I'll be out of town this weekend at a friend's wedding, so here's something to work on while I'm gone. This is a closed-book test, and you may not discuss your answers with other students. Show all work for full credit. Exam papers are due at the beginning of class on Monday.
Fall 2003, Exam 1
1) A clever physics student decides that we need better demonstrations of Special Relativity, and sets out to provide one. He takes a cheap digital alarm clock (without a seconds display), and hops a convenient rocket ship into a frame that is stationary with respect to the Sun.
(Helpful facts: The Earth orbits the Sun at a radius of approximately 1.5 1011 m, taking 365.25 days to complete a single orbit. For the purposes of this exam, we can assume that the orbit is circular and that the Earth and the Sun are each in inertial frames.)
(Hint: The binomial expansion is your friend.)
a) Approximately how long does he need to wait for his clock and an identical clock left on Earth to end up one minute out of synch?
b) To pass the time, he carefully observes the Earth. If the Earth in its own rest frame is a perfect sphere, what shape would our hapless colleague see as the Earth passes (ignore the rotation of the Earth), and why?
c) If the radius of the Earth in its own rest frame is 6.37 106 m, what is the difference (in meters) between the maximum and minimum radii he measures?
2) A badly underpaid professor of English drives a 1975 Oldsmobuick, with a rest length of 6 m. Unfortunately, his garage is only 5 m deep. Overhearing a conversation between two Physics 19 students, though, he decides that he can use Relativity to make the car fit. He tells his wife to close the garage door at the instant when the car is completely within the garage, and then stomps on the gas.
a) How fast would he need to go for his (stationary) wife to see the car fit in the garage?
b) At this speed, does he see the car fitting in the garage? If not, how much of the car does he see sticking out of the garage at the instant when the front bumper first touches the back wall?
c) At this speed, how much force is required to accelerate the 1000-kg car at 1 m/s2?
1) The Sun delivers energy to the Earth at an average rate of 1.7 1017 W. How much mass (in kg) must be converted to energy each second in order to supply energy to the Earth?
2) A muon (mass 105.66 MeV/c2) with a kinetic energy of 70.44 MeV collides with a second particle of unknown mass, initially at rest. The two particles stick together after the collision.
a) What is the initial speed of the muon?
b) If the final speed of the two particles is v = 0.6 c, what is the mass of the unknown particle?
3) As in one of the homework problems, a spaceship of rest length LR is flying alongside a very long stick floating in space, at a speed of 0.8 c. Two space cadets, Alice and Bob, want to cut a piece of the stick that is exactly the length of the ship as seen in the stick frame. If Alice is in the front of the ship, and Bob in the back, how should they time the firing of their lasers to make their cuts simultaneous in the stick frame? (That is, who fires first, and how long does the second person wait before firing?)
I'm Largely Indifferent to the 70's
As I confessed back in July, I'm weirdly fascinated by VH1's "I Love the 80's" series. If I run across an episode while channel-surfing, I tend to lock in on it, no matter how much Kate rolls her eyes.
On the other hand, last night, I ran across an episode of "I Love the 70's", which doesn't really have the same hypnotic quality. I watched them when they first came out, but they're just not as compelling.
The problem is summed up pretty well by one of the few really good lines from the first episode. One of the deadpan weird guys (either Michael Ian Black or Mo Rocca, I forget which) said "I cried when the Beatles broke up. Of course, I was nine months old at the time, and nine-month-old babies cry all the time."
The problem with "I Love the 70's" is the same thing that made "I Love the 80's" such genius-- most of the B-list celibrities they got for the show are about my age, give or take a few years. They're not really old enough to remember the early 70's clearly, much less know anything about "key parties" that they didn't learn from The Ice Storm. The jokes are a little stiffer, and more scripted, and a lot of them clearly don't recognize the kitsch items they're handed in the various product nostalgia bits.
What makes the first one brilliant, and keeps me re-watching the shows, is the "Oh, COOL!" nostalgia kick most of the stuff provokes. It's interesting to watch them talk about items and fads I don't really remember (I grew up in a hick town in Central New York-- clothing fads lagged six months to a year behind big cities, if they even made it out that far), because of the way their faces light up when they remember stupid things from the 80's (Rob Thomas from Matchbox 20 is probably the best example-- when they talk to him about Star Wars stuff, his eyes bug out like he's an eight-year-old geek all over again. His band still sucks, but he's pretty entertaining).
The prequel series doesn't offer as much of that, particularly for the first half of the decade. There's some in the later years (and I start to recognize a few things myself), but prior to about 1977, the only really good bits are the Beatles line quoted above, and the revelation that Morgan Freeman was on "The Electric Company."
There's good news on the pop-nostalgia front, though: a second round of "I Love the 80's" is coming soon...
Web Radio, Hallelujah!
Part of my bad Windows karma over the past several years has been the utter inability of my desktop computer at work to deal with streaming audio files. I don't know why, but it just couldn't deal with them at all-- if I sat back in my chair, with my hands well away from the keyboard and mouse, it could just about play an audio stream without choking. If I attempted to actually use the computer for anything else-- to work on lecture notes, check my email, move the mouse in a circular pattern to amuse myself-- the sound would break up and get all garbled.
As you might imagine, this was somewhat irritating when I had to work on the first weekend of the NCAA tournament... Worse yet, though, it meant that despite sitting here with a 1.2 GHz machine connected to the Information Supercollider through Ethernet and a T3 line, and theoretical access to all manner of fascinating audio files via the web, I had to listen to either local radio, or my own CD's. The problem is that I need music playing more or less constantly, as background noise, and I get sick of listening to the same CD's over and over. And local radio sucks, now more than ever, as the 80's station mentioned in that post has changed formats to become all but indistinguishable from the Clear Channel candy-ass adult alternative station. Never in my life have I heard so much Elton goddamn John.
So, there was great rejoicing a day or so ago when one of my advisees, talking about music, pointed me to Digitally Imported. Not because I'm particularly fond of techno/trance/whatever (I'm not), but because it actually worked. The audio comes through fine, and I can still surf the web, grade papers, make up new assignments, and all the rest of the things I use the computer for.
Which means I have access to KEXP in Seattle, a web broadcast featuring truly excellent variety. I get to actually hear songs by bands I've previously known only through reviews in music magazines and CD's bought sound unheard. While I've been typing this, they played a track off the Old 97's Too Far to Care, a candidate for Perfect Album status, but hardly the best-known album ever. They're playing Kings of Leon now, one of those bands I've read about, but never heard played. They snuck in some James Brown and Al Green yesterday, and I've started keeping a list of new bands to check out on scrap paper on my desk.
This will undoubtedly lead to a collision between my audiophilia and Kate's frugality, but I don't care. Finally, I have decent music to listen to at work.
Great Moments in Advertising
Electioneering Division: Local elections are coming up soon, so those annoying yard signs have popped up all over the neighborhood. One common sign, seen many times while walking the dog, exhorts us to vote "Bill Chapman for Councilmen." Sadly, it doesn't provide any details to explain Bill's plurality. Is he just such a good guy the he deserves two votes, or is he really fat, and needs more than one chair? The voters have a right to know...
Spam Division: In the past couple of weeks, I've started getting a new kind of spam: unsolicited resumes. Two "engineers" and one "electrician" have graced my inbox with poorly-formatted copies of their employment histories, sent from free Yahoo accounts.
Let me just take a moment to assure you, Richard Siek, that if ever I find myself in need of "Automation equipment and machine control using for machines and robots precision mechanisms motion programming, fluid mechanics, pneumatics systems, mounting and positioning devices, electro-mechanical and vacuum mechanisms, design and analysis of structures, castings, welded frames, mechanical detailing, redesigning mechanical components for manufacturing," or any of the other eight hundred things you list as skills and abilities, I still won't hire some anonymous jackass who spammed me from a Yahoo account.
We'll keep your resume on file, though, and if any openings come up, we'll be sure to call you.
What does all this inertial frame stuff have to do with statutes of limitations? Well, for one thing, it (hopefully) explains my teaser comment about inertial frames. When you start talking about prosecution, you'll eventually need to have all the parties in the same frame of reference (that, or really short trials). This will necessarily involve a bunch of shifting between frames, which means people will be accelerating and decelerating, and that'll play hell with the timing.
The other issue that needs addressing is some apparent confusion about what the different clock speeds actually entail. For example, Diana Jarvis asks:
Can anyone think of how two parties could affect each other who were not in the same relative time when they interacted?
Since the different clock speeds result from relative motion of the two parties, it's actually very easy to imagine cases where two people whose clocks run at different speeds would be able to affect one another. A hit-and-run accident, for example, or a drive-by shooting. As long as the two parties are in relative motion, their clocks will run at different speeds, and they'll experience the timing of events differently.
The other fact to emphasize here is that moving back and forth between "relative times" is trivial-- just change your speed, and you're in a different "relative time." That's the glory of relativity-- to make time run at different rates, you don't need to do anything more drastic than speed up or slow down.
Granted, you've got to be moving pretty fast if you hope to use time dilation to your advantage, but it's not like hopping between parallel universes. All you need to do is move.
The Inconceivable Conception
A few comments in response to the post about relativity and the law made me realize that I need to provide a little more explanation of some aspects of the theory. I glossed over a few points last year when I discussed this stuff, and I need to say a bit more about the idea of inertial frames. Which, oddly, will lead to an analogy between physics theory and Catholic doctrine, so stick around.
The central bit of Special Relativity is the idea of the absolute equivalence of all inertial frames. As I wrote last year:
Einsteinian Relativity is the idea that all the laws of physics are the same for all observers moving with constant velocity; or, stated another way, that there's absolutely no way to distinguish between frames of reference. You're not allowed to determine unequivocally that one person is moving, while another is stationary-- relative motion is all that matters.
The caveats "inertial" and "constant velocity" are very important there. Special Relativity applies only to observers in inertial frames, which are frames of reference moving with constant velocity relative to one another. Observers who are changing speed are subject to a slightly different set of rules.
The concept of "inertial frames" pre-dates Einstein by a good bit. Initially, it just meant any frame where Newton's Laws appear to hold-- objects at rest remain at rest unless acted on by an external force; objects accelerate in direct proportion to the force acting on them and in inverse proportion to their mass, and so on. Special Relativity extends this to say that all the laws of physics are the same in any inertial frame, but the basic definition-- frames with constant relative velocity-- remains the same. The easiest way to understand this sort of thing is by reference to everyday objects, because everybody has experienced inertial and non-inertial frames of reference.
The classic example of an inertial frame usually involves objects thrown in the air. From the point of view of a small boy throwing a ball up into the air and catching it, the ball appears to obey Newton's Laws. It moves straight up and down, accelerating only in the direction in which a force acts on it (vertically). It doesn't go shooting off to one side or another, because there's no force that makes it move side to side.
The situation looks different to an observer moving relative to the boy, though-- say, to his mother, watching him float downstream on a garbage barge. In that case, the path of the ball appears as a parabolic arc, not a straight up-and-down line, simply because the car moves some horizontal distance while the ball is in the air. Still, Newton's Laws hold: the ball only accelerates vertically, because that's the only direction in which a force acts, while the horizontal velocity (set by the river) remains constant.
According to Einstein, those two inertial frames are absolutely equivalent. There's no experiment you can do that will tell you unambiguously that one of the two is moving, while the other is stationary. All the laws of physics work exactly the same way for both observers. This is why you'll sometimes hear explanations of relativity invoking the indistinguishability of two frames-- any time you can determine which is which, one of the two is not an inertial frame, and then Special Relativity doesn't really apply.
(This is the point where people always object that, of course you can tell the difference-- if you look out the window, and see trees moving by at constant velocity, it's a safe bet that you're moving and they're not. Which is usually true, but the situation is mathematically equivalent to the case where you're stationary and the world is rushing past you in the opposite direction. The immobility of trees is simply a matter of convention.)
Non-inertial frames are pretty common, as well. Any time you step on the gas as a light changes, or slam on the brakes for a speed trap, or even take a corner in your car, you're experiencing a non-inertial frame. It's perhaps not immediately obvious that Newton's Laws don't hold in these cases, but it's true. What gives it away is the fictitious forces you feel in those cases-- there's no real force pushing you back into the seat when you accelerate, or throwing you into the seat belts when you break, and there's no such thing as a "centrifugal force" pushing you to the outside as you round a corner. Those are all misperceptions by accelerating observers-- in every case, what you experience is actually a result of inertia. You're not being pushed back, forward, or to the outside of the turn, you're actually resisting being pushed forward, backward, and to the inside of the turn.
The key feature here is that non-inertial frames are distinguishable. You feel the fictitious forces when you're driving, but people watching on the street do not. They're in an inertial frame, and the laws of physics don't appear to change for them.
The really baffling features of relativity all have to do with the indistinguishability of inertial frames. While it's strange to think that moving clocks run slow, it's even harder to grasp the fact that this is true no matter which frame you're in. If I'm sitting calmly in my lab, watching a student fly by at a good fraction of the speed of light, I'll note to myself that his clock is running slow. At the very same time, that student will be thinking "Stupid professor. He doesn't even realize that his clock is running slow." The situations are exactly parallel-- any observer will be convinced that his own clock is right, while the clocks of any and all observers moving relative to him are slow. There's no way to distinguish between a stationary professor and a moving student, as long as the velocity is constant.
This brings us around to the Twin Paradox, the gedankenexperiment that kicked off this discussion in the first place. The Twin Paradox, it turns out, is sort of like the doctrine of Immaculate Conception: it does not mean what most people think it means.
The paradox is set up like this: Take two twins, send one off to Alpha Centauri and back at close to the speed of light, and keep the other one here on Earth. Since moving clocks run slow, one twin ends up experiencing more time passing than the other one does.
Most people think that's the whole story-- twins of different ages being somewhat paradoxical. That's not the paradox physicists have in mind, though (after all, twins are never exactly the same age...). The real paradox is this: How do you tell which is which? According to Relativity, after all, inertial frames are all equivalent. So, from the standpoint of the twin in the rocket ship (call her "Mary,"), it looks like her brother ("Frank"-- did I mention that they're fraternal twins?) is the one who went on a trip, taking the entire planet Earth with him. Frank, of course, sees things differently. Each twin, by this logic, should see the other's clock running slow, and arrive home expecting to be older. That's the paradox-- they each ought to be older than the other.
The answer here is that Mary isn't in an inertial frame the whole time. She accelerates on leaving Earth, after all, and decelerates at the other end of the trip, then accelerates to come back, and decelerates when she reaches Earth again. You can make the acceleration and deceleration periods as short as you like (providing you've got some good way to keep Mary from being squashed), but during those intervals, she's not in an inertial frame. And because of that, you have a clear way to distinguish which of them was moving, and which was stationary. Which resolves the paradox, and leaves Frank the older of the twins (Mary has all the luck).
(The above is not, I hasten to add, completely without glossing-over. It would be more correct to say that the differential aging is the result of clock synchronization issues that occur when Mary switches from one frame to another, but it adds up to basically the same thing (that's also what gets you this head-scratcher from last year). I should also note that accelerating frames can, in fact, be handled theoretically-- it's just Special Relativity that doesn't work. General Relativity covers what happens to accelerating clocks, among other things.)