travelling at a constant speed relative to the ground, a spaceship coasting in a region free from fields, a railroad car rolling on straight tracks at constant speed are also inertial frames. A (coordinate system inside a) car that is accelerating (say by going around a curve), a spaceship that is accelerating, a freight car that is speeding up or slowing down – these are all examples of non-inertial frames. All of Newton’s laws suppose an inertial reference frame (yes, the third law too) and are generally false for accelerations evaluated in an accelerating frame as we will prove and discuss next week.

In the meantime, please be sure to learn the statements of the laws including the condition “in an inertial reference frame”, in spite of the fact that you don’t yet really understand what this means and why we include it. Eventually, it will be the other important meaning and use of Newton’s First Law – it is the law that defines an inertial reference frame as any frame where an object remains in a state of uniform motion if no forces act on it!

You’ll be glad that you did.

1.5: Forces

Classical dynamics at this level, in a nutshell, is very simple. Find the total force on an object. Use Newton’s second law to obtain its acceleration (as a di erential equation of motion). Solve the equation of motion by direct integration or otherwise for the position and velocity.

That’s it!

Well, except for answering those pesky questions that we typically ask in a physics problem, but we’ll get to that later. For the moment, the next most important problem is: how do we evaluate the total force?

To answer it, we need a knowledge of the forces at our disposal, the force laws and rules that we are likely to encounter in our everyday experience of the world. Some of these forces are fundamental forces – elementary forces that we call “laws of nature” because the forces themselves aren’t caused by some other force, they are themselves the actual causes of dynamical action in the visible Universe. Other force laws aren’t quite so fundamental – they are more like “approximate rules” and aren’t exactly correct. They are also usually derivable from (or at least understandable from) the elementary natural laws, although it may be quite a lot of work to do so.

We quickly review the forces we will be working with in the first part of the course, both the forces of nature and the force rules that apply to our everyday existence in approximate form.

1.5.1: The Forces of Nature

At this point in your life, you almost certainly know that all normal matter of your everyday experience is made up of atoms. Most of you also know that an atom itself is made up of a positively charged atomic nucleus that is very tiny indeed surrounded by a cloud of negatively charged electrons that are much lighter. Atoms in turn bond together to make molecules, atoms or molecules in turn bind together (or not) to form gases, liquids, solids – “things”, including those macroscopic things that we are so far treating as particles.

The actual elementary particles from which they are made are much tinier than atoms. It is worth providing a greatly simplified table of the “stu ” from which normal atoms (and hence molecules, and hence we ourselves) are made:

In this table, up and down quarks and electrons are so-called elementary particles – things that are not made up of something else but are fundamental components of nature. Quarks bond together three at a time to form nucleons – a proton is made up of “up-up-down” quarks and has a charge of +e, where e is the elementary electric charge. A neutron is made up of “up-down-down” and has

Table 1: Basic building blocks of normal matter as of 2011, subject to change as we discover and understand more about the Universe, ignoring pesky things like neutrinos, photons, gluons, heavy vector bosons, heavier leptons that physics majors (at least) will have to learn about later...

no charge.

Neutrons and protons, in turn, bond together to make an atomic nucleus. The simplest atomic nucleus is the hydrogen nucleus, which can be as small as a single proton, or can be a proton bound to one neutron (deuterium) or two neutrons (tritium). No matter how many protons and neutrons are bound together, the atomic nucleus is small – order of 10−15 meters in diameter40. The quarks, protons and neutrons are bound together by means of a force of nature called the strong nuclear interaction, which is the strongest force we know of relative to the mass of the interacting particles.

The positive nucleus combines with electrons (which are negatively charged and around 2000 times lighter than a proton) to form an atom. The force responsible for this binding is the electromagnetic force, another force of nature (although in truth nearly all of the interaction is electrostatic in nature, just one part of the overall electromagnetic interaction).

The light electrons repel one another electrostatically almost as strongly as they are attracted to the nucleus that anchors them. They also obey the Pauli exclusion principle which causes them to avoid one another’s company. These things together cause atoms to be much larger than a nucleus, and to have interesting “structure” that gives rise to chemistry and molecular bonding and (eventually) life.

Inside the nucleus (and its nucleons) there is another force that acts at very short range. This force can cause e.g. neutrons to give o an electron and turn into a proton or other strange things like that. This kind of event changes the atomic number of the atom in question and is usually accompanied by nuclear radiation. We call this force the weak nuclear force. The two nuclear forces thus both exist only at very short length scales, basically in the quantum regime inside an atomic nucleus, where we cannot easily see them using the kinds of things we’ll talk about this semester. For our purposes it is enough that they exist and bind stable nuclei together so that those nuclei in turn can form atoms, molecules, objects, us.

Our picture of normal matter, then, is that it is made up of atoms that may or may not be bonded together into molecules, with three forces all significantly contributing to what goes on inside a nucleus and only one that is predominantly relevant to the electronic structure of the atoms themselves.

There is, however, a fourth force (that we know of – there may be more still, but four is all that we’ve been able to positively identify and understand). That force is gravity. Gravity is a bit “odd”. It is a very long range, but very weak force – by far the weakest force of the four forces of nature. It only is signficant when one builds planet or star sized objects, where it can do anything from simply bind an atmosphere to a planet and cause moons and satellites to go around it in nice orbits to bring about the catastrophic collapse of a dying star. The physical law for gravitation will

40...with the possible exception of neutrons bound together by gravity to form neutron stars. Those can be thought of, very crudely, as very large nuclei.

be studied over an entire week of work – later in the course. I put it down now just for completeness, but initially we’ll focus on the force rules in the following section.

Don’t worry too much about what all of these symbols mean and what the value of G is – we’ll get to all of that but not now.

Since we live on the surface of a planet, to us gravity will be an important force, but the forces we experience every day and we ourselves are primarily electromagnetic phenomena, with a bit of help from quantum mechanics to give all that electromagnetic stu just the right structure.

Let’s summarize this in a short table of forces of nature, strongest to weakest:

a)Strong Nuclear

b)Electromagnetic

c)Weak Nuclear

d)Gravity

Note well: It is possible that there are more forces of nature waiting to be discovered. Because physics is not a dogma, this presents no real problem. If a new force of nature (or radically di erent way to view the ones we’ve got) emerges as being consistent with observation and predictive, and hence possibly/plausibly true and correct, we’ll simply give the discoverer a Nobel Prize, add their name to the “pantheon of great physicists”, add the force itself to the list above, and move on. Science, as noted above, is a self-correcting system of reasoning, at least when it is done right.

1.5.2: Force Rules

The following set of force rules will be used both in this chapter and throughout this course. All of these rules can be derived or understood (with some e ort) from the forces of nature, that is to say from “elementary” natural laws, but are not quite laws themselves.

a) Gravity (near the surface of the earth):

~

The direction of this force is down, so one could write this in vector form as F g = −mgyˆ in a coordinate system such that up is the +y direction. This rule follows from Newton’s Law of Gravitation, the elementary law of nature in the list above, evaluated “near” the surface of the earth where it is varies only very slowly with height above the surface (and hence is “constant”) as long as that height is small compared to the radius of the Earth.

The measured value of g (the gravitational “constant” or gravitational field close to the Earth’s surface) thus isn’t really constant – it actually varies weakly with latitude and height and the local density of the earth immediately under your feet and is pretty complicated41 . Some “constant”, eh?

Most physics books (and the wikipedia page I just linked) give g’s value as something like:

meters

g ≈ 9.81 (25) second2

41Wikipedia: http://www.wikipedia.org/wiki/Gravity of Earth. There is a very cool “rotating earth” graphic on this page that shows the field variation in a color map. This page goes into much more detail than I will about the causes of variation of “apparent gravity”.

(which is sort of an average of the variation) but in this class to the extent that we do arithmetic with it we’ll just use

because hey, so it makes a 2% error. That’s not very big, really – you will be lucky to measure

g in your labs to within 2%, and it is so much easier to multiply or divide by 10 than 9.80665.

b)The Spring (Hooke’s Law) in one dimension:

This force is directed back to the equilibrium point (the end of the unstretched spring where the mass is attached) in the opposite direction to x, the displacement of the mass on the spring away from this equilibrium position. This rule arises from the primarily electrostatic forces holding the atoms or molecules of the spring material together, which tend to linearly oppose small forces that pull them apart or push them together (for reasons we will understand in some detail later).

c) The Normal Force:

This points perpendicular and away from solid surface, magnitude su cient to oppose the force of contact whatever it might be! This is an example of a force of constraint – a force whose magnitude is determined by the constraint that one solid object cannot generally interpenetrate another solid object, so that the solid surfaces exert whatever force is needed to prevent it (up to the point where the “solid” property itself fails). The physical basis is once again the electrostatic molecular forces holding the solid object together, and microscopically the surface deforms, however slightly, more or less like a spring to create the force.

d) Tension in an Acme (massless, unstretchable, unbreakable) string:

This force simply transmits an attractive force between two objects on opposite ends of the string, in the directions of the taut string at the points of contact. It is another constraint force with no fixed value. Physically, the string is like a spring once again – it microscopically is made of bound atoms or molecules that pull ever so slightly apart when the string is stretched until the restoring force balances the applied force.

e) Static Friction

(directed opposite towards net force parallel to surface to contact). This is another force of constraint, as large as it needs to be to keep the object in question travelling at the same speed as the surface it is in contact with, up to the maximum value static friction can exert before the object starts to slide. This force arises from mechanical interlocking at the microscopic level plus the electrostatic molecular forces that hold the surfaces themselves together.

f) Kinetic Friction

(opposite to direction of relative sliding motion of surfaces and parallel to surface of contact). This force does have a fixed value when the right conditions (sliding) hold. This force arises from the forming and breaking of microscopic adhesive bonds between atoms on the surfaces plus some mechanical linkage between the small irregularities on the surfaces.