bowel

and that starts with b ...

(Apologies to the The Music Man.)

But where exactly is the kinetic energy?

It's reasonable to say that it is in a moving object and spread distributed throughout it.

But there are subtelties.

For example, say you had two observers, Moe and Larry, in relative motion.

Moe says ``I'm at rest and Larry is moving and has KE.''

But Larry says ``NYAAAH, I'm at rest and Moe is moving and has KE.''

There seems to be a paradox.

The resolution---which we won't explore in mathematical detail---is that both observers will agree on changes in energy (thinking of more than just kinetic energy) and on physical predictions.

They will both also agree that energy is conserved in their own frame of reference---but they disagree on where it is and in some cases how much there is.

There are transformation relations for changing between reference frames that will acount for changes in amounts of energy

But is there some absolute way of counting up energy. Some true reference-frame-independent count of a href="http://en.wikipedia.org/wiki/Energy">energy.

I'm not sure at this moment---the question is more subtle than I thought.

Perhaps it's true that absolute amounts of energy are hard to define, but energy changes which come into observable changes are not.

Is there a simple definition of energy?

Yes and no.

One can offer simple definitions, but they can only be of limited adequacy: e.g.,

1. ``Energy is everything'':

This is a common aphorism (whose origin I cannot trace) rather than a definition, but it expresses the fact that energy, if not everything, is everywhere and is always a consideration.

2. ``Energy is the capacity for change'' or ``energy is the capacity to do work (in the physics sense)''. The two definitions are much the same since work brings about change.

Well yes.

But a bit more precisely one can say ``Energy is the quantified capacity for change'' or ``energy is the quantified capacity to do work (in the physics sense)''.

These definitions help since energy is conserved, the amount of change that can occur to an isolated physical system and to the associated forms of energy is limited by the amount of energy available.

The change that occurs is sometimes described as happening through work which in physics jargon has the meaning of a force acting on an object over (or as we say through) a distance.

3. ``Energy is a scalar quantity that is associated with the state of one or more objects'' (Halliday et al. 2001, p. 117):

A scalar is a quantity described by a single real-number value like temperature, density, and energy.

A vector is a quantity with a magnitude and direction.

Momentum is a vector. When a football player slams into you, which direction you are thrown depends on which direction you were hit, not just on the magnitude of the momentum.

For the record, the formula for momentum is

```
p_vec=mv_vec ,

where p_vec is momentum,

m is mass,

and

v_vec is velocity which in physics
is also a vector.

We won't deal with vectors and vector nature much explicitly
in this module.

```
``Associated''? Well remember what was said above. You don't see energy as directly as we see length and motion.

You see more direct observables (e.g., length and speed) and calculate energy from them.

So saying energy is ``associated'' with systems is often a favored locution.

But saying a system ``has'' energy is used too and more often.

Energy actually takes a long explanation that includes dealing with its particular forms in order to understand it.

This lecture is sort of a long explanation for example.

But it is not nearly long enough for complete understanding.

Another even longer explanation is given by Wikipedia: Energy. I found it enlightening, but somewhat confusing---which is probably because the article tries to cover a lot of ground without being an entire book.

# Conservation of Energy

Is there a theoretical proof for the
conservation of energy?

Having a theoretical proof would mean that we understand conservation of energy in terms of more basic or fundamental law.

Which would be satisfying in our endless quest of reductionism---which began at an early stage:

``But Mama why?''

``Just because.''

The answer is yes: there is a theoretical proof---but the instructor has NOT made the effort to understand it or whether it is really adequate for everything energy is or is supposed to be.

The proof is Noether's theorem (1918) which shows that energy will be conserved if the laws of physics are invariant with time: i.e., they don't change with time.

Remember a proof is only as good as it's underlying assumptions and those may not be guaranteed---and I don't know if they are.

Mathematician Emmy Noether (1882--1935) when young.

Credit: Unknown it seems. The image is obviously from the early 20th century, probably the before circa 1910. The copyright is probably ended and it is even likely that no one knows who held it.

But evidently Noether's theorem is not considered sufficient even by great physicists.

Here is a quote from Richard Feynman (1918--1988), a famous 20th century physicist whose was also a popular scientist du jour in the 1980s:

There is a fact, or if you wish, a law, governing all natural phenomena that are known to date.

There is no known exception to this law---it is exact so far as we know.

The law is called conservation of energy; it states that there is a certain quantity, which we call energy that does not change in manifold changes which nature undergoes.

That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity, which does not change when something happens.

It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same.

--- Feynman et al. 1963, p. 4-1, quoted in Wikipedia: Energy but with a couple of typos as of 2007aug17.

Maybe Feynman just didn't want to delve into the gore of Noether's theorem in his undergraduate physics lectures.

A somewhat similar perspective is offered by another authority A. P. French (1920--).

Although we may not be able to define energy in general, that does not mean that it is only a vague qualitative idea.

We have set up quantitative measures of various specific kinds of energy: gravitational, electrical, magnetic, elastic, kinetic, and so on.

And whenever a situation has arisen in which it seemed that energy had disappeared, it has always been possible to recognize and define a new form of energy that permits us to save the conservation law.

The French quote suggests that we have just invented forms of energy to preserve conservation of energy and that somehow energy is not a real thing.

This is NOT the case as I or anyone else would say.

The very fact that we've always found new forms of energy---they've been there for us to discover---suggests that energy is a real thing.

Also we have Noether's theorem.

And also there is ONE KIND of measurement that in principle measures all kinds of energy indiscriminately which we will mention in the section Mass-Energy Equivalence or E=mc**2.

We cannot make this kind of measurement in practice directly in many cases, but all of inextricably-linked modern physics tells us it should work in principle.

In many/almost all modern physics branches, the well established results of other branches are used as givens.

It's logically like a house of cards or nearly.

If some well established result was wrong, the house would fall logically.

But the house doesn't fall.

So even if some results cannot be very directly tested in some regimes, we often strongly believe they should hold in those regimes.

Sometimes we're wrong, but that's pretty rare nowadays.

And when it does happen, it's pretty colossal.

That ONE KIND of measurement detects all kinds of energy strongly suggests that energy is a real thing.

# Utility of the Energy Concept

Various physics techniques (with varying amounts of effort) allow you to calculate the DETAILED future and past of systems using (along with many other things) the energy concept (or, to be brief,
energy).

Equally importantly, using energy one can often calculate LIMITED predictions of past and future behavior very easily.

To give a homely example, you know how many gallons of gasoline you have in your car and thus you know how far you can drive without a refill.

Number of gallons is a measure of available energy since there is so much chemical energy per gallon and some fraction of that gets turned into kinetic energy of your car and that kinetic energy moves your car and gets lost to waste heat ultimately.

Note the prediction is easy, but LIMITED: you know how far you can go, but not where you will go.

It takes more than physics to determine where you will go.

We will be seeing lots more examples of the utility of energy as we go along in this course.

# The Empire of Energy

The
word energy or rather energeia was coined in the Metaphysics of Aristotle from a compound of ``en'' (at or in) and ``ergon'' (work).

It is not clear whether Aristotle (384--322 BCE) or some later editor did the coining (Wikipedia: History of Energy).

"Beginning of book 7 of Aristotle's Metaphysics, translated into latin by William of Moerbeke. 14th century manuscript.": Linked source: Wikipedia

Aristotle the supreme authority.

He gave it a philosophical meaning which is not at all clear (to me).

Energiea Attempts to say what Aristotle meant.

1. ``Aristotle's concept of act or actuality'' (see Greek Philosophical Terms).
2. ``actuality identified with movement'' that he connected (whatever that means) with ``complete reality'' (Smil 2006, p. 1).
3. ``It comes from the Greek energeia, or activity, with the first technical definition of the word being provided by Aristotle. His definition was, however, different from the one that we use today. Every existing thing, he said, has an energeia that maintains it in being and is related to its end or function, or telos. He called a body's potential or capacity for action its dynamis, and used en-ergeia to refer to the body being "at work" en route to - or at - that telos. As the philosopher Stephen Toulmin has shown, Aristotle's views derive from the everyday phenomena that he was seeking to explain - in which an agent (such as a horse) faces obstacles (the resistance of road and cart) to keeping a body (the cart) in motion.'' ("What does energy really mean?" by Robert P Crease).

From these statements, Aristotle's energeia seems closer to what we mean by energy transformation.

Imprecisely defined ideas that seem related to energy as we use it qualitatively seem to arise in many cultures, but often with a spiritual dimension.

Such ideas amount to what can be called life-energy: e.g., mana in Pacific Ocean cultures.

Mana and other related concepts seem hard to define precisely.

But as we've seen energy is in the same boat.

But to get back on the historical track, the expression vis viva (Latin for living force) was introduced by Gottfried Wilhelm von Leibniz (1646--1716).

Philosopher, mathematician, physicist Gottfried Wilhelm von Leibniz (1646--1716) with big hair.

I think the hair is phoney.

Credit: Bernhard Christoph Francke (?--1729). No copyright existed at the time of this painting and Wikepedia maintains that the image is in the public domain.

Leibniz gave a precise definition for it: mass times speed squared.

He noticed in some mechanical systems (e.g., metal balls colliding on a smooth, flat surface) that the SUM of the vis vivas of the parts remained constant as the system evolved in time.

This is conservation of vis viva or as we would now say conservation of kinetic energy which is valid if there no forces do net work (in the physics sense): they may do work, but not net work.

Remember work in physics is W=Fd where W is work, F is the force doing the work, and d is the component of displacement in the direction of the force.

For example, consider the a system of perfectly elastic balls sliding on a frictionless surfaces and interacting through collisions.

There is no work done by the surface since no surface force acts parallel to the surface

The balls do work on each other through the collisions.

But since the balls are perfectly elastic, in a collisions the net work is zero.

```                       W_12=F_12d  work of 1 on 2.

W_21=F_21d  work of 2 on 1.

But by Newton's 3rd law F_21=-F_12,

and thus W=W_12+W_21=0.

```

In the elastic ball system, collisions can redistribute the kinetic energy among the balls, but they don't change the overall total kinetic energy.

Every statement in physics and science always seems to demand a qualification---until you give up from exhaustion.

In this case, it is that it is the sum of translational kinetic energy and rotational kinetic energy that is conserved.

Unlike ideal point-masses, the balls can rotate.

Also during collisions, one may imagine that some kinetic energy gets transformed into elastic energy (the potential energy of compression) for short or ideally zero periods of time.

Question: What force/forces commonly acts/act to reduce kinetic energy in everyday life?

1. Friction
2. Gravity.
3. None of the above.
4. Both of the above.

Answer 1 and 2 are right, but 4 is rightest.

Friction is pretty much always acting even when you don't think of it much. For example, the internal friction of your body that is constantly resisting motion.

But friction is good. Without it we'd be slipping all over the place.

Gravity takes away kinetic energy, for example, when a ball is thrown up.

But it can also give it back, for example, when a ball falls down.

But you can imagine making friction less and less for the elastic balls system and then you see the conservation of kinetic energy as the ideal case.

Leibniz must have been making idealization argument to discover the conservation of vis viva.

No actual experiment he could have done could have shown it and, in fact, I rather think he would have had a hard time getting very close given the experimental equipement available to him---and given the fact that he was more of a mathematician and philosopher than an experimenter.

He would have always seen loss of vis viva to waste heat due to friction.

Imagining such ideal cases or thought experiments is the standard procedure in physics since Galileo.

Idealization probably goes back way longer intermittantly.

Archimedes (287?--212? BCE) probably consciously used idealization in his science.

"Bildbeschreibung: Bronzeskulptur Archimedes by Gerhard Thieme (1972) Berlin-Treptow, Berlin/Germany; Standort: Berlin-Treptow, im Garten der Archenhold-Sternwarte, photo taken by SpreeTom on 2006-12-14": Linked source: Wikipedia.

In fact, in order to obtain exact mathematical physical laws, one must believe idealization is valid to explain away discrepancies.

Archimedes did obtain such laws, and so he idealized.

Einstein was also much addicted to Gedunkenexperimenten as he called them in his German way.

The point is to grasp the underlying basic principles undeterred by the complications of actual systems.

Actual experiments are designed to approach the ideal cases by eliminating uncontrolled factors.

The concept of vis viva stuck around and in 1807 in a lecture at the Royal Institution in London Thomas Young (1773--1829) changed the name of vis viva to energy which thus began its career as a scientific term (Smil 2006, p. 1; Wikipedia: vis viva).

Young was using energy just for kinetic energy.

In the course, of the 19th and 20th centuries, the concept of energy has been extended to many realms---which required inventing/discovering new forms of energy.

1. As mentioned above kinetic energy can just be derived from Newtonian physics and the physical concept of work.

2. Elastic potential energy gravitational potential energy probably followed quickly from Newtonian physics, the physical concept of work, and the elastic and gravitational force laws.

3. Now what about heat (really thermal energy)?

In the 18th century, heat was hypothesized to be an indestructible substance called caloric.

But American expatriot Benjamin Thompson (1753--1814) noticed when boring cannon---causing cannon ennui---that heat seemed to be endlessly generated and concluded that it seemed arise from motion.

This was probably not the first time this had been noticed, but apparently this did kind of eliminate the caloric theory.

"Englischer Garten, Muenchen (Munich), Deutschland (Germany); Source Own work; Date 2002; Author Fritz Geller-Grimm Permission (Reusing this image); CC-By-SA-2.5": Linked source: Wikipedia.

Benjamin Thompson (AKA Count Rumford) also designed the Englisher Garten in Munich where I used to occasionally quaff a beer in die guten alten tagen.

James Joule (1818--1889) in the 1840s, then showed that what one measures for heat changes corresponded to changes in mechanical energy (which is the sum of kinetic energy and potential energy).

"Picture of James Joule": Linked source: Wikipedia.

18th century scientists shaved; 19th century scientists tended to be furry.

In fact, mechanical energy could be converted into heat and vice versa conserved the total of heat and mechanical energy.

It was clear heat was another form of energy.

4. Others forms of energy, were developed.

To make a long story, very short, in the 19th and 20th centuries the energy concept was extended to include:

1. Chemical energy.

This is the energy of chemical bonds and is actually a combination of electric potential energy, kinetic energy, and probably some magnetic field energy when looked at the microscopic level.

In the jargon of physicists ``microscopic'' means molecular scale or smaller.

Anything much bigger is ``macroscopic'' and this includes actual microbes.

In between, we have ``mesoscopic'' which may (but I don't) include the nanometer scale.

But actually these terms are used rather loosely with a lot of context dependence.

2. Elastic energy.

This is potential energy of distortions of bodies: e.g., stretched springs.

At the microscopic level it is electrical potential energy.

3. The energy of the electromagnetic fields (Wikipedia: electromagnetism).

Actually, this energy overlaps with electrical potential energy.

The energy of electromagnetic radiation is a form of this form of energy.

4. Kinetic energy.

As noted above this is the energy associated with motion has the simple formula

```
KE = (1/2)*m*v**2

```
But this is only for speeds much less than the vacuum speed of light actually.

5. Mass. We will discuss this in the section Mass-Energy Equivalence or E=mc**2.

6. Mechanical energy.

This is the sum of macroscopic kinetic and potential energies in a macroscopic system.

7. Nuclear energy. There are two nuclear forces: the strong nuclear force and the weak nuclear force.

They both have associated potential energies.

8. Potential energy.

All the fundamental forces have associated potential energies.

We mentioned the nuclear potential energies above.

There is also electrical potential energy and gravitational potential energy.

9. Rest mass energy. We discuss this in section Mass-Energy Equivalence or E=mc**2.

10. Thermal energy.

This is actually a combination of microscopic electric potential energy, kinetic energy, and probably some magnetic field energy when looked at the microscopic level.

If you are not a pedantic physicist, you can call this heat or heat energy.

As the above list makes clear, there are lots of forms or categories of energy and the energy categories overlap actually.

The overlapping is because even if two forms of energy are intrinsically the same, they may manifest themselves in such different ways or appear in such different contexts that two names are convenient.

The above list not exhaustive, because special names for forms of energy are invented for special cases: e.g., sound energy for the kinetic energy and electrical potential energy associated with sound waves.

But all forms of energy are energy, because---among other things---any form can be changed into any other form.

Nature can do that alone and we can make nature do it.

But those transformations are not necessarily easy for us---at least not at the macroscopic scale.

Some are: you can change kinetic energy into thermal energy (i.e., waste heat) very easily.

In fact, you can do it completely.

Drop an object its gravitational potential energy is converted to kinetic energy by the force of gravity which accelerates the body downward. But that kinetic energy is converted to thermal energy shortly after it hits the ground.

There could be some bouncing and some sound both of which have associated energies, but they die out and soon and only thermal energy is left.

Friction internal to the object and the surface it lands changes the kinetic energy and elastic energy (stored in deformation of the object and surface) into thermal energy pretty quickly usually.

On the other hand, as we will see, thermodynamics forbids the changing of some amount of thermal energy entirely into kinetic energy in any cyclic process (e.g., a heat engine process).

All the forms of energy are needed to to preserve conservation of energy.

But as we argued in section Conservation of Energy, nature seems have forced us to this invention.

The very fact that we've always found new forms of energy suggests that energy is a real thing and it is conserved.

# Mass-Energy Equivalence or E=mc**2

Albert Einstein (1879--1955) circa 1905.

He was always a flashy dresser.

Credit: unknown.

This picture was obviously taken in the first decade of the 20th century and by U.S. copyright law is now out of copyright. According to the informative, but not authoritative, source WebMuseum, Paris copyright in all other jurisdictions would have expired if the holder died more than 70 years ago.

In 1905, Einstein published three famous papers including the one on special relativity with all of its predictions for the dependence of time and length on relative motion (or more exactly dependence on inertial frame of reference).

Here is not the place to go into all that, except to say that inertial frame of reference is an unaccelerated coordinate system.

Oh well, we can say a bit about inertial frames.

Interactions as it turns out are not just with respect to bodies or fields.

They are also with respect to space or spacetime in relativistic physics

But only space described by inertial frames.

For example, the ground is for many purposes---but not all---approximately an inertial frame

To accelerate relative to the ground, a force must act on you.

But if a car accelerates relative to the ground and goes by you.

You are accelerated relative to the car, but no force causes that.

Question: Is Newton's 2nd law (F=ma) wrong?

1. No.
2. Yes.
3. Maybe.

Newton's laws are defined relative to inertial frames.

For pedagogical (or sloppiness) reasons, this is often not made clear to students.

The car does not define an inertial frame.

The set of frames that participate in the mean expansion of the universe are the primary inertial frames in modern theory.

Frames unaccelerated relative to a primary set member and local to the set are also inertial frames

All other local frames are not exactly inertial frames.

Almost all physical bodies that rotate (like planets and stars) do NOT exactly define inertial frames.

To be explicit almost every large material body is rotating and therefore accelerating: clusters of galaxies, galaxies, stars, planets in orbit and on their axes, all things on planets.

Rotating frames are always NON-INERTIAL strictly speaking.

Local Group of galaxies Local Group The rotation means they are accelerated relative to primay set of inertial frames.

But like the Earth's surface such frames may be approximately inertial for most purposes.

For some purposes one does have to correct for non-inertial frame effects.

Einstein also published a 4th famous paper in 1905 or a 2nd one on special relativity.

In this 4th paper he considered its further consequences of special relativity for mass (Bernstein 1973, p. 97--98).

He had already shown that mass should depend on relative motion in his first special-relativity paper.

The mass of a body measured in the inertial frame of reference where the body is at rest is the rest mass.

Since mass changes with motion are minute in most human contexts and a lot of others too, we usually just say mass for rest mass.

But sometimes as in this section, one needs to be clear and say rest mass when there is any chance of ambiguity.

In the 2nd special-relativity paper or, as we will call it, the E=mc**2 PAPER, Einstein deduced that all mass including that of matter in its own frame of rest should have an associated energy.

The derivation from special relativity was not in strict axiomatic mathematical style, but involved invoking reasonable physical principles (Lawden 1975, p. 46).

The derivation doesn't have the rigor of pure math.

The formula which we call the EINSTEIN EQUATION or the mass-energy equivalence equation or just rattle off in lieu of any name is

```
E = m*c**2

where E is the total energy of an object,

m is its mass,

and

c is the speed of light in a vacuum.

```
Question: What is c?

1. 10 m/s.
2. 344 m/s.
3. 2.99792458*10**8 m/s = approximately 3*10**8 m/s.

The vacuum speed of light is very high and the funny effects of special relativity manifest themselves only when the speed of objects approaches c to within a factor of 10 or so.

Then they show up depends on the sensitivity of your measurement, of course. The effects turn on gradually as speed is increased.

Answer 2 is the speed of sound in air at sea level with normal conditions and T=20 degrees C (Wikipedia: Speed of sound).

Answer 1 is about the maximum running speed of a human---an Olympic human that is.

Question: At firework displays, the sight and sound of an explosion are:

1. simultaneous.
2. in order sight then sound.
3. in order sound then sight.

Answer 2 is right. Light is faster than sound.

You-all should remember those endless 4th of July firework displays---John Philip Sousa, etc.

It seems as if you are watching a film with the picture and sound not synchronized properly.

In special relativity, the vacuum speed of light is the ultimate physical speed.

All observers in inertial frames, no matter what their relative motion, measure the speed of a light beam in a vacuum to be c.

What about those in non-inertial frames?

Textbooks are elusive on this very simple point.

I'm not sure.

This, of course, has weird consequences for time and length, etc., that we won't go into that here.

Well we can say that time, length, mass, and energy all become dependent on the frame of reference.

The fact that time flows differently in different frames of reference is the most mind-blowing.

We don't notice this or other special relativity weirdness in everyday life, because these effects only become noticeable with relative speeds near the vacuum light speed which we don't encounter in everyday life---except for light itself which is a special case.

But the effects are quite measurable and special relativity is a well verified theory.

Actually, E=mc**2 has two readings both of which are correct.

1. The first reading is that all energy has an associated mass (Lawden 1975, p. 46).

For example light has mass since it has energy, but it has no rest mass which is why it can travel at the vacuum speed of light.

When people say light is massless that is just a shorthand for for saying light has no rest mass.

Thermal energy has an associated mass: heat a body and its mass will increase. Heat flows into the body from somewhere and its energy and mass increase.

A body in motion relative to you has kinetic energy and therefore more mass than relative to you than if it were at rest.

These associated masses or mass changes were too small to notice before special relativity came along and even now are hard to detect. But they are detected when we have sensitive enough equipment.

Actually in this reading of E=mc**2, a measurement of mass is a measurement of energy.

One can even say that the distinction between mass and energy has disappeared.

The characteristics of mass (resistance to acceleration and gravitational attraction) are just characteristics of energy.

You can say mass and energy are the same thing seen in different aspects usually.

In fact, especially when speaking in the jargon of special relativity people sometimes stop making distinction between mass and energy and just say MASS-ENERGY.

Another point about mass-energy equivalence is that since all energy has mass a measurement of the mass of a system is a measurement of the energy of the system.

Thus all forms of energy can be detected by one kind of measurement in principle as discussed in Section Conservation of Energy.

But actually, many changes in amounts of energy that occur in everyday life though very large in energy effects are too small to detect changes in mass.

This is why before 1905 people thought one did have conservation of mass in thermal and chemical reactions.

2. The second reading of E=mc**2 is what has really caught people's attention.

There is a form of energy associated with the rest mass.

This could be called the rest-mass energy though it seems people seldom do.

The most striking thing about rest-mass energy is how large it is for small objects. For example,
```

E = 1 kg * ( 3.00*10**8 m/s )**2

= 9*10**16 J  = approximately 0.1 EJ = approximately 1/5000 of world
commercial energy consumption per year

= approximately 20 megatons of TNT (explosion energy).

See

Wikipedia:  Energy units
and
Wikipedia:  TNT equivalent.

```

Caption: Castle Bravo test---The Castle Bravo test "was an experimental thermonuclear device, 15-megaton weapons related surface event. Detonated 28 February 1954 on Bikini Atoll."

This was scary---it still is.

Click on image and on the next one for the high resolution image.

Permission: Public domain at least in USA.

So in principle one could change small amounts of rest mass energy by human standards into large amounts of other kinds of energy by human standards.

However, macroscopic amounts of rest mass are pretty stable in the human context.

In fact, it is hard to change much rest mass to other energy forms in the context of most places in the universe including where we live.

You could do it with the right ingredients.

Say you had a kilogram of matter and a kilogram of antimatter.

Ramming together they would tend to annihilate and create intense electromagnetic radiation.

Eventually, all the antimatter would be gone (and about 1 kg of ordinary matter too) and about 18*10**16 J of radiation and heat would be left.

But antimatter on Earth and in observable universe (so far as we can tell) only exists in microscopic traces as a result of certain nuclear and high-energy particle reactions.

Neither nature or we can more than build up microscopic traces.

Antimatter keeps annihilating with ordinary matter before it can accumulate.

Other imaginable processes for direct conversion of ordinary lumps of rest mass. into other form of energy have their difficulties.

But on the whole this good---we don't want people or nature setting off megaton explosions everywhere.

Minute changes in rest mass happen all the time since all energy has associated mass.

For example, since all energy has mass, chemical reactions that release or absorb energy change rest mass, but by such minute amounts that they are almost undetectable.

Before E=mc**2 came along people took rest mass (then just called mass) as an absolutely conserved quantity.

The conservation of (rest) mass is now seen as only as an approximate result for cases where energy changes for an object of (rest) mass m are much less than mc**2. It is still very useful of course.

Say a chemical reaction in a sample released 1 joule of energy from chemical bond energy to heat energy.

If the heat all stayed in the sample, the sample mass would not change since the heat energy just has the mass previously had by the chemical bonds.

If all the heat flowed out of the sample, the sample mass would decrease by

```                   Delta m = E/c**2 = 1 J/(3*10**8)**2 =approximately 10**(-17) kg  .
```
Such mass changes were unmeasurably small before 1900 and may not be measurable even today.

I imagine chemical-reaction mass changes can be measured nowadays, but that's just a guess.

In our environment, the biggest relative changes in rest mass are occur for nuclear reactions.

Chemical reactions change the chemical bonds of molecules.

Nuclear reactions change the nuclear force bonds of the atomic nuclei.

There is some analogy between the two cases, but there are many differences and one of them is the scale of the energy released or absorbed.

Nuclear reactions are typically of order 10**6 (or a million) times more energetic than chemical reactions.

That factor of 10**6 has mesmerized people ever since early days after the discovery of radioactivity in 1896 which preceded the discovery of the atomic nucleus in 1911.

People were effectively doing nuclear physics before they knew they were doing that.

It is WRONG to say that Einstein's discovery of E=mc**2 was the singularly important ingredient in development of nuclear reactions for commercial nuclear power and for nuclear weapons.

There is a whole complex of important ingredients which are inextricably interconnected.

But E=mc**2 is certainly one of the ingredients as well as being an immensely important discovery of pure science.

Almost all of modern physics is inextricably interconnected. So it's impossible to imagine taking away a key result like E=mc**2 without the whole edifice collapsing.

So E=mc**2 and many other results stand or fall together.

And they do stand.

It's like a house of cards---but one that doesn't fall down.

Actually, Einstein rather than immediately seeing E=mc**2 as a key to energy from rest mass saw that the huge energy changes that happen in radioactive decay (then unrecognized as nuclear reactions) compared to chemical reactions meant that changes of masses could possibly be measured.

An observation of these changes would be an experimental verification of E=mc**2.

``It is not impossible that with bodies whose energy-content is variable to high degree (e.g., with radium salts) the theory may be successfully put to the test.''

---Einstein's penultimate sentence in his E=mc**2 PAPER as quoted by Bernstein (1973, p. 98).

Caption: Albert Einstein (1879--1955) with friends Conrad Habicht and Maurice Solovine, ca. 1903. The Olympian Academy.

Or Einstein hanging out with the guys.

``In my youth, I despised all authority---and I've been punished for it by having been turned into an authority myself.''

----Einstein quoted from memory.

Credit: Unknown photographer.

Permission: Public domain at least in USA.