A Note of Caution. . . .
Remember, this study guide is only an aid: a guide to studying your books and notes. I will try to mention all of the important themes, and will illustrate these with some of the details -- but I certainly will omit some details. Nevertheless, these details may turn up on the exam.
If You Read Nothing Else . . .
The special relativity principle, time dilation, and length contraction
. . . then take this one crucial piece of advice:
Note down all of the basic quantities we've learned about (work, power, etc.),
memorize their definitions, and
memorize their measurement units!
Trying to think about conceptual physics without first learning the basic definitions is like trying to analyze Shakespeare without first learning how to read. There's much more to Shakespeare than simply knowing what the words mean -- but if you don't know what the words mean, you can't even begin to address the more complex issues. So it is with physics: There's much more to it than memorizing definitions, but memorize you must.
Introductory physics students are often tempted to make vague use of precisely defined physical quantities, based on the imprecise use of these words in ordinary English. Resist temptation. Energy isn't power, wavelength isn't amplitude, and speed isn't frequency.
Galileo realized in the early seventeenth century that the difference between being at rest and being in uniform motion -- motion in a straight line at constant speed -- is arbitrary, at least when considering the motion of material bodies (i.e., mechanics). It's psychologically natural for someone sitting on the beach to consider herself to be at rest and to consider a boat sailing by to be uniformly moving, but from the boat's perspective (or "reference frame"), it's the person on the shore who is moving while the sailor and the boat sit still. As we discussed in class, every mechanics experiment that the sailor can perform, every moving object that the sailor can observe, appears to obey the known laws of physics: there's nothing at all wrong with the boat's reference frame, no experiment that yields a "wrong" result and thus reveals that it's "wrong" to think of the boat as being stationary.
It's the principle of inertia that permits both reference frames to be valid ones. If I sit in a moving car and toss an apple straight upward into the air in front of me, the apple doesn't curve backwards and smash me on the chin, because it continues moving forward even after it's no longer in contact with me or the car. Since it moves forward at the same rate I do (as viewed by someone on the side of the road), it doesn't move horizontally at all in my reference frame, just up and down. Inertia says that you don't have to exert a force on something to make it keep moving however it was already moving, you only need a force to change an object's state of motion.
Just for completeness, I note that Galileo didn't have inertia quite right, since he thought that material bodies naturally move in a curved path along the (spherical) Earth's surface. It took Descartes and finally Newton to realize that natural motion is straight-line motion (at constant speed).
There are two restrictions to Galilean relativity: the physics involved must be mechanics, not electricity or magnetism or optics (light); and the observers must be in uniform motion. Don't think that the observed material objects must be in uniform motion! For example, my apple two paragraphs ago was accelerating in the vertical direction due to gravity, even though it moved at constant speed in the horizontal dimension. But Galilean relativity still holds here, because neither of the two observers (me or the person on the side of the road) is accelerating.
The distinction between uniform motion and accelerated motion does not seem to be arbitrary. (Note that speeding up, slowing down, and changing direction are all "acceleration" to a physicist.) If I'm too busy tossing my apple and drive into a stopped car, the person on the side of the road says that I accelerated. In my own reference frame, I was stationary and it's the guy on the side of the road who suddenly stopped moving (i.e., who accelerated). But I get whiplash and he doesn't. That is, my claim to have been in uniform motion (at zero speed) seems to be a false claim, since the observation from my reference frame -- the guy on the side of the road suddenly slows from 60 to 0 mph without having been pushed by anything, and he comes to no physical harm -- defies the known laws of physics and biophysics.
Since Newton's most famous work dealt with forces and accelerations, it's understandable that he took a step back from Galilean relativity and talked about "absolute acceleration" vs. "absolute uniform motion," depending on whether or not one was accelerating relative to the invisible-but-real 3D grid of "absolute space." In class we did some common-sense demonstrations that showed the need for absolute space: for example, we saw that when I walked into a wall, I was the one who "really" accelerated and not you (why?). Newton also defined "absolute time," which passes at the same rate for everyone, everywhere; but that was just common sense -- until 1905.
Waves are periodic disturbances in time and space. How's that again? First, something (the medium) is being disturbed (pushed around). The maximum disturbance is called the amplitude, and it's a measure of how much energy is in the wave. This disturbance is periodic, meaning that it repeats.
If you focus on one particular part of the medium -- imagine watching a piece of seaweed bobbing up and down on account of a water wave -- then the disturbance forms a repeating pattern in time. In other words, that bit of the medium goes back and forth every so many seconds. The time required for one full cycle is called the period, and the number of full cycles completed every second is the frequency.
So far, everything we've said is equally true of a single oscillating object such as a pendulum. The difficult thing about waves is that there's also a repeating pattern in space. If you look at the entire medium at a single instant of time -- imagine taking a snapshot -- you'll see that different parts of the medium are disturbed by different amounts, and that these disturbances form a pattern. The repetition distance for this pattern is called the wavelength, which you can think of as the "crest-to-adjacent-crest" distance. Pendulums don't have wavelengths; only waves do.
Note that since a wave is a pattern in time and space, it exists throughout time and throughout space. (A pattern, after all, is by definition a spread-out entity.) This means that a single back-and-forth motion of the medium is not a wave; the repeated oscillations of a single section of the medium are not waves; a single crest of water traveling toward the shore is not a wave. The wave is all of the crests and valleys taken together, all of the oscillations of every part of the medium considered as a single extended "thing." Abstract, no?
As a result of the material oscillations, energy is carried from one place to another at wave speed v. A powerboat, for example, can transfer energy to a small rowboat by producing water waves as it zooms by. Less annoyingly, I can transfer energy to your eardrum by producing sound waves with my larynx.
Remember, the material -- the medium -- just moves back and forth, while energy moves steadily forward. People who haven't studied physics are sometimes fooled into thinking that the medium is moving along with the energy, because we see a pattern of crests and troughs move steadily along when we shake a spring or splash in the water. If you take a snapshot of the medium, then another, then another, and so on, you'll see what appears to be motion at speed v in one particular direction. But this is an illusion, a result of each part of the medium vibrating just a bit out of step with the neighboring parts. (This is what happens when sports fans do "the wave.")
There's a mathematical relationship between wave speed, wavelength, and frequency; you should be able to do simple manipulations with this equation.
Things get more complicated when there's more than one source of wave energy trying to "shake" the same medium. Suppose you're listening to a pair of musical instruments -- perhaps a Mozart duet for tuning forks. I think you'll agree that your left eardrum can't be pushed inward and stretched outward at the same time! It can only do one thing at a time, so the influences of the two tuning forks must be combined in order to determine this single response. The "combination" of wave influences to produce a single wave pattern is called interference.
If the two waves combine so that the crests of one "add" to the troughs of the other, we have "destructive interference": the two cancel each other, producing a low-amplitude (or even zero-amplitude) oscillation. If the two are trying to do the same thing to your eardrum at the same time, we instead have "constructive interference": a large-amplitude vibration results. You typically get a complicated "interference pattern" of places with constructive vs. destructive interference.
We saw various electromagnetic phenomena in rapid succession. A current through a wire (preferably a coil of wire) produces a magnetic field; we call this an electromagnet. An electric motor runs on this principle: When a current is passed through a coil of wire, the coil becomes an electromagnet with a north pole and a south pole. If the coil ("armature") is free to spin, and if it is in the vicinity of a strong stationary magnet, it will indeed spin so as to get its north pole as close as possible to the stationary magnet's south pole. There's a bit more to the story (at least for a DC motor, where the current is always in the same direction), but this is the main idea.
Just as an electric motor converts electrical energy to kinetic energy (energy of motion of a spinning shaft), an electric generator (or "dynamo") converts the kinetic energy of a spinning shaft into electrical energy. It accomplishes this via electromagnetic induction, the fundamental behavior of nature by which an electric field is induced (generated) whenever a magnetic field is in the process of changing its strength or its direction. So if a coil of wire is made to spin in the presence of a strong stationary magnet, from the coil's "perspective" it's the magnet which is spinning, with the result that the magnetic field produced by the magnet is constantly changing direction: an electric field is induced in the coil. Since the coil is a conductor, electrons in the coil can move in response to this electric field: an AC current is generated.
In the 1860s the Scottish theoretical physicist James Clerk Maxwell put together the four equations that described the known laws of electricity and magnetism -- including the two described in the preceding two paragraphs -- and he decided to make an important addition to the equation that describes how electric currents produce magnetic fields. He boldly hypothesized a second method of producing a magnetic field. Since changing a magnetic field can induce an electric field, Maxwell decided that changing an electric field would probably induce a magnetic field.
This was a bold step because no experimental physicist had observed any such thing in the lab. But Maxwell indirectly verified his hypothesis by showing mathematically that if it were correct, electromagnetic waves should exist, electric fields oscillating in tandem with magnetic fields, the changes in each field inducing the changes in the other field. Furthermore, Maxwell was able to show that such a wave should travel at the very high speed of 3.0 x 108 m/s, better known as c. He realized that this speed was, to within a few percent, the same speed that experimental physicists had already measured for light waves. Light, realized Maxwell, is an electromagnetic wave. Direct experimental verification came in the 1880s, when Heinrich Hertz was able to record low-frequency light waves (radio waves) emanating from a spark on the other side of his lab. (Today you recreate Hertz's experiment every time you hear a burst of radio static caused by a nearby lightning strike.)
The electromagnetic spectrum is simply an ordered listing of all the different frequencies of light, from the lowest (radio) to the highest (gamma rays). Equivalently, this listing runs from long to short wavelength. You should memorize the electromagnetic spectrum, including the visible portion (ROYGBIV).
Naturally (said nineteenth-century physicists) every wave needs a medium, a material that "waves" in order to carry energy from one place to another. So of course electromagnetic waves require a medium. Clearly the medium isn't air, because we see the stars despite the fact that there's virtually no air beyond Earth's atmosphere. (NOTE: It's a common misconception to think that our atmosphere extends far into space, and that planets and stars are in our outer atmosphere. In fact our atmosphere is extremely thin, like an apple's skin compared to the apple, and the Moon and planets and stars lie far beyond it.) This medium exists even where there's no other material. It doesn't slow down the planets in their orbits, so it must have very low density; but it allows light to travel very quickly, so it must be extremely elastic (hard to stretch or compress). This strange, ubiquitous, hypothetical medium was the lumiferous ether.
But besides the practical fact that ether had rather strange properties, ether had the philosophical drawback that it constituted a special reference frame. Maxwell showed mathematically that light is an electromagnetic wave that travels at speed c. Travels at speed c relative to what? Relative to the wave's medium, to the ether. So anyone who is stationary relative to the ether will measure the correct speed for light, but anyone who moves uniformly relative to the ether will measure some other speed, the wrong speed, a speed that violates the laws of electricity and magnetism embodied in Maxwell's four equations. Electricity and magnetism (and optics), that is, do not obey Galilean relativity. If two different uniformly moving observers observe the motion of a cannonball (said Galileo), neither one of them will obtain "wrong" results, so there is no basis for claiming that either one of them is "really" moving rather than "really" at rest; but if they instead observe a beam of light (says Maxwell), at least one of them will obtain "wrong" results, thus revealing that she is "really" moving rather than "really" at rest. Why on Earth should physics be divided in two like this, mechanics vs. electromagnetism? This philosophical conflict, this "dichotomy in physics" (as Wolfson calls it in his book), is what bothered Einstein and motivated him to put forth his solution in 1905.
OK, enough philosophy, time to observe and experiment and measure. What did astronomical observations of stellar aberration and of double stars tell us about light's behavior? Next, how was the famous Michelson-Morley experiment set up? What did these two physicists expect to see, and why? What did they actually see when the ran the experiment?
Those two astronomical observations and that one famous experiment together seemed to rule out every reasonable possibility as to how light behaves and how our planet is moving relative to the ether. So after 18 years of the world's leading physicists fumbling around for an explanation, finally a Swiss patent clerk published a paper describing an unreasonable possibility.
This unreasonable possibility doesn't seem very unreasonable at first glance. The principle of special relativity is just like Galilean relativity, except that the restriction to mechanics is lifted: All uniformly moving observers will obtain the same laws of physics. Another wording: No physics experiment of any kind can reveal that an observer is really stationary vs. really uniformly moving.
The problem, of course, is that light seems to violate this principle, as discussed in the preceding section: There's just one uniformly moving reference frame, the frame in which the ether is stationary, where light will behave "correctly" by traveling at the correct speed c. So Einstein simply stated that (a) all uniformly moving observers will measure the same speed for light, and (b) ether is "superfluous" so let's just assume it doesn't actually exist.
How can light have the same speed no matter who does the measuring??? It certainly would explain the null result of the Michelson-Morley experiment (although Einstein in later life claimed that he hadn't really paid any attention to this experiment -- which may or may not have been an accurate recollection). Speed is distance over time, so if speed is behaving weirdly, it must be the case that distance and/or time behaves weirdly as well.
The "photon clock" thought experiment showed us that time must be weird if the special relativity principle is correct: Moving clocks are always observed to run slow. (Remember that "moving clocks" means "clocks that are moving as viewed in the observer's reference frame": don't violate the special relativity principle by thinking that clocks can be moving in an absolute sense.) It's not really just mechanical "clocks" that run slow, but radioactive decays and heartbeats and aging and time itself. If that weren't true, you could compare your watch's ticking rate to your heart rate (say) to determine whether you're really at rest (the two rates agree) vs. really moving (the two don't agree), and the special relativity principle says that no such determination is ever possible. Remember as well that if you see my clock running slow as I zip by you, I simultaneously see your clock running slow (not fast!) as you zip by me.
We also illustrated this "time dilation" via the famous "twin paradox" thought experiment. What's the basic story? Why is the result paradoxical? How do we resolve the paradox?
And we saw that in order to understand this situation from the "space twin's" perspective, we must also have distances behaving weirdly: length contraction. Moving objects are squashed along the direction of motion, relative to what someone would observe if the object were stationary. Remember, this "contraction" isn't a matter of the object crunching in on itself, squeezing a person's innards or stressing a car's chassis; no, the person feels just fine, the car's chassis is in no danger of cracking, it's space itself that shrinks from our perspective. And again, in that object's reference frame the object has its normal length and you are squashed.
The muon experiment -- a real experiment this time, not a thought experiment -- also illustrates the fact that a result due to time dilation in one reference frame is due to length contraction in the other reference frame. You should understand this experiment. Ditto the Hafele-Keating experiment with atomic clocks.
Even something as simple as simultaneity -- the claim that two events occur in two different places at the same time -- is relative. Two events that are simultaneous for you are not simultaneous for me, if we are moving relative to each other. (Think of the "pole-barn paradox" for an interesting illustration of this.) We are both right in our conflicting claims.
In 1908, three years after Einstein presented special relativity to the world, his former math professor Hermann Minkowski recast the theory in terms of 4D spacetime geometry. On this view, time is the fourth dimension (or c multiplied by time, if you want all four dimensions to have units of distance). The time dimension is not, however, exactly the same as the x, y, and z spatial dimensions, and this difference shows up as an innocent-looking minus sign in the expression for the hypotenuse of a right triangle: the square of the hypotenuse's length is x2 + y2 + z2 - t2. (This quantity, the spacetime interval, is discussed in Chapter 13.) Although different observers will disagree on the elapsed time between two events (time dilation) and on the distance between two events (length contraction) they will always agree on the spacetime interval between the two events. It's an odd kind of "hyperbolic geometry" (or "Minkowski space") in which, for example, the hypotenuse is not the longest side of a right triangle.
(It was one of the triumphs of nineteenth-century mathematics that people freed themselves from thinking only about the geometry of the physical world, about Euclid's geometry, and worked out results for non-Euclidean geometries that existed only in the creative minds of mathematicians. Then Einstein and Minkowski showed that these geometries are relevant to the physical world after all. Einstein would take this idea a major step further with general relativity. . . .)
We talked about "light cones" and how they divide spacetime into the "absolute past" vs. the "absolute present" vs. the "absolute elsewhere." How do these three regions differ from each other in terms of cause-and-effect relationships? Under what circumstances can the order of spacetime events be different for different observers?
What's a "worldline"?
We saw that velocities don't add according to common sense: If a spaceship traveling at 0.9 c relative to us fires a torpedo forward at 0.4 c, we do not observe the torpedo to move at 1.3c. (You should know how to use the formula for computing how fast we really do see it move.) No matter how much energy we give something, we never see it move at the speed of light, although it can get very close.
It follows that the expression for an object's energy must get larger and larger even when the object's speed is close to c and is barely increasing. This expression is E = γmc2, where γ (gamma) is the relativistic factor we've seen repeatedly, the reciprocal of the square root of 1 - (v/c)2. This factor shoots upward toward infinity as v inches ever closer to c. Hence it would take an infinite amount of energy to accelerate an object with mass to the speed of light or beyond -- so it never happens.
What about the other extreme, a stationary object with mass? Since γ = 1 when v = 0, our expression for rest energy is just E0 = mc2. This famous expression tells us that mass isn't conserved -- it doesn't remain the same at all times -- but instead represents a form of energy that can be converted to other forms of energy. Similarly, adding energy to something -- say, by heating it -- increases its mass. Why don't we notice this if it's true? You should be able to discuss rest energy in the context of, say, pair creation (or pair production), pair annihilation, nuclear fission or fusion, a cooling cup of coffee, or a particle accelerator.