Antz Larngstrom surprised everyone in the sports world by racing his bike again this year in the Tore De Pants marathon; he had finished last for seven straight years and had announced retirement. Rumour has it that he evaluated other professional athletes and decided to use performance enhancing substances, copying the best baseball player of all time; hot dogs and beer. I am unable to provide a reference for a specific quote on this so you just have to take it from me. This year he locked his gearbox in ultra high gear so as to always maintain the same gear ratio between the pedal sprocket and the rear wheel drive sprocket because previously he often crashed in the confusion of gear shifting. He hired me to analyze the physics of his riding style; he may be encouraged to race still another time if he finishes higher than dead last. I have chosen my favorite assistants Gigi and Yvonne to help, one to observe his pedal sprocket RPM and the other to observe his bike velocity relative to the embankment. Being the only one left, I was forced into the task of monitoring the quality versus time curve of our observation team provisions of chilled white wine and cheese. We were all stationary in a rest station area while making our observations. On the first day the racers fought a headwind and a constant uphill road grade; Gigi reported the bike speed as .45c. while Yvonne reported observing the pedal sprocket showing 30 RPM. Next day the racers enjoyed a strong tailwind and a long downhill road grade. We expected a higher speed, but also got an unexpected surprise. Gigi reported a bike speed of .90c. ( I must interject here to explain that Gigi was observing velocity by holding her favorite rubber ruler at arm's length in front of her and sighting over it to determine the optical apparent length of Antz's bike; simple arithmetic comparison to its stationary length in the station area and the expected Lorentz contraction enabled her to solve its velocity. ). Yvonne had innocently expected to see a direct linear relationship between pedal sprocket RPM and bike velocity; i.e. she thought 30 RPM @ .45c would extrapolate to 60 RPM @ .90c. ( I must interject here to explain that Yvonne was measuring RPM by seeing the pedal sprocket as a clock face and the pedal as a clock hand; when the pedal went from straight up all the way around to straight up again it had indicated one revolution like a clock's sweep second hand. She was timing it with a stopwatch bought when she had been looking for a certain kind of specialty store, asked directions from someone who was hearing challenged, and found herself in a pawn shop. It is the same stopwatch that Foucault used to time the speed of light, so she was told. Since it was an impulse purchase, she calls it her crazy clock.). You might unfairly think that Gigi and Yvonne are a little unreliable, but if you ever saw their assistance like I have, you would have to agree that you could not get better assistance anywhere. Yvonne's raw data did not seem to match her expectation of the RPM @ .90c. She keeps babbling something about time dilution and that only clocks go slow when they go fast. We are burning the midnight oil to double check her field notes and correct the mistakes. We are self taught experts at watching clocks but just do not understand Einstein's time delusion. We are pretty much certain that Antz observed 30 RPM @ .45c and 60 RPM @ .90c. Yvonne claims that her observation of the RPM @ .90c was nowhere near 60 RPM. She claims that she observed Antz's Rolex, once personally owned by Einstein himself and bought in the same pawn shop, going about half as fast when he was going twice as fast. Please, before we correct her field notes, can anyone help us figure out what the corrections should be; what RPM should we expect to see @ .90c?
What you and your assistants failed to notice is that length contraction turned the circular wheels into elipses. Therefor ground speed and wheel rpm don't track linearly.
kelvinalm, your response is giving our theoretical department some much needed motivation. However, due to my alledged tyranical attitude toward them, they were already as motivated as possible, so they say. We are at the mercy of wiser Relativity minds than ours. In order to correct the raw data field notes, in the usual fashion, we desperately need Relativity experts to inform our ignorant selves of what RPM stationary observers should have seen at the bike speed of .90c.
Very entertaining reading, CANGAS! Tore De Pants Marathon, lol! I think that you have presented a challenging question for relativity as long as we neglect the curve-balls you've thrown, such as relying upon a measurement of optical apparent length to calculate speed. I have read that length-contraction is not visually detectable, and if this is true, then your methodology for measuring speed would have to be changed to a more direct approach. No big deal. So far so good! To reduce this to a workable gedanken, I would suggest assuming that Antz is just pedaling directly to the front wheel, like those original bicycles that had huge front wheels and no drive chains. This meets your requirement for a fixed gear ratio, and can match your RPM variables as long as the front wheel diameter is large enough. It will not matter to Antz either way, because the hot dogs and beer have given him super-fast, super-strong legs anyway, so they might as well be long enough to reach pedals at the center of a huge front wheel. The relative velocity between the bottom of the tires and the road is zero, so there will be no time dilation or length contraction at that theoretical, mathematical point. However, the relative velocity between the top of the tires and the road would be 2v classically (where v is the velocity of the bicycle relative to the road), so there will be time dilation and length contraction there. The relativistic velocity addition formula should probably be substituted for that classical expression, but regardless, it is clear that there will be varying amounts of length-contraction, time-dilation, and relative-simultanaety at various places around the perimeter of the wheel. This is as far as I can go right now, but if I had to try to approach it, my first thought would be that a 3-dimensional Lorentz transform is required (two spatial and one time dimension), or perhaps it cannot be completed without some calculus, I am not sure. Anyway, I would be interested in learning from this if someone does complete the calculations. Day 1 measurements: v<sub>1</sub> = 0.45c w<sub>1</sub> = 30 RPM Day 2 expected measurements: v<sub>2</sub> = 0.90c w<sub>2</sub> = 60 RPM Day 2 measurements: v<sub>2</sub> = 0.90c w<sub>2</sub> = nowhere near 60 RPM (?) By edit: After a few more minutes thought, I think the solution is actually quite simple. Take any point on the tire, and consider its path relative to the embankment frame. It forms a cycloid, and so the measured period (RPMs) should be just whatever is expected classically in the embankment frame (w<sub>2</sub> = 60 RPM). It is only when we wish to transform the "per minute" part of the RPMs to Antz's frame must we use relativity. If so, then this is much easier than it looks at first!
Actually, it might be easier to analyze in the rest frame of the rider, Antz. Then you can use the fact that the course is Lorentz contracted to conclude that the total number of revolutions over the course is less at .9c. I think. But then the rim of the tire is length contracted such that pi doesn't hold. Maybe. I could use some assistance. Are Gigi and Yvonne available? I would very much like to see... ummm... have their assistance. Please Register or Log in to view the hidden image!
I finally got around to doing some calculations here for this interesting problem. For convenience I used units where c=1 and where the radius of the bike tire = 1. In these units the angular frequency ω in Antz's frame is also his linear velocity in Gigi's frame. For convenience I also made the substitution, t=θ/ω. If we paint a spot on Antz wheel at some angle, φ, then the worldline of that spot in Antz's rest frame is: s = (θ/ω, cos(θ+φ), sin(θ+φ), 0) Lorentz transforming to Gigi's frame: s' = (γ (θ/ω + ω cos(θ+φ)), γ (θ + cos(θ+φ)), sin(θ+φ), 0) So, to find the period and the displacement in each frame we simply evaluate s and s' at θ=0 and θ=2π and take the difference. This gives: Δs = (2π/ω, 0, 0, 0) Δs' = (γ 2π/ω, γ 2π, 0, 0) So, given a wheel of a particular radius in Antz's frame everything Lorentz transforms consistently into Gigi's frame. Unfortunately, this does not really answer CANGAS' original question because I don't know how to determine the proper size of a spinning wheel. So it could be that a r=1 wheel for ω=.9 is a different proper size than a r=1 wheel for ω=.5 Anyone have any ideas about how to determine the proper size of a spinning wheel? -Dale
This whole thing has really been generally vexing to everyone concerned. Exacerbated by the fact that Antz is rarely coherent. He has been mimicing not only the type but also the quantity of performance enhancing substances alledgedly used by his target best baseball player of all time; large amounts of hot dogs and beer. We are guessing that the large amount of mustard is having a toxic effect and keeping him chronically intoxicated. So, attempts to gain intel by debriefing him about his speedometer readings and tachometer readings have been minimally successful. After the initial flap wherein we seemed to get an anamolous stationary observation of Antz's pedal RPM, we decided to run some tests. First, we brought Antz and his bike up to our temporary headquarters in the Hilton penthouse. We put a wine case under the bike frame to hold the rear wheel up off the floor, and instructed Antz to pedal to hold his speedometer on .45c, and to be careful to notice the exact pedal sprocket tachometer reading. Next we instructed him to repeat the drill at a speedometer reading of .90c. With both the speedometer and the tachometer running off of the drive train, the immobile front wheel would not matter. Gigi and Yvonne independently observed the pedal sprocket RPM while being relatively stationary on the easy chair and the couch, respectively. The assistants' RPM observations were in close agreement with Antz's. Antz casually reported seeing ABOUT 35RPM @ .45c and ABOUT 70RPM @ .90c Next, we scoured the neighborhood and were able to locate and "borrow" a flatbed truck with a turbocharged motor. We put the bike up on the bed and strapped it down, again with the frame blocked up with one of our empty wine cases to hold the rear wheel up off the floor of the flatbed. Since traffic was non existent in the middle of the night, we easily ran two more tests. We first instructed Antz to pedal the bike at .45c on his speedometer and observe the trachometer reading while Gigi drove the truck past us at .45c. Then he was to repeat the drill at his speedometer reading of .90c corresponding to a truck drive-by at .90c. during these passes, Yvonne was sitting motionless on the embankment and making a stationary observation of his pedal sprocket RPM. And as usual I was carrying out the hard job of monitoring our refreshment supply. In his usual nonchalant fashion, Antz reported seeing his tachometer having shown ABOUT 35RPM @ .45c and ABOUT 70RPM @ .90c. Honestly, if he hadn't been my boss, I would have fired him on the spot for his sloppy observations. I wanted exact readings. However, my assistant again reported 30RPM @ .45c and nowhere near 70RPM @ .90c. While we admit that we just don't understand time donation in Special Relativity, we have it figured that the constant gearing forces the RPM to exactly double as the bike speed increases exactly double. But we suspect that Special Relativity may make us station area observers see undouble. But if the pedal sprocket RPM must be double, how could anything make it be seen as nowhere near double? HELP! All this seemed to jog Antz's memory and he reported that on race day he had seen ABOUT 35RPM @ .45c. No amount of begging or threatening was successful to get him to remember his race day RPM @ .90c, however. He did say that it might have been abo t My k ybo r is cra ing ga n. I'l c mplet t is ost s so n as pos le.
Put Antz back on the wine case and have Gigi watch the speedometer and tachometer while Yvonne measures the radius of the wheel. Once you have a radius v. tachometer curve in the bike frame you can use the transform I provided above to determine speed and RPM in the embankment frame. If it is constant then it is particularly easy. -Dale
If you are hoping to calculate the circumference of the rotating wheel from the measured radius, you should remember that, according to relativity, circumference = 2πr does not hold for rotating disks. Euclidean geometery fails in this case. As for the circumference of the rotating disk, relativity predicts that it should increase with the rotation rate (in the rotating frame) and decrease with the rotation rate (in the inertial frame fixed at the center of the disk). Just consider kevinalm's suggestion that, in Antz's frame of reference, the embankment is length-contracted as it moves under Antz's wheels. This allows him to cover a different "distance-per-wheel-cycle" than would be expected using Euclidean geometery. In the bike frame, you should be able to calculate the "rotating-circumference" of the circular wheel by comparing the rotation-rate (in that frame) to the length-contracted embankment traversed in one cycle of the wheel. Likewise, In the embankment frame, you should be able to calculate the "rotating-circumference" of the elliptical wheel by comparing the rotation-rate (in that frame) to the proper length of embankment traversed in one cycle of the wheel. All frames must agree as to the location of the beginning and end of one rotation cycle of the wheel. However, they can disagree on the amount of time and space seperating the endpoints of the cycle.
Yes, I knew that, which is why I set up the math to figure out the period and distance without ever explicitly calculating the circumference. The question is not about the circumference, but about the radius as it determines the length scale. The reason is that, for convenience, I used c=1 and r=1. The c is obviously not a problem since it is independent of the proper RPM. On the other hand, r may not be independent of the proper RPM. If that were true then e.g. they could be using meters at .5c and yards at .9c: I essentially need the conversion factor. -Dale
Don't let me discourage you from persuing your most interesting approach to the math. I am simply wondering why the "length scale", as you call it, not a simple (x, t, x', t') type of Lorenz transform? In Antz's frame, the rotating disk is a circle, but in the embankment frame, as he pedals along at v the rotating disk is geometrically an ellipse, while temporally being a "something-oid" with an even larger circumference than expected. That is why I thought calculating the circumference would be more predictive than the radius. What use is knowing the radius for a something-oid which does not adhere to 2πr?
This is a beautiful puzzle. Did you come up with it yourself? A simple calculation suggests that according to SR, the pedal rpm at 0.9c is less than the pedal rpm at 0.45c. Specifically, SR says that the pedal rpm at 0.9c is only 29.3rpm: <table border cellpadding=5><tr><th>Speed</th><th>Gamma</th><th>Antz</th><th>Yvonne</th></tr><tr><th>0.45c</th><td>1.12</td><td>33.6 rpm</td><td>30 rpm</td></tr><tr><th>0.90c</th><td>2.29</td><td>67.2 rpm</td><td>29.3 rpm</td></tr></table> But how can this be? The gear ratio is frame independant, and the distance traversed by the bike per wheel revolution in Yvonne's frame should also be independant of the bike's speed (consider a ribbon wrapped around the wheel, unrolling as the bike progresses). Therefore the distance traversed by the bike per pedal revolution in Yvonne's frame should be independent of the bike's speed, and the pedal rpm should indeed track linearly with bike speed, as Yvonne expected. I am sorely puzzled, will think more, and will read up on the rigid rotating disk literature.
Well, actually I did both. I did a simple Lorentz transform from Antz's frame to Gigi's frame, but I simply chose the units in Antz's frame to make the equations prettier. By setting r=1 all of the r terms dropped out and it made the expression simpler. I always like to do all of the simplification I can prior to transformation, because the transformation always makes expressions more complicated. Exactly, the wheel is a circle in Antz's frame, but I am not sure that the radius is independent of the RPM. Once you have the circle in Antz's frame, transforming to any other inertial frame is pretty easy. I hadn't planned on it, but I will see if I can figure out how to explicitly calculate the circumference in Gigi's frame. If nobody has any objections then I am going to go under the assumption that the radius is independent of RPM for all further calculations. Perhaps this requires some actively controlled material that expands or contracts based on the centripetal force applied. -Dale
Oh now I understand. Yes, I agree with you, the radius is independant of the RPM. The circumference, on the other hand, is not independant of the RPM, which you already knew. I just thought that the thread would ultimately come down to the non-intuitive fact that the circumference is larger when the bicycle speed is greater, leading to predictions of slower RPM's at .90c, and a faster RPM's at .45c. This tends to be confusing -- even Pete seems to be a bit puzzled about it in his above post. So I tried to get the explanation out there in advance. PS: The calculation for the circumference in Gigi's frame is just the distance covered in one cycle.
After a little research and much cogitation, I believe I have it. According to SR, the distance traversed per wheel revolution does in fact change with the bike speed. Assuming a constant wheel radius (very strong spokes!), the tire and rim are physically stretched by a factor of gamma. So the previously posted table is the actual SR prediction, and Yvonne should expect to measure 29.3 rpm for the pedals when Antz is cruising at 0.9c. To check this explanation, I recommend testing if the tire and rim are in fact stretched. I can think of two simple ways: 1) Measure the thickness of the tyre at rest and at speed. if the bike is in motion, Yvonne can measure this thickness by sight at the top of the wheel. 2) Attach strain gauges at various places around the rim. Obtaining untainted telemetry may prove dificult, but I'm sure the team can work out a way... perhaps small LED displays could be mounted with the gauges and read by sight? Postscript: I am not really comfortable with the physicalness of this explanation. It doesn't sit right with me that the circumference of the wheel can be larger when it is spinning. It is internally consistent, but it just doesn't feel right. Does anyone know if anything like this has been observed experimentally?
When working it through, it helped a lot to think of a similar situation with a tank. Consider the length of the tank treads in both frames at different speeds. Just focus on the straight sections.
Well if you give it some thought you would see that the tread on the ground has "0" velocity and the upper tread must have 2 times the axle velocity. However at relavistic speeds you cannot have 2 x 0.9c = 1.8c but only 0.994475c. Something has to happen to the wheel which is non-linear around the circumference. That is no change at the bottom to increasing change which balances the 0.994475c velocity of the upper tread with the non-relavistic fact that the upper tread must have a velocity 2x the axle velocity.
Hi Pete. That is a good way to look at it, because it shows that we can arrive at essentially the same result without having to consider the frame-forces on the rotating disk. Einstein talks about this as a GR problem, and he equates the frame-forces on the rotating disk with a gravitational field. Here is a quote from chapter 23 of Relativity: The Special and General Theory. 1920. Albert Einstein <blockquote>But the observer on the disc may regard his disc as a reference-body which is “at rest”; on the basis of the general principle of relativity he is justified in doing this. The force acting on himself, and in fact on all other bodies which are at rest relative to the disc, he regards as the effect of a gravitational field. </blockquote> http://www.bartleby.com/173/23.html There you can also read about the failure of Euclidean geometry in the rotating frame, and the inadequacy of π in calculating the circumference based on the radius. I do not know whether any experimental evidence has been found for this effect in particular. That is a good question!
Only if the tread is the same density throughout its length in the ground frame, which it isn't. In the ground frame, the bottom section is stretched, the top section is contracted. That's not a fact... Give it some more thought. Imagine a rubber band around two axles, with one section stretched more than the other.
You really think this arguement holds water? I don't. Stretching and contracting compensating for 0.805525c velocity differential? Hardly. Please Register or Log in to view the hidden image!
