Analyzing Waves on a String

Michael Fowler 5/30/08

From Newton’s Laws to the Wave Equation

Everything there is to know about waves on a uniform string can be found by applying Newton’s Second Law, , to one tiny bit of the string.  Well, at least this is true of the small amplitude waves we shall be studying—we’ll be assuming the deviation of the string from its rest position is small compared with the wavelength of the waves being studied.  This makes the math simpler, and is an excellent approximation for musical instruments, etc.   Having said that, we’ll draw diagrams, like the one below, with rather large amplitude waves, to show more clearly what’s going on.

 

 

Let’s write down  for the small length of string between x and x + Dx in the diagram above. 

 

Taking the st ... Read more »

Remarks on General Relativity

Michael Fowler 
University of Virginia

Einstein’s Parable

In Einstein’s little book Relativity: the Special and the General Theory, he introduces general relativity with a parable.  He imagines going into deep space, far away from gravitational fields, where any body moving at steady speed in a straight line will continue in that state for a very long time.  He imagines building a space station out there - in his words, “a spacious chest resembling a ... Read more »

Transforming Energy into Mass: Particle Creation

Michael Fowler, University of Virginia

Pion Production

We have mentioned how, using a synchrocyclotron, it is possible to accelerate protons to relativistic speeds.  The rest energy of a proton m_pc² is 938 MeV, using here the standard high energy physics energy unit: 1 MeV = 10⁶ eV.  The neutron is a bit heavier—m_nc² = 940 MeV.  (The electron is 0.51 MeV).  Thus to accelerate a proton to relativistic speeds implies giving it a K.E. of order 1,000 MeV, or 1 GeV. 

The standard operating procedure of high energy physicists is to accelerate particles to relativistic speeds, then smash them into other particles to see what happens.  For example, fast protons will be aimed at protons at rest (hydrogen atoms, in other words—the electron can be neglected).  These proton-proton collisions take place inside some kind of detection apparatus, so the results can be observed.  One widely-used detector is the bubble chamber: a transparent container filled with a superheated liquid.  The electric field of a rapidly moving charged particle passing close to a molecule can dislodge an electron, so an energetic particle moving through the liquid leaves a trail of ionized molecules. These give centers about which bubbles can n ... Read more »

Energy and Momentum in Lorentz Transformations

Michael Fowler, University of Virginia

How Does the Total Energy of a Particle Depend on Speed?

We have a formula for the total energy E = K.E. + rest energy,

so we can see how total energy varies with speed.

The momentum varies with speed as

.

How Does the Total Energy of a Particle Depend on Momentum?

It turns out to be useful to have a formula for E in terms of p.

Now

so

hence using p = mv we find

_{... Read more »}

How Relativity Connects Electric and Magnetic Fields

Michael Fowler, University of Virginia

A Magnetic Puzzle…

Suppose we have an infinitely long straight wire, having a charge density of electrons of –λ coulombs per meter, all moving at speed v to the right (recall typical speeds are centimeters per minute) and a neutralizing fixed background of positive charge, also of course λ coulombs per meter.  The current in the wire has magnitude I = λv (and actually is flowing to the left, since the moving electrons carry negative charge). 

Suppose also that a positive charge q is outside the wire, a distance r from the axis, and this outside charge is moving at the sam ... Read more »

Relativistic Dynamics

Michael Fowler, UVa Physics,  3/1/2008

The Story So Far: A Brief Review

The first coherent statement of what physicists now call relativity was Galileo’s observation almost four hundred years ago that if you were in a large closed room, you could not tell by observing how things move-living things, thrown things, dripping liquids-whether the room was at rest in a building, say, or below decks in a large ship moving with a steady velocity.  More technically (but really saying the same thing!) we would put it that the laws of motion are the same in any inertial frame.  That is, these laws really only describe relative positions and velocities.  In particular, they do not single out a special inertial frame as the one that’s “really at rest”.  This was later all written down more formally, in ... Read more »

Mass and Energy

Michael Fowler, University of Virginia   3/1/2008

Rest Energy

The fact that feeding energy into a body increases its mass suggests that the mass m₀ of a body at rest, multiplied by c², can be considered as a quantity of energy.  The truth of this is best seen in interactions between elementary particles.  For example, there is a particle called a positron which is exactly like an electron except that it has positive charge.  If a positron and an electron collide at low speed (so there is very little kinetic energy) they both disappear in a flash of electromagnetic radiation.  This can be detected and its energy measured.  It turns out to be ... Read more »

More Relativity: The Train and the Twins

Michael Fowler, UVa Physics  2/27/08

Einstein’s Definition of Common Sense

As you can see from the lectures so far, although Einstein’s theory of special relativity solves the problem posed by the Michelson-Morley experiment—the nonexistence of an ether—it is at a price.  The simple assertion that the speed of a flash of light is always c in any inertial frame leads to consequences that defy common sense.  When this was pointed out somewhat forcefully to Einstein, his response was that common sense is the layer of prejudices put down before the age of eighteen.  All our intuition about space, time and motion is based on childhood observation of a world in which no objects move at speeds comparable to that of light.  Perhaps if we had been raised in a civilization zipping around the universe in spaceships ... Read more »

Adding Velocities: A Walk on the Train

Michael Fowler, UVa Physics, 12/1/07

The Formula

If I walk from the back to the front of a train at 3 m.p.h., and the train is traveling at 60 m.p.h., then common sense tells me that my speed relative to the ground is 63 m.p.h. As we have seen, this obvious truth, the simple addition of velocities, follows from the Galilean transformations.  Unfortunately, it can’t be quite right for high speeds!  We know that for a flash of light going from the back of the train to the front, the speed of the light relative to the ground is exactly the same as its speed relative to the train, not 60 m.p.h. different.  Hence it is necessary to do a careful analysis of a fairly speedy person moving from the back of the train to the front as viewed from the ground, to see how velocities reallyadd.

... Read more »

The Lorentz Transformations

Michael Fowler, UVa Physics.  2/26/08

Problems with the Galilean Transformations

We have already seen that Newtonian mechanics is invariant under the Galilean transformations relating two inertial frames moving with relative speed v in the x-direction,

However, these transformations presuppose that time is a well-defined universal concept, that is to say, it’s the same time everywhere, and all o ... Read more »

Time Dilation: A Worked Example

Michael Fowler, UVa Physics, 12/1/07

“Moving Clocks Run Slow” plus “Moving Clocks Lose Synchronization” plus “Length Contraction” leads to consistency!

The object of this exercise is to show explicitly how it is possible for two observers in inertial frames moving relative to each other at a relativistic speed to each see the other’s clocks as running slow and as being unsynchronized, and yet if they both look at the same clock at the same time from the same place (which may be far from the clock), they willagree on what time it shows!

Suppose that in Jack’s frame we have two synchronized clocks C₁ and C₂ set 18 x 10⁸ mete ... Read more »

Special Relativity: Synchronizing Clocks

Michael Fowler, UVa Physics  2/29/08

Suppose we want to synchronize two clocks that are some distance apart.

We could stand beside one of them and look at the other through a telescope, but we’d have to remember in that case that we are seeing the clock as it was when the light left it, and correct accordingly.

Another way to be sure the clocks are synchronized, assuming they are both accurate, is to start them together. How can we do that? We could, for example, attach a photocell to each clock, so when a flash of light reaches the clock, it begins running.

... Read more »

Special Relativity

Michael Fowler, UVa Physics  3/3/08

Galilean Relativity again

At this point in the course, we finally enter the twentieth century—Albert Einstein wrote his first paper on relativity in 1905.  To put his work in context, let us first review just what is meant by “relativity” in physics.  The first example, mentioned in a previous lecture, is what is called “Galilean relativity” and is nothing but Galileo’s perception that by observing the motion of objects, alive or dead, in a closed room there is no way to tell if the room is at rest or is in fact in a boat moving at a steady speed in a fixed direction.  (You can tell if the room is accelerating or turning around.)  Everything looks the same in a room in steady motion as it does in a room at rest.  After Newton formulated his Laws of Motion, desc ... Read more »

Special Relativity: What Time is it?

Michael Fowler, Physics Department, UVa.

Special Relativity in a Nutshell

Einstein’s Theory of Special Relativity, discussed in the last lecture, may be summarized as follows:

The Laws of Physics are the same in any Inertial Frame of Reference.  (Such frames move at steady velocities with respect to each other.) 

These Laws include in particular Maxwell’s Equations describing electric and magnetic fields, which predict that light always travels at a particular speed c, equal to about 3×10⁸ meters per ... Read more »

The Michelson-Morley Experiment

Translation: Spanish   Flashlet of the Experiment
 

Michael Fowler  U. Va. Physics  3/13/08

The Nature of Light

As a result of Michelson’s efforts in 1879, the speed of light was known to be 186,350 miles per second with a likely error of around 30 miles per second.  This measurement, made by timing ... Read more »

Frames of Reference and Newton’s Laws

Michael Fowler  University of Virginia  3/14/08

The cornerstone of the theory of special relativity is the Principle of Relativity:

The Laws of Physics are the same in all inertial frames of reference.

We shall see that many surprising consequences follow from this innocuous looking statement.

Let us first, however, briefly review Newton’s mechanics in terms of frames of reference.

... Read more »

The Speed of Light

Michael Fowler  Physics Dept., U. Va.  3/14/08

Early Ideas about Light Propagation

As we shall soon see, attempts to measure the speed of light played an important part in the development of the theory of special relativity, and, indeed, the speed of light is central to the theory.

The first recorded discussion of the speed of light (I think) is in Aristotle, where he quotes Empedocles as saying the light from the sun must take some time to reach the earth, but Aristotle himself apparently disagrees, and even Descartes thought that light traveled instantaneously.  Galileo, unfairly as usual, in Two New Sciences (page 42) has Simplicio stating the Aristotelian position,

... Read more »