Historians and others have made lists of the most influential people in the history of the world. The lists all tend to include Jesus of Nazareth, Isaac Newton, Muhammad, Confucius, Albert Einstein, and others. Apart from religious figures in those lists, Isaac Newton tends to be regarded as the most influential man who ever walked the earth.[br][br]Joseph Louis Lagrange - a brilliant man in his own right who changed the study of motion forever - said of Newton: "Newton was the greatest genius that ever existed, and the most fortunate, for we cannot find more than once a system of the world to establish." In modern language this might be written: "Newton was the greatest genius who ever lived, and a most fortunate man, for the laws of nature can only be discovered once, and Newton had that privilege." [br][br]Here is a list of Newton's major contributions: [br][list][*]Three laws of motion [/*][*]The law of universal gravitation with which he calculated planetary motions [/*][*]The mathematics of calculus that he invented for those calculations [/*][*]A long treatise on optics in which lens aberration and other issues are discussed[/*][*]Invented the reflecting telescope to avoid lens aberrations[/*][/list][br]While this isn't a long list, and isn't comprehensive, these contributions really laid the foundations for all of modern science.

[u]Newton's First Law [/u] [br]We already discussed Newton's first law in the previous chapter.  Recall that it is synonymous with the law of inertia.  In the words of our kinematic variables, the simplest way to write the first law is:[b] Insofar as we can ignore external influences, the velocity of an object will be constant[/b]... and make sure you realize that a zero vector ([math]\vec{v}=\vec{0}[/math]) is a valid constant velocity.[br][br][color=#1e84cc]An Historical Aside:[br]A recent (2023) scientific publication pointed out that the common translation of Newton's first law of motion from Latin to English three centuries ago that has been referenced ever since, has a slight error in it. [br][br]The common statement that every student memorizes tends to go something like: "Objects at rest remain at rest, and objects in motion remain in motion, [b]unless[/b] acted on by an external force." Let's instead look at the original text. It is in Latin and reads: "Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare." The best translation of that is "Every body perseveres in its state of rest or of uniform motion in a straight line,[b][i] except insofar as[/i][/b] it is compelled to change that state by forces impressed upon it." (bold italics added for emphasis). [br][br]The replacement of "except insofar as" with "unless" is the issue at hand. The latter is a weaker statement that would imply that many things are in an inertial state. It sounds a lot like "this thing will be in an inertial state unless something comes along and messes things up. The "except insofar as" version is more suggestive of objects almost always being perturbed by external influences. It reads more like "objects would be going in straight-line, constant-speed trajectories except to the extent that all the other things in nature change that outcome by affecting them". This is a better view of reality since NOTHING IS TRULY ISOLATED. If you are thinking of your computer monitor as being in a state of inertia, think again. It's going in circles as the earth spins on its axis and as that earth carries it on a curved trajectory around the sun. That sun is in a spiral galaxy accelerating toward a galactic cluster. There is no true state of inertia. I think Newton recognized this.[/color][br][br]Expect to understand after studying this chapter that the first law is encompassed by the second law.  So if you understand the second law correctly you will see the first law pop out of it.  I will therefore not spend any additional time in this chapter on the topic of the first law, with the exception of this one example:[br][br]Suppose you know the mass of a rubber ball that is dropped from a tall tower.  At first we'd expect the ball to speed up at a rate given by g=10m/s[sup]2[/sup].  After a while, however, air drag will become so large that the gravitational force - while still pulling - will not manage to speed the ball up any further.  When this occurs, the ball is said to be at [b]terminal speed[/b]. [br][br]While the force of air drag might seem hard to determine, in such cases it's easy.  If we are willing to suppose that the force of gravity trying to speed up the ball is perfectly balanced by the upward resistance of air drag trying to fight the ball's progress, then all we need to know to find drag is the gravitational force acting on the ball. If treated as an interial case, then forces must cancel. In other words, the effect of gravity pulling downward is exactly cancelled out by the effect of air drag pushing upward.  Added as force vectors, they'd give a zero vector.  If, for instance, the mass of the ball was 0.50kg, then the force of gravity would be 0.50kg(10m/s[sup]2[/sup])=5N, and the air drag would also be 5N, except upward and in opposition to the gravitational force.[br][br][br][u]Newton's Second Law [/u] [br]While kinematics from the last chapter is just an exercise in defining useful variables to describe motion (like position, velocity, etc) and the mathematics that relates them, we now will focus on dynamics.[br][br][b]Dynamics is the study of forces and torques (twisting forces) and the subsequent motion that results.[/b]  So in this section we are discussing the [i]cause[/i] of acceleration.  Isaac Newton gave us a very concise way to do this in the form of his second law of motion.  It is a waste of time to use words describing his second law since math does a much better job.  [b]Newton's 2[sup]nd[/sup] law[/b] states: [br][br][math]\sum\vec{F}=m\vec{a}[/math].[br]The left side represents the sum of all the forces acting on an object of interest.  The right side expresses the consequence, which is acceleration of the object. The [i]m[/i] is the mass of the object on which the forces are acting.  Recall that it must be expressed in kilograms. [b]Notice that the forces the object itself may be exerting on other objects in its surroundings do not get included in this sum.[/b] This is important. For example, as an airplane flies through the atmosphere, it shoves air all around - generally downward and backward. While this is interesting, those forces don't cause the plane to fly and stay aloft. Instead the forces exerted BY the air on the plane cause it to fly and stay aloft. One very important fact Newton gave us in his third law which follows next is that the forces on the atmosphere by the plane and those on the plane by the atmosphere are equal and opposite. Being true, added together they make a zero vector like any vectors that are equal and opposite would do. [br][br]Keep in mind that forces are vectors and they must be added by components as we must always add vectors.  To say that a force is a vector simply indicates that it both entails how hard something is being pushed or pulled as well as the direction in which that push or pull is directed.  The unit of force is the newton, which naturally commemorates the man.  [math]1N\equiv1kg\cdot\frac{m}{s^2}[/math]  The triple equal sign is often used to state a definition or an equivalence.  It is worth noting that the units in this equation (and all equations) must match on both sides.  Here we have newtons on the left and kilograms times the acceleration in meters per second squared on the right side.[br][br][color=#b45f06]AN ASIDE [/color][br][i][color=#b45f06]It's worth noting that while it would have been impossible in Newton's time to realize the limitations of this equation, that it is in fact only true when we are dealing with objects that travel a fair bit slower than the speed of light.  (A common rule is v<10% of light speed)  While this is not much of a limitation for ordinary applications, keep in mind that at all of the particle laboratories throughout the world this law would be rather useless.  An equation that works for objects at all speeds - including those particles in accelerator labs - must be written in terms of momentum, and the momentum must be defined a little differently than most first year students or high school students typically see.  But for now, just keep in the back of your mind that in the present form [b]Newton's second law is a non-relativistic relation[/b], a fancy way of saying it doesn't work at really high speeds.[/color][br][/i][br]In ordinary applications where Newton's second law does apply, the most challenging part of applying it is usually related to determining correctly all the force vectors.  So as a first order of business, we need to be able to correctly determine all of the external forces acting on a system.  Many types of candidate forces exist. We already discussed several of these, and will take time to look into more details shortly.[br][list][*]fundamental forces like gravitational or electrostatic forces.[/*][*]contact forces between objects - none of which are really "contact" if you look closely enough.[/*][*]force of air drag (which is really a "contact" force between many, many molecules of air and our system[/*][*]tension forces in ropes/ cables[br][/*][/list][br][u]Newton's Third Law[/u] [br]We discussed how Newton's first law really adds nothing to the second law in the sense that it is a special case of the second law.  The third law, however, stands as a completely different and powerful statement in its own right.  Consider two objects labeled A and B.  By writing [math]\vec{F}_{AB}[/math] I will mean in words "the force [b]on[/b] object A [b]by[/b] object B".  If you reverse the order it will be wrong.  So if a person was in the act of pushing a shopping cart we might think of the shopping cart as A and the person as B.  [br][br]What Newton realized about all interactions in nature is that when A pushes B, that B will [b]always[/b] push back on A just as hard but in an opposite direction.  There is no time delay to this returned push.  Another way to think of this is that [b]all forces in nature come in pairs.[br][br][/b]If you ever measure a force in lab, or if you ever draw a force in a diagram, you are guaranteed that there is somewhere another force just as strong and pointing in the opposite direction.[br][br][br][color=#b45f06]ANOTHER ASIDE [/color][br][color=#b45f06][i]Having mentioned the shortcomings of Newton's second law at speeds approaching the speed of light, I would be careless if I didn't tell you that Newton's third law can be violated in nature as well.  Once you learn a bit more physics you will see that Newton's third law can be seen as a condition that guarantees conservation of momentum in systems of interacting particles isolated from all the rest of creation.  This being the case, if momentum can be carried by entities other than massive particles, then we might run into problems with the third law.  As it turns out, massless, invisible fields (light, for instance) can carry momentum.  If these fields carry away enough momentum to be meaningful, we will find the third law to be compromised.  Rest assured that this is not a problem to physics.  You should just take note again in the back of your mind that a more fundamental statement than Newton's third law, is the conservation of momentum which we will discuss soon enough.[/i]  [i]There are other complications related to the third law and the concept of simultaneity of measurement which arise in special relativity as well which I will not discuss right now.[/i][/color][br][br][br][u]Mathematical Statement of the Third Law[/u] [br]The clearest way to write Newton's third law is this: Given a force on A by B, there will always be a paired force on B by A that is equal and opposite.  Mathematically we may write:[br][br][math]\vec{F}_{AB}=-\vec{F}_{BA}.[/math][br]Since forces always come in pairs, it is a common test of understanding to see if students can correctly identify the [b]Newton third law pairs[/b], as they are called.  [br][br]Before we set out identifying Newton third law pairs, I wonder if any of you is having a crisis right now with the third law.  After all, what I just told you is that in nature there are no single forces.  Forces always come in pairs, and those forces in each pair are always equal and opposite to one another.  That being true, those paired forces add up to zero vectors.  Naturally this can't be suggesting that every time you try to move something that an equal and opposite force arises to cancel your efforts and make objects fundamentally immovable, but it's worth thinking clearly about this issue and being able to clearly demonstrate how both Newton's third law can be true and objects can be movable, as we know them to be in our world.  Take some time to consider this dilemma before moving on.[br][br]Did you find the solution?  I hope after some reflection that you realized that [i]the paired forces never act on the same object[/i].  One force acts on one object the other paired and equal, but oppositely directed force pushes on some other object.  Phew.  So things [i]are[/i] allowed to move.[br][br]Two forces that we will commonly draw in our diagrams are the force of gravity on an object and a normal force pointing upward exerted by the level surface on which the object rests.  Students very often mistake these two forces as a Newton third law pairing.  THEY ARE NOT.  I understand the mistake.  After all, there are two forces and they happen to be equal and opposite in some cases.  Realize in light of our discussion just a moment ago that these forces are BOTH acting on the same object.  That's all you need to hear to realize that these can't be a Newton third law pair.  It would be useful to see if you can find the actual third law paired forces for these two forces (the gravitational force and the normal force). [br]

Third law pairs need to involve only two objects. The forces are not approximately equal, but exactly equal, at all times, period. Here are some examples: [br][list][*]The force on the moon by earth is equal and opposite to the force on earth by the moon.[/*][*]The force on the soles of my shoes by the ground is equal and opposite to the force on the ground by the soles of my shoes.[/*][*]The force on my car's windshield by the air as I drive along a freeway is equal and opposite to the force on the air by my windshield.[/*][*]The force on a bungee jumper by the elastic band is equal and opposite to the force on the elastic band by the bungee jumper.[/*][*]The force on the head of a nail by a hammer is equal and opposite to the force on the hammer by the head of the nail. [br][/*][/list]