The Physics
Opus in profectus


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The first chapter of this book dealt with the topic of kinematics — the mathematical description of motion. With the exception of falling bodies and projectiles (which involve some mysterious thing called gravity) the factors affecting this motion were never discussed. It is now time to expand our studies to include the quantities that affect motion — mass and force. The mathematical description of motion that includes these quantities is called dynamics.

Many introductory textbooks often define a force as "a push or a pull". This is a reasonable informal definition to help you conceptualize a force, but it is a terrible operational definition. What is "a push or a pull"? How would you measure such a thing? Most importantly, how does "a push or a pull" relate to the other quantities already defined in this book?

Physics, like mathematics, is axiomatic. Each new topic begins with elemental concepts, called axioms, that are so simple that they cannot be made any simpler or are so generally well understood that an explanation would not help people to understand them any better. The two quantities that play this role in kinematics are distance and time. No real attempt was made to define either of these quantities formally in this book (so far) and none was needed. Nearly everyone on the planet knows what distance and time mean.


How about we build up the concept of force with real world examples? Here we go…

free body diagrams

Physics is a simple subject taught by simpleminded folk. When physicists look at an object, their first instinct is to simplify that object. A book isn't made up of pages of paper bound together with glue and twine, it's a box. A car doesn't have rubber tires that rotate, six-way adjustable seats, ample cup holders, and a rear window defogger; it's a box. A person doesn't have two arms, two legs, and a head; they aren't made of bone, muscle, skin, and hair; they're a box. This is the beginning of a type of drawing used by physicists and engineers called a free body diagram.

Physics is built on the logical process of analysis — breaking complex situations down into a set of simpler ones. This is how we generate our initial understanding of a situation. In many cases this first approximation of reality is good enough. When it isn't, we add another layer to our analysis. We keep repeating the process until we reach a level of understanding that suits our needs.

Just drawing a box is not going to tell us anything. Objects don't exist in isolation. They interact with the world around them. A force is one type of interaction. The forces acting on an object are represented by arrows coming out of the box — out of the center of the box. This means that in essence, every object is a point — a thing with no dimensions whatsoever. The box we initially drew is just a place to put a dot and the dot is just a place to start the arrows. This process is called point approximation and results in the simplest type of free body diagram.

Let's apply this technique to a series of examples. Draw a free body diagram of…

  • a book lying on a level table
  • a person floating in still water
  • a wrecking ball hanging vertically from a cable
  • a helicopter hovering in place
  • a child pushing a wagon on level ground

a book lying on a level table

Free body diagram

First example: Let's start with the archetypal example that all physics teachers begin with — a demonstration so simple it requires no preparation. Reach into the drawer, pull out the textbook, and lay it on top in a manner befitting its importance. Behold! A book lying on a level table. Is there anything more grand? Now watch as we reduce it to its essence. Draw a box to represent the book. Draw a horizontal line under the box to represent the table if you're feeling bold. Then identify the forces acting on it.

Something keeps the book down. We need to draw an arrow coming out of the center pointing down to represent that force. Thousands of years ago, there was no name for that force. "Books lie on tables because that's what they do," was the thinking. We now have a more sophisticated understanding of the world. Books lie on tables because gravity pulls them down. We could label this arrow Fg for "force of gravity" or W for it's more prosaic name, weight. (Prosaic means non-poetic, by the way. Prosaic is a poetic way to say common. Prosaic is a non-prosaic word. Back to the diagram.)

Gravity pulls the book down, but it doesn't fall down. Therefore there has to be some force that also pushes the book up. What do we call this force? The "table force"? No that sounds silly and besides, it's not the act of being a table that makes the force. It's some characteristic the table has. Place a book in water or in the air and down it goes. The thing about a table that makes it work is that it's solid. So what do we call this force? The "solid force"? That actually doesn't sound half bad, but it's not the name that's used. Think about it this way. Rest on a table and there's an upward force. Lean against a wall and there's a sideways force. Jump on a trampoline high enough to hit your head on the ceiling and you'll feel a downward force. The direction of the force always seems to be coming out of the solid surface. A direction which is perpendicular to the plane of a surface is said to be normal. The force that a solid surface exerts on anything in the normal direction is called the normal force.

Calling a force "normal" may seem a little odd since we generally think of the word normal as meaning ordinary, usual, or expected. If there's a normal force, shoudn't there also be an abnormal force? The origin of the Modern English word normal is the Latin word for a carpenter's square — norma. The word didn't acquire its current meaning until the 19th century. Normal force is closer to the original meaning of the word normal than normal behavior (behavior at a right angle?), normal use (use only at a right angle?), or normal body temperature (take your temperature at a right angle?).

Are we done? Well in terms of identifying forces, yes we are. This is a pretty simple problem. You've got a book, a table, and the Earth. The Earth exerts a force on the book called gravity or weight. The table exerts a force on the book called normal or the normal force. What else is there? Forces come from the interaction between things. When you run out of things, you run out of forces.

The last word for this simple problem is about length. How long should we draw the arrow representing each force. There are two ways to answer this question. One is, "Who cares?" We've identified all the forces and got their directions right, let's move on and let the algebra take care of the rest. This is a reasonable reply. Directions are what really matter since they determine the algebraic sign when we start combining forces. The algebra really will take care of it all. The second answer is, "Who cares is not an acceptable answer." We should make an effort and determine which force is greater given the situation described. Knowing the relative size of the forces may tell us something interesting or useful and help us understand what's going on.

So what is going on? In essence, a whole lot of nothing. Our book isn't going anywhere or doing anything physically interesting. Wait long enough and the paper will decompose (that's chemistry) and decomposers will help decompose it (that's biology). Given the lack of any activity, I think it's safe to say that the downward gravitational force is balanced by the upward normal force.

W = N

In summary, draw a box with two arrows of equal lengths coming out of the center, one pointing up and one pointing down. Label the one pointing down weight (or use the symbol W or Fg) and label the one pointing up normal (or use the symbol N or Fn).

It may seem like I've said a lot for such a simple question, but I rambled with a reason. There were quite a few concepts that needed to be explained: identifying the forces of weight and normal, determining their directions and relative sizes, knowing when to quit drawing, and knowing when to quit adding forces.

a person floating in still water

Free body diagram

Second example: a person floating in still water. We could draw a stick figure, but that has too much unnecessary detail. Remember, analysis is about breaking up complex situations into a set of simple things. Draw a box to represent the person. Draw a wavy line to represent water if you feel like being fancy. Identify the forces acting on the person. They're on Earth and they have mass, therefore they have weight. But we all know what it's like to float in water. You feel weightless. There must be a second force to counteract the weight. The force experienced by objects immersed in a fluid is called buoyancy. The person is pulled down by gravity and buoyed up by buoyancy. Since the person is neither rising nor sinking nor moving in any other direction, these forces must cancel

W = B

In summary, draw a box with two arrows of equal lengths coming out of the center, one pointing up and one pointing down. Label the one pointing down weight (or W or Fg) and the one pointing up buoyancy (or B or Fb).

Buoyancy is the forced that objects experience when they are immersed in a fluid. Fluids are substances that can flow. All liquids and gases are fluids. Air is a gas, therefore air is a fluid. But wait, wasn't the book in the previous example immersed in the air. I said there were only three objects in that problem: the book, the table, and the Earth. What about the air? Shouldn't we draw a second upward arrow on the book to represent the buoyant force of the air on the book?

The air does indeed exist and it does indeed exert an upward force on the book, but does adding an extra arrow to the previous example really help us understand the situation in any way? Probably not. People float in water and even when they sink they feel lighter in water. The buoyant force in this example is significant. It's what the problem's probably all about. Books in the air just feel like books. Whatever buoyant force is exerted on them is imperceptible and quite difficult to measure.

Analysis is a skill. It isn't a set of procedures one follows. When you reduce a situation to its essence you have to make a judgment call. Sometimes small effects are worth studying and sometimes they aren't. An observant person deals with the details that are significant and quietly ignores the rest. An obsessive person pays attention to all details equally. The former are mentally healthy. The latter are mentally ill.

a wrecking ball hanging vertically from a cable

Free body diagram

Third example: a wrecking ball hanging vertically from a cable. Start by drawing a box. No wait, that's silly. Draw a circle. It's a simple shape and it's the shape of the actual thing itself. Draw a line coming out the top if you feel so inclined. Keep it light, however. You don't want to be distracted by it when you add in the forces.

The wrecking ball has mass. It's on the Earth (in the Earth's gravitational field to be more precise). Therefore it has weight. Weight points down. One vector done.

The wrecking ball is suspended. It isn't falling. Therefore something is acting against gravity. That thing is the cable which suspends the ball. The force it exerts is called tension. The cable is vertical. Therefore the force is vertical. Gravity down. Tension up. Size?

Nothing's going anywhere. This sounds like the previous two questions. Tension and weight cancel.

W = T

In summary, draw a circle with two arrows of equal length coming out of the center, one pointing up and one pointing down. Label the one pointing down weight (or W or Fg) and the one pointing up tension (or T or Ft).

a helicopter hovering in place

Free body diagram

Fourth example: a helicopter hovering in place. How do you draw a helicopter? A box. What if you're tired of drawing boxes? A circle is a good alternative. What if even that's too much effort? Draw a small circle, I suppose. What if I want to try drawing a helicopter? Extra credit will not be awarded.

You know the rest of the story. All objects have weight. Draw an arrow pointing down and label it. The helicopter is neither rising nor falling. What keeps it up? The rotor. What force does the rotor apply? A rotor is a kind of wing and wings provide lift. Draw an arrow pointing up and label it.

The helicopter isn't sitting on the ground, so there is no normal force. It's not a hot air balloon or a ship at sea, so buoyancy isn't significant. There are no strings attached, so tension is nonexistent. In other words, stop drawing forces. Have I mentioned that knowing when to quit is an important skill? If not, I probably should have.

Once again, we have an object going nowhere fast. When this happens it should be somewhat obvious that the forces must cancel.

W = L

In summary, draw a rectangle with two arrows of equal lengths coming out of the center, one pointing up and one pointing down. Label the one pointing down weight (or W or Fg) and the one pointing up lift (or L or F).

and now… the law

Let's do one more free body diagram for practice.

a child pushing a wagon on level ground

First, establish what the problem is about. This is somewhat ambiguous. Are we being asked to draw the child or the wagon or both? The long answer is, "it depends." The short answer is, "I am telling you that I want you to deal with the wagon." Draw a rectangle to represent the wagon.

Next, identify the forces. Gravity pulls everything down, so draw an arrow pointing down and label it weight (or W or Fg according to your preference). It is not falling, but lies on solid ground. That means a normal force is present. The ground is level (i.e., horizontal), so the normal force points up. Draw an arrow pointing up and label it normal (or N or Fn). The wagon is not moving vertically so these forces are equal. Draw the arows representing normal and weight with equal length.

W = N

The child is pushing the wagon. We have to assume he's using the wagon for its intended purpose and is pushing it horizontally. I read left to right, which means I prefer using right for the forward direction on paper, blackboards, whiteboards, and computer displays. Draw an arrow to the right coming out of the center of the block. I see no reason to give this force a technical name so let's just call it push (P). If you disagree with me, there is an option. You could call it the applied force (Fa). That has the benefit of making you sound well-educated, but also has the drawback of being less precise. Calling a force an applied force says nothing about it since all forces have to be applied to exist. The word push is also a bit vague since all forces are a kind of push or pull, but pushing is something we generally think of as being done by hands. Since there is no benefit to using technobabble and the plain word push actually describes what the child is doing, we'll use the word push.

Motion on the Earth does not take place in a vacuum. When one thing moves, it moves through or across another. When a wheel turns on an axle, the two surfaces rub against one another. This is called dry friction. Grease can be used to separate the solid metal parts, but this just reduces the problem to layers within the grease sliding past one another. This is called viscous friction. Pushing a wagon forward means pushing the air out of the way. This is another kind of viscous friction called drag. Round wheels sag when loaded, which makes them difficult to rotate. This is called rolling resistance. These resistive forces are often collectively called friction and they are everywhere. A real world analysis of any situation that involves motion must include friction. Draw an arrow to the left (opposite the assumed direction of motion) and label it friction (or f or Ff).

Now for the tricky part. How do the horizontal forces compare? Is the push greater than or less than the friction? To answer this question, we first need to do something that physicists are famous for. We are going to exit the real world and enter a fantasy realm. We are going to pretend that friction doesn't exist.

Watch the swinging pendulum. Your eyes are getting heavy. You are getting sleepy. Sleepy. I am going to count to three. When I say the word three you will awake in a world without friction. One. Two. Three. Welcome to the real world. No wait, that's a line from the Matrix.

Assuming hypnosis worked, you should now slide off whatever it is your sitting on and fall to the ground. While you're down there I'd like you to answer this seemingly simple question. What does it take to make something move? More precisely, what does it take to make something move with a constant velocity?

In the real world where friction is everywhere, motion winds down. Hit the brakes of your car and you'll come to a stop rather quickly. Turn the engine of your car off and you'll come to a stop gradually. Bowl a bowling ball down your lane and you probably won't perceive much of a change in speed. (If you're a good bowler, however, you're probably used to seeing the ball curve into the pocket. Remember, velocity is speed plus direction. Whenever either one changes, velocity changes.) Slap a hockey puck with a hockey stick and you'll basically see it move with one speed in one direction. I've chosen these examples and presented them in this order for a reason. There's less friction in coasting to a stop than braking to a stop. There's less friction in a hockey puck on ice than a bowling ball on a wooden lane.

How about an example that's a little less everyday? Push a railroad car on a level track. Think you can't do it? Well think again. I'm not asking you push an entire train or even a locomotive — just a nice empty boxcar or subway car. I'm also not saying it's going to be easy. You may need a friend or two to help. This is something that is routinely done by railroad maitenance crews.

Video screen shot of workers moving a subway car in a maintenance shopWorkers moving a subway car. Source: 所さんの目がテン!


Three free body diagrams


Heaven is a place where nothing ever happens.

Isaac Newton (1642–1727) England. Did most of the work during the plague years of 1665 & 1666. Philosophiæ Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy) published in 1687 (20+ year lag!) at Halley's expense.

Lex. I. Law I.
Corpus omne perſeverare in ſtatu ſuo quieſcendi vel movendi uniformiter in directum, niſi quatennus illud a viribus impreſſi cogitur ſtatum suummutare. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.
Projectilia perſeverant in motibus ſuis, niſi quatenus a reſiſtentia aëris retardantur, & vi gravitatis impelluntur deorſum. Trochus, cujus partes cohærendo perpetuo retrahunt ſeſe a motibus rectilineis, non ceſſat rotari, niſi quatenus ab aëre retardantur. Majora autem planetarum & cometarum corpora motus ſuos & progreſſivos & circulares in ſpatiis minus reſiſtentibus factos conſervant diutius. Projectiles continue in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are continually drawn aside from rectilinear motions, does not cease its rotations, otherwise than it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, persevere in their motions both progressive and circular for a much longer time.

(Newton, interpreted by Elert)

An object at rest tends to remain at rest and an object in motion tends to continue moving with constant velocity unless compelled by a net external force to act otherwise.

This rather complicated sentence says quite a bit. A common misconception is that moving objects contain a quantity called "go" (or something like that — in the old days they called it "impetus") and they eventually stop since they run out of "go".

If no forces act on a body, its speed and direction of motion remain constant.

Motion is just as natural a state as is rest.

Motion (or the lack of motion) doesn't need a cause, but a change in motion does.

Definitio. III. Definition III.
Materiæ vis insita est potentia resistendi, qua corpus unumquodque, quantum in se est, perseverat in statu suo vel quiescendi vel movendi uniformiter in directum. The vis insita, or innate force of matter, is a power of resisting, by which every body endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.
Definitio. IV. Definition IV.
Vis impressa est actio in corpus exercita, ad mutandum ejus statum vel quiescendi vel movendi uniformiter in directum. An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.
Consistit hæc vis in actione sola, neque post actionem permanet in corpore. Perserverat enim corpus in statu omni novo per solam vim inertiæ. Est autem vis impresa diversarum originum, ut ex ictu, ex pressione, ex vi centripeta. This force consists in the action only; and remains no longer in the body when the action is over. For a body maintains every new state it acquires, by its vis inertiæ only. Impressed forces are of different origins as from percussion, from pressure, from centripetal force.

In general, inertia is resistance to change. In mechanics, inertia is the resistance to change in velocity or, if you prefer, the resistance to acceleration.

In general, a force is an interaction that causes a change. In mechanics, a force is that which causes a change in velocity or, if you prefer, that which causes an acceleration.

When more than one force acts on an object it is the net force that is important. Since force is a vector quantity, use geometry instead of arithmetic when combining forces.

External force: For a force to accelerate an object it must come from outside it. You can't pull yourself up by your own bootstraps. Anyone who says you can is literally wrong.