The Physics
Opus in profectus


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what is work?

The target audience of this book is people with some amount of education. This isn't intended to be a children's book; and by children, I don't mean the opposite of adults. I consider adolescents (or teenagers, if you prefer) to be proto-adults. If this describes you, then you've had some formal science education (good, bad, or ugly). Somewhere along the line, you should have been introduced to the concept of energy. If you haven't, then stop reading this and go get yourself some education (or at least some life experience).

Those of you with a bit of formal education were probably given a lesson on energy at some point in your life. If so, then the chances are pretty good that you were given a definition of energy as "the ability to do work". If you were a good student or you just wanted to please your teacher, you probably heard this and said to yourself, "OK, energy is the ability to do work." If you were a really good student with a desire to learn or a really bad student with a desire to point out your teacher's intellectual shortcomings, you should have then asked the next logical question. What is work?

Hopefully you were given the right answer, but chances are fifty-fifty you were shrugged off. Not because the right answer is so difficult to know, but rather because the right answer is so difficult to explain, or at least difficult to explain in a way that can be grasped quickly. I think this is mostly due to the fact that the word work has two meanings: the ordinary one of everyday life and the technical one of physics.

Technically, work is the force-displacement product (for those of you who prefer algebra)

W = Δs cos θ

or the force-displacement path integral (for those of you who prefer calculus).

W = 
F · ds

I understand that for many of you this is a meaningless definition. So many words and so little said, no? Actually, quite the contrary. This definition is so compact it's like poetry. It says as much as it can in as few words as possible. It's so compact that explaining it in ordinary language makes the half dozen words of the technical definition expand to nearly a hundred words of so-called "natural language". Let me explain what work is though a series of mental images. Whenever an example is presented, remember that work is done whenever a force causes a displacement.

Imagine that a physics teacher is standing motionless before a class of students. Since he isn't exerting any forces that will displace anything outside of his body he isn't doing any work. Obviously. But doing this for any length of time will certainly drain him of energy just as if he pushed papers across his desk all day (an example where a force does result in a displacement). Surely, you could now convince him that his definition of work must be wrong. Maybe a lesser teacher would cave under the pressure, but not a physics teacher.

Most certainly, a physics teacher or any other person standing is doing work, but the work being done isn't easily visible. Inside the body the heart is pumping blood, the digestive system is grinding away on breakfast, receptors are driving molecules across cell membranes. We do work even as we sleep. Forces causing displacements are happening everywhere under our skins. The human body is a busy place.

If a system as a whole exerts a force on its surroundings and a displacement occurs, the work done is called external work. A physics teacher pushing papers across his desk is doing external work. A physics teacher standing motionless is not doing any significant external work.

If a part of a system exerts a force on another part of the same system and a displacement occurs, the work done is called internal work. A physics teacher thinking deeply or lying in a coma is doing internal work. (Extra credit if you can tell the difference between the two.) A physics teacher doing anything — or nothing for that matter — is doing internal work. A physics teacher who is dead is not doing any work, internal or external. In mechanics, when we say work has been done we are often referring to external work.

Now that we've decided that a teacher standing still isn't doing any work, let's imagine a teacher moving around and ask if work was done. Hmm, well anytime arms and legs get moving the situation is moderately complex. This makes it hard to identify what it is about the motion that involves work and what doesn't. We need to simplify things a little bit more. Give the teacher a book (like a physics textbook) and ask him to move the book around in a few simple ways. The question now is, "Did the teacher do any work on the book?" This is much narrower than asking if the teacher did any work, which means it's easier to answer and better suited to introducing the concept.

For a teacher holding a book, or any other system for that matter, work is done whenever a force results in a displacement. Consider the following six examples presented three at a time.

The first example makes obvious sense. Holding a book without moving it surely results in no work being done on the book. Replace the teacher with a table or the floor. A book lies on the floor. What work is the floor doing? Nothing is going anywhere. Nothing is happening. Nothing is being done — not even work.

The second and third examples also make sense. The teacher pushes on the book and it moves. A force resulted in a displacement. Work was done. This agrees with our everyday notion of work. All is right with the world.

Let's look at three more examples.

The first one in this set is bothersome. It's counterintuitive. It basically says that no work is done carrying a book across level ground. It's so patently stupid it must be wrong — right? Wrong! It is right. (You have to read this last bit as an internal dialog for it to make sense.) Work is done on an object whenever a force causes a displacement. In this example, the force applied is vertical but the displacement is horizontal. How does a vertical force affect horizontal motion? Short answer, "It doesn't."

Vertical forces affect vertical motion. Horizontal forces affect horizontal motion. When motion and force are parallel, life is simple. When motion and force are not parallel, life is not simple. The angels leave and the demons take over. And by demons I mean vectors — in particular, vector components. Work is done whenever a force or a component of a force results in a displacement. No component of the force is acting in the direction of motion when the book is moved horizontally with a constant velocity. The force and the displacement are independent. No work is done by the hand on the book.

Take a look at the last two examples in this set of six. Here we see negative work being done. Given what I said about components, this may or may not make sense to you. Once again, when force and displacement are parallel, life is simple.

Apply a force → Displace an object → Do work

When force is not quite parallel to displacement, it's like less force is being used to do the work.

Apply less force → Do less work

That's also pretty simple. When the angle between force and displacement reaches 90°, the component of the force parallel to the displacement reduces to zero.

Apply no force → Do no work

OK, that was counterintuitive at first, but now it makes some sense.

The farther the two vectors get from parallel, the less work is done. Expand the angle beyond 90°. Force and displacement are starting to point in opposite directions. At 90°, no work was done. Beyond 90°, less than no work must be done. This is negative work.

Apply less than no force → Do less than no work

There is another reason to embrace negative work. The sign of work indicates the direction of a change. A negative sign indicates a loss of something. In the case of lowering a book, it means lowering its ability to do work — lowering its energy.

Follow this line of reasoning. Raising a book takes work. Raising a book raises its energy. I can now use the energy stored in the book to do work — and by "work" I mean physical labor, not educating America's youth. I can pound stuff with it — walnuts, insects, square pegs into round holes. The way I do this work is by lowering the book. This also lowers its energy. It can't do anymore work once it's back on the table. Raising the book does work on it. Lowering it undoes work on it. From a work or energy standpoint, the book has returned to its initial state. Numerically the positive work done raising it was cancelled by the negative work done lowering it resulting in zero work being done overall on the book. (The situation is different for the smashed walnut, insect, or square peg.)


Work is done whenever a force results in a displacement. All other things being equal, applying a greater force should result in more work being done. Likewise, exerting a given force over a greater distance should result in more work being done. And as we discussed in the dozen or so paragraphs preceding this one, the component of the force parallel to the displacement is what matters. Work is directly proportional to the first two factors: force and displacement. Direction is handled with the cosine function. Cosine is greatest when the angle is zero (the angle between two vectors pointing in the same direction is zero), zero at ninety degrees (forces perpendicular to displacement do no work), and negative for obtuse angles (forces acting opposite displacement undo work).

Work is best defined by an equation. Here's one common version…

W = Δs cos θ


W =  work done
 =  average force exerted
Δs =  displacement caused by force
θ =  force-displacement angle

This equation assumes that the force is constant in both magnitude and direction relative to the displacement at all times. For many problems this assumption is reasonable, which is why it's written here.


For those cases where changes in magnitude or direction are significant, we introduce our friend calculus. Over some finite displacement, force may be changing in magnitude and direction. Over a smaller displacement, it will surely change less. Cut the displacement up into a series of small displacements, compute the work done on each step, and add the results together. For best results let the steps approach an infinitesimally small size.

While we're at it, let's also replace the cosine function with the more compact dot product notation. There are two ways to multiply vectors — the dot product  · and the cross product × . The dot product is a scalar product that increases with increasing similarity of direction. The trig function that does this is cosine. The cross product is a vector product that increases with increasing perpendicularity and points out of the plane containing the two vectors. The trig function that does this is sine. Since we previously identified cosine as the correct function, we'll be going with the dot product.

W = 
F · Δs

In the limit, the finite Δs becomes the infinitesimal ds and the finite becomes the infinite . A finite sum of finite quantities is always finite. An infinite integral of infinitesimal differentials can also be finite. The magic of calculus is that the latter can be true at all.

Work is best defined by an equation. Here's another common version…

W = 
F · ds

This equation is an example of a path integral (or line integral). When most students are introduced to integration, they are told that integration is the way to find the area under a curve. The way this is done is by mathematically chopping the curve up into infinitesimal segments of uniform width, measuring the area of the rectangular strip that fits between every segment of the curve and the horizontal axis, and then adding the areas of the segments together. There's nothing wrong with this as an introduction to integration, but sometimes students get stuck on the notion that integration is just about "finding the area". Integration is really about putting parts together to make a whole. It's the primary meaning of the word in English and the primary meaning of the word in calculus. Integration can be used to find the area under a curve (I'll call that a traditional integral) but it can also be used to find the amount of some quantity accumulated over a path (a path integral), to find the amount of some quantity captured by a surface (a surface integral), or the amount of some quantity contained in a volume (a volume integral).


The SI unit of work is the joule.

[J = Nm = kg m2/s2]

Work and energy can be expressed in the same units. Unfortunately, there are a lot of units for energy beside the joule. (This is discussed in another section of this book.) The ones most commonly seen in the US in the early Twenty-first Century are probably calorie (diet and nutrition), Btu (heating and cooling), kilowatt hour (electric bills), therm (natural gas bills), quad (macroeconomics), ton of TNT (nuclear weapons), erg (older scientists), and foot pound (older engineers). The first two in this list, the calorie and the Btu, were first introduced by Nineteenth Century scientists studying calorimetry. (The French gave us the calorie and the English gave us the British thermal unit or Btu.) The last one in the list, the foot pound, was introduced by Nineteenth Century scientists studying mechanics. In the Nineteenth Century, calorimetry and mechanics were separate disciplines. Calorimetry is the study of heat. Mechanics is the study of motion and forces. A learned gentleman (and they usually were men at this time) might study both, but he probably didn't link them in any significant way. That is, unless his name was Joule.

James Joule (1818–1889) was a wealthy English brewer who dabbled in various aspects of science and economics. Sometimes these endeavors overlapped. He invented the foot pound as a unit of work. (Foot being the unit of displacement and pound being the unit of force.) This enabled him to quantitatively compare the "economical duty" of different mechanical systems. Coal-fired steam engines were the primary source of industrial might at the time, but electricity was then emerging on the high tech horizon. Joule realized that mechanical work, heat, and electric energy were all somehow interconvertible. Heat can do work. Work can make heat. Work can make electricity, Electricity can do work, Electricity can make heat. Heat can make electricity. Energy is a versatile actor.

Joule's most famous experiment is probably the determination of the mechanical equivalent of heat (to be discussed in more detail elsewhere in this book, I hope). Heat was measured in British thermal units (by the British at least) and work was measured in foot pounds (which Joule invented). Joule established that one British thermal unit of heat was equivalent to approximately 770 foot pounds of mechanical work. (Very close to today's value of 778 ft lb/Btu.) This result was essential in the realization that, despite appearing in multiple forms, energy was one thing.

The International System of Units, which began to dominate the scientific world in the mid-Twentieth Century, was French in origin. Foot pounds and British thermal units had no place in this much more logical system. 12 inches in a foot. 16 ounces in a pound. 128 ounces in a gallon in the US and who knows how many in the UK. The math was much too difficult. Parlez-vous les unités métriques? The SI was French in origin, but international in nature. When the call went out to name the unit of energy, the answer was resounding: Joule! Absolument!

Some notes on units.

conservative vs. nonconservative forces




work energy theorem


W = ΔE

Sign conventions:

Thomas Young (1773–1829) was the first to use this formula. ← Is this right?