Good detective work, SYA.
The above shows the new spline curve after including that last data point at 6.9. It looks a little smoother now. Kind of reminds me of Salvador Dali's painting, "The Persistence of Memory."

Good detective work, SYA.
The above shows the new spline curve after including that last data point at 6.9. It looks a little smoother now. Kind of reminds me of Salvador Dali's painting, "The Persistence of Memory."
Dear SYA,
I have an idea for how we might be able to solve for the optical shape of a translating, but nonrotating disk. (It could be rotating also, but this method focuses on the shape, not the locations of any particular spoke tips.)
Let the camera be located at (x,y)=(0,0). Imagine a circle of radius r centered on that point. If all points on that circle emit light rays simultaneously, they would all arrive at the camera simultaneously. By choosing length and time units such that the speed of light c=1, we also know that the light would arrive at the camera after a period of time equal to r. So, if the light rays were emitted at time t=r, they would all arrive at the camera at t=0.
We also know where the elliptical wheel would be located at t=r. If we could solve for the intersection between the ellipse and the circle, we could know as many as two points which were located on the perimeter of the ellipse, which also emitted light that arrived at the camera at time t=0.
For reference, here is a link:
Find the Points of Intersection of two Circles
An ellipse is basically a lengthcontracted circle. So, do you think we could solve this directly? We might have to process a number of different r values. What do you think?
I've got the equation for the circle...
(where is the radius of the circle, and also the time elapsed between the light being emitted from the circle and then subsequently entering the camera at the center of the circle)
...and I've got the equation for the ellipse...
(where is the radius of the wheel in the axle frame)
...but I'm not having much luck solving for the intersection points...
I agree those eqns are good. I'll take a look at the simultaneous eqns to solve for the ellipse/circle intercepts.
Thank You,
SinceYouAsked
I get these ...
x^{2} + y^{2} = (ct)^{2}
(γR(x+vt))^{2}+ (yR)^{2} = R^{2}
My first attempt to solve the 2 simultaneous equations fails, because both the x^{2} and y^{2} terms do not cancel out. Pretty much a mess. Thus, it would seem we need to find a 3rd simultaneous equation, to cancel out the required term to solve it. Not sure what eqn that might be as yet. Does look to be messy, and possibly undo'able using your reference web site's procedure. I'll be on the road tomorrow, so may be a day or 2 before I can post.
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 03262014 at 07:11 PM.
Just to test your equation, I tried gamma=2, R=2, and vt=1:
x^2 + y^2 = 1
(2*2*(x+1))^2 + (y2)^2 =2^2
As you can see, that produces a much narrower ellipse than it should be for gamma=2. So, something is amiss there. Also, it looks like the t term should be subtracted instead of added, (my error).
Here is a link to the above webpage:
Equation Grapher
I agree, there's not much hope in solving the simultaneous equations. There are graph plotting programs which can find points of intersection, so all we have to do is plot the equations and let the software do the work.
Here is a link to the above webpage:
Online Multiple Equations Plotter
Maybe this will help. Putting the circle and the ellipse into the exact same ellipse form, I think they look like this:
Circle:
Ellipse:
Which, for reference, look like these in nonlatex language:
(x  0)^2/(ct/1)^2 + (y  0)^2/(ct/1)^2 = 1
(x  vt)^2/(R/γ)^2 + (y  R)^2/(R/1)^2 = 1
I think...
So, now can we get some things to cancel out? Doesn't look like it...
I thought along those lines as well, however, I don't think that works either. Ultimately the same fundamental problem remains. We need to add a 3rd equation. With the right equation, that should do it.
Thank You,
SinceYouAsked
EDIT: I just figured it out. Hint: In that messy simultaneous eqns soln, place y in terms of v, t, r, and gamma. These are either constants, or variables held constant, as x & y are then solved for. Then the quadratic soln is attained straight from the coefficients of the quadratic eqn. Having the x solns, the y solns are then attained.
Sorry JT,
You posted just prior to my edit of my prior post. Here's the edit ...
EDIT 1: I just figured it out. Hint: In that messy simultaneous eqns soln, place y in terms of v, t, r, and gamma. These are either constants, or variables held constant, and so the quadratic eqn is then in terms of x^{2} and x alone. Then the quadratic soln is attained straight from the coefficients of the quadratic eqn. Having the x solns first, the y solns are then attained.
If you plot it in Excel, you could tell it to label only the "integer spoketip values", which should validate the Gron figure 9 part C as accurate (or not). EDIT 2: Actually, labeling only the spoketips every 22.5 degrees apart would produce the 16 equally spaced radial elements of Gron's rolling disk in his figure 9.
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 03272014 at 07:17 AM.
SYA, here is part C with part A emission point intersections in red and part B intersection points in blue. They have similar patterns on the x axis but neither of them appears to represent equal spacings even if the timing order is not sequential. I was surprised that there are 2 sets of 2 emission points on the same part A or B wheel as each has 13 center lines on the x axis.
EDIT: It seems that the analysis below will not get the job done after all (per the link reference below), and so a new method will require determination. Sigh.
Relativistic Rolling Wheel II
Thank You,
SinceYouAsked
************************************
JTyesthatJT,
Could you please review this for correctness? Then at your leisure, if you think the analysis below is OK, please do that wonderful spreadsheet thing you do and plot the part C figure, if you will. Varying the velocity should eventually tell us what Gron's part C was really intended to do, as it seems something went amiss there. Thanx.
The circle …
x²/(radius)² + y²/(radius)² = 1
x²/(ct)² + y²/(ct)² = 1
x² + y² = (ct)²
Eqn 1 ... x² + y² = (ct)²
The ellipse …
x²/a² + y²/b² = 1 … ellipse (a for semiminor axis, b for semimajor axis)
x²/(r/γ)² + y²/r² = 1 ... r is the disk's radius per axle system
(γ/r)²x² + y²/r² = 1
γ²x² + y² = r²
but for an ellipse offcenter wrt origin, with offcentering defined by x=vt and y=r, then ...
γ²(x+x_{offset})² + (y+y_{offset})² = r²
γ²(xvt)² + (yr)² = r² ... (where t can be negative)
γ²(x²2vtx+v²t²) + (y²2ry+r²) = r²
γ²(x²2γ²vtx+γ²v²t² + y²2ry+r² = r²
(γ²)x²  (2γ²vt)x + y² – (2r)y = (γvt)²
Eqn 2 ... (γ²)x²  (2γ²vt)x + y²  (2r)y = (γvt)²
above, where γ = 1/√(1v²/c²)
Subtracting the simultaneous equations, ie Eqn 1  Eqn 2 …
so Eqn 1 ...
x² + y² = (ct)²
minus Eqn 2 below ...
(γ²)x²  (2γ²vt)x + y² – (2r)y = (γvt)²
equals ...
x²  (γ²)x² + (2γ²vt)x + (2r)y = (γvt)² + (ct)²
(1γ²)x² + (2γ²vt)x + (2r)y = (γvt)² + (ct)²
Eqn 3 ... (1γ²)x² + (2γ²vt)x + (2r)y = (γvt)² + (ct)²
Place y into terms of "constants, variables held constant, or time considered in the instant" …
F = 360(vt'/(2πr)) … angular orientation of spoke per axle frame
F = 360(v(t/γ)/(2πr)) … in terms of ground time
F = 360vt/(γ2πr)
y = y’ = rrcos(F) … spoketip’s ylocation wrt angular orientation of spoketip per axle
y = rrcos(360vt/(γ2πr)) … spoketip’s ylocation in terms of r, v, γ, and ground time t
y = r(1cos(360vt/(γ2πr)))
Eqn 4 ... y = r(1cos(360vt/(γ2πr)))
substituting for y its (above) value …
(1γ²)x² + (2γ²vt)x + (2r)y = (γvt)² + (ct)²
(1γ²)x² + (2γ²vt)x + (2r)( r(1cos(360vt/γ2πr)) ) = (γvt)² + (ct)²
(1γ²)x² + (2γ²vt)x + (2r²(1cos(360vt/γ2πr)) = (γvt)² + (ct)²
(1γ²)x² + (2γ²vt)x + (2r²(1cos(360vt/γ2πr))(γvt)²(ct)²) = 0
so our quadratic eqn is this ...
Eqn 5 ... (1γ²)x² + (2γ²vt)x + (2r²(1cos(360vt/γ2πr))(γvt)²(ct)²) = 0
For a standard quadratic eqn …
ax² + bx + c = 0
The 2 solns for x are …
x = (b±√(b²4ac)) / (2a)
Our quadratic Eqn 5 here was …
(1γ²)x² + (2γ²vt)x + (2r²(1cos(360vt/γ2πr))(γvt)²(ct)²) = 0
Defining the coefficients …
a = 1γ²
b = 2γ²vt
c = 2r²(1cos(360vt/γ2πr))(γvt)²(ct)²
Substitute those coefficient values into the eqn below to attain the two x solns …
Eqn 6 ... x = (b±√(b²4ac)) / (2a)
Substitute those solns into the eqn below to attain the corresponding two y solns …
x² + y² = (ct)²
y² = (ct)²  x²
y = √[(ct)²  x²]
Eqn 7 ... y = √[(ct)²x²]
*************************************************
OK, so labeling spoketips that are equally spaced by ( 360/16 = ) 22.5 degrees would produce the 16 perimeter points labeled by Gron in his figure 9 part C. The spoketip degree can be divided by 22.5 to produce Gron's 1 thru 16 perimeter points. Just start from the spoketip 1 ground contact event as x,y,t = 0,0,0 (as the reference) and work backwards (toward left along x) using negative t values for input (wrt that reference). Remember that ... F = 360vt/(γ2πr), so t = γ2πrF/(360vt). Plug in the desired F (multiples of 22.5) and you have the associated ground time t, and using that time t that should produced Gron's 16 designated perimeter events. And all this fun cincirob is missing out on since he vanished!
Thank you,
SinceYouAsked
Last edited by SinceYouAsked; 03282014 at 09:30 AM.
Wow, SYA, that's pretty cool. I'm doing it a completely different way, but I'm not done yet. I'm looking forward to seeing if I get the same answer that you got... Thanks.
Hi laurieag,
Well, Gron's disk rolls from left to right, in the direction of +x, at a steady roll rate (translation velocity) of v. We can assume the spoketip 1 ground contact event (near the right side of fig 9 part C) occurs at the origin of both systems, axle (primed system) and ground (unprimed) ... so at x,y,t = x',y',t' = 0,0,0. The left edge of the part C figure occurs at some negative time t. As the wheel continues to roll toward the origin, time is increasing, so the magnitude of time t ( ie t ) is decreasing toward 0. As such, the first firing event occurs back (about) when the left edge of the moving contracted ellipse intersects near the part C slanted oval (near its left edge). One firing event occurs there. After that, there are 2 simultaneous firing events for each time t, with the lower event always offset (leftwards) wrt the higher event, until a final firing event occurs near the rightmost edge of the part C slanted oval. So wrt to those 2 boundary events, the firing is left to right, sequentially. I'm hoping all those events are defined by my prior post, assuming I did not screw that up By "firing event", I mean EM emission from a point of the rolling disks perimeter, such that all such events produce photons that arrive at the ground contact event for spoketip 1 at the colocated origins.
Thank You,
SinceYouAsked
Each of your blue contracted ellipses (per ground system) have only 1 pair of simultaneous emission events. The red round wheel (per axle system) will hold those same 2 emission events as asynchronous. By emission events, I mean "those emission events that produce photons that arrive at spoketip 1's ground contact event", at x,y,t = x',y',t' = 0,0,0.
Thank you,
SinceYouAsked
Thanks SinceYouAsked,
Sorry to be a pain, I just found the following wikipedia page looking for 'translational velocity'.
Rigid body  Wikipedia, the free encyclopedia
The linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion (motion with no rotation), all points on a rigid body move with the same velocity. However, when motion involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation.
Circle:
Ellipse:
To which we can apply the quadratic formula... ugh...
Or maybe I should change the last few lines to these...
...and then substitute that into the original circle equation... ugh...
After further review, my prior soln here ...
Relativistic Rolling Wheel II
will not work either JT. So unfortunately, I see no need to model that by spreadsheet.
Where I substitute for y its value, it is the ylocation of the spoketip 1 at the prior time t, in terms of gamma, r, v, and t. However, the 2 ysolns we are looking for via the quadratic soln are not the location of the spoketip 1 at that time t. It's the 2 intercept points we are looking for at that time t. I cannot substitute a value for y at a time t unless that yvalue represents the location of one of the two intercept points. Sigh. You agree?
Back to the drawing board.
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 03272014 at 08:33 PM.
Yeah, I agree. I can't see any way to solve it directly. I'm wondering if it would be worthwhile to build a multiple function grapher in Excel, and have it check for (approximate) intersection points by brute force. That would be a neat thing to have, but it would be a whole lot easier just to use the graphing website that I linked to.
As far as Gron's figure 9C, there is no velocity which can fix the problems it has. A whole new figure 9C would have to be made, similar to the one I made for 0.866025c. Even after all that work, we would not necessarily get the individual spoke tip locations. We'd likely get mostly inbetween numbers like the ones I got in my C diagram, (1, 1.9, 2.4, 3.2, 5, 6.9, 9.1, 10.6, 11.8). So, it doesn't look like we're going to get any bananas on this one.
I suppose, but I'd like to find a direct soln method. Any good mathematician should be able to do that. I'm gonna persist a little longer on this, just for the challenge, see what happens.
Does seem to be the case, yes. We'll see.
Thank You,
SinceYouAsked
Good news! I've got the next best thing to an actual solution to the simultaneous equations! I built a function plotter in Excel, and it seems to work pretty well for finding the intersections we were looking for. You can download it here:
http://www.filedropper.com/functionplotter_1
Once downloaded, all you have to do is hover your mouse over any intersection point you might see on the graph, and Excel will display its x and y coordinates for you. (Excel actually defaults to this behavior, standard.) Next, try typing different values of t in yellow cell labeled t, and a new graph is drawn! No need to type the negative sign on the times (t). You can also change the value of v in the other yellow cell. Enjoy!
Cool, thanx JT.
I must say though, that I have found this exercise rather frustrating. I have just looked around the web, and there seems to be nothing for the intersection of an ellipse and circle when the ellipse is offset wrt x & y, while the circle is not. The brute force method seems the required course. At least the relations involved in the part C figure have been presented clearly, IMO. The Gron figure is most definitely in error, and I wonder whether that was an oversight during editing and publishing? I agree, the spoketip events depicted on Gron's fig 9 part C are not correct, although the general aspects look appropriate. The labeled points do look to have been hand placed. If you look carefully, on the rightmost contracted ellipse (http://areeweb.polito.it/ricerca/rel...los/gron_d.pdf see pg 39), you'll note that the spoketips on the right side of the wheel are higher up than on the left side, which should never be the case for an even number of equally spaced spokes in the instant when one spoketip is at ground contact and its respective spoke in the vertical 6:00 orientation.
While it would be nice to produce the correct part C image for Gron's figure 9, I don't think it's worth the effort. I think enough has been presented here. It's a bummer that the quadratic soln doesn't work here, but as you said, we'd still have to figure out where each spoke resides on that part C figure anyways, so.
Thank You,
SinceYouAsked
Hi JT and SYA,
I'd like to thank you both for having perservered through two threads, with 737 and counting posts, to try get to the bottom of things and I also would like to see a correct part C plot.
Considering that the majority of the visible sources in our universe rotate, seeing how an accurate solution really works will most likely be an essential step in resolving the galactic rotation curve problem among other things.
You're welcome, and thank you.
I'll take another look at it, see if I can tweek my old spreadsheet analysis to produce Gron's snapshot image. JT's hasn't posted very recently, so he may be looking at it as well, using his own spreadsheet analysis? Either of us can certainly do it, but it's just a matter of the effort and time required to do so. For myself, time is not as available as late. JT and I agree that Gron's part C figure is the correct general shape, presenting the correct general concepts. However, the spoketip placements are most certainly off kilter. His part C spoketip placements do not seem reasonable for any steady velocity.
I figure it likely that Gron's figure 9 was added in haste during the editing/publishing process, and the spoketip events quickly handplaced to present the general aspects of the scenario. While no one is perfect, Gron was not the type to publish incorrect math. He is a top rate physicist, amongst other things. I only mention this because of the antirelativists who may argue that relativity is in error, because relativists (like Gron) produce unreasonable figures using relativity. None of that's true of course. I'd like to see the figure as well, so I'll take a look at it. BTW, JT's figure he presented here ...
see link > Relativistic Rolling Wheel II
covers all the bases. While it does not present 12 (or Gron's 16) evenly spaced spoketip events (per axle) in the snapshot (per ground), it does present nonevenly spaced spoketips (per axle) in the snapshot (per ground), which is just as good. The spoketip placement and spoketips specified are accurate, and the general shape is correct (which looks very much like Gron's).
Check into the thread periodically, just to see if that figure is posted yet. If I should decide not to tackle it at present, because of time constraints, I'll post that as well. I will eventually post it later, if not sooner. Thanx laurieag.
Thank You,
SinceYouAsked
JTyesthatJT,
Do you have a decent idea of what approach you would use if you tackled recreating the part C figure from scratch?
On my approach, I plan to consider the cycloid of each Gron's 16 equally spaced spoke tips independently, over a full rotation of the wheel, and then capture the t=0 solns alone. The spoke 0 ends at ground contact in the 6:00 orientation, and that'll the event for solns per ground. Then plot them. I'll let you know how it goes.
I figure I'll go with v = 0.866025c the first time, although Gron's contracted ellipse looks more like v=0.82164c or so. Since Gron's part C is certainly inaccurate (for any velocity), maybe 0.866025c is the way to go. That way, we can compare that soln with your prior posted part C soln here ... Relativistic Rolling Wheel II
Thank You,
SinceYouAsked
*************************
EDIT: wrt the above, if using the spoketip 1 contact event as the colocated origins, I think I need to determine the ratio of ...
current_axle_range / current spoketip range = 0.866025 , so ...
vt = sqrt(x_spoketip^2 + y_spoketip^2)
as per your prior posted figure. So t=0 at the contact event, and t begins at negative values for any other spoketip's prior ground contact event. Think that'll work?
Last edited by SinceYouAsked; 03312014 at 04:29 AM.
Some things to think about:
1. If I recall correctly, cincirob found a plugin for Excel that performed iterations/interpolations automatically. I know there is a plugin called "Solver" but that might not be the one he used. Anyway, having something like that could potentially save a lot of time and work. Imagine typing in a spoke tip number, and getting instant results for its location at road time t=0. That would help.
2. If you start with v=0.866025c you might end up doing a lot of work just to find out that it does not match Gron's figure 9B. I would say to use the velocity you think matches Gron's best, v=0.82164c., and then solve a minimum number of points (maybe 3 or 4). Keep comparing the results to Gron's figure 9B before proceeding. That way, if you notice things starting to look wrong, you can tweak the velocity a little and start over.
3. Once you have the velocity that creates a figure nearly identical to Gron's figure 9B, you will actually need to use that velocity to find the locations of a lot more than 16 points. Probably at least five times that. If you could do ten times that, it would be even better. The reason you need so many is because if you solve for figure 9C using only 16 points, you're going to get a lot of inbetween numbers like I did. By starting with more points, you have more likelihood of getting results which are whole number points, instead of the inbetween numbers.
4. My most recent spreadsheet can help you solve for the intersection points between the circle and the ellipse. So, I would suggest you might want to take the same approach I did in my part C type diagram. You will imagine the ellipses equal distances apart, and then simply shift the numbers on your 160 labeled points by one digit. It will be too messy to do it all on one diagram, (you'll need a lot of separate diagrams). Next you use my spreadsheet to find the intersection points. Finally you can make a single diagram like Gron's figure 9C, perhaps fudging some points to make them whole numbers. Like if you get a location for 5.9, and another one for 6.2, you can probably guess where 6 would be, based on what you found.
If you have any questions, just ask. Good luck!
Quick example:
For my diagram at the top of this page, I used v=0.866025c and 12 spoke tip points. I placed some ellipses along the road, equallyspaced. The horizontal distance between the ellipses, centertocenter, is gamma*2piR/12 which in this case is equal to 1.047197551. The amount of road time that elapses between those snapshots is gamma*2piR/12v which in this case is equal to 1.209199576 . So, in my "function plotter" spreadsheet, I leave the velocity as it is, and the first time I put in the yellow cell is t=1.209199576. Next, I hover my mouse over the intersection points. I find the lower intersection point to be located at approximately (x,y)=(1.208000,0.053848) and I find the higher intersection point to be located at approximately (x,y)=(0.540000,1.081926). Those would be the points which I labeled 11.8 and 1.9, (respectively), on my diagram at the top of this page. Granted these are approximations upon approximations, but if I had started with about 5 times more data points, a very clear picture should start to emerge. For example, I might have found 12.0 and 2.0 as well as 11.8 and 1.9, (or at least I would have been in a much better position to estimate where 12.0 and 2.0 should be located).
Dear SYA,
Check out this video. Be sure to watch to the end (it's only 3 1/2 minutes).
Iterative Solutions/Excel  YouTube
That method would probably let us solve for all the spoke tip locations, as well as the intersection points with the light circles. We have it made! All we have to do is rebuild our spreadsheet around this concept. That will take some work, but the end result is automatic solutions and graphs of everything! Whoo Hooo!
JTyesthatJT,
Thanx for the Excel technique. Yes, there are multiple ways the figure can be produced. I stuck with my spreadsheet, and tweeked it. It was not difficult, once I had the approach square ...
Laurieag,
Below ...
The FIGURE ... is a remake of Gron's figure 9 Part C using v=0.866025c. As he did, the wonder camera takes a snapshot at the colocated origins at ground time t=0, and so that event is x,y,t = x',y',t' = 0,0,0. Call this event of interest "the EVENT", a ground contact event for the tip of spoke 1. The disk rolls left to right along x in the direction of increasing x, the axle traveling at an inertial v. For Gron's 16 equally spaced spokes (of the axle frame), the ground image at t=0 is the product of all the emission events that result in photons arriving at the camera lens at "the EVENT", as follows ...
where gamma = 2 since the disk rolls with an axle translation rate of v = 0.866025c.
EDIT: The EVENT FIRING ORDER (requested later by Laurieag) is posted here ... post link > Relativistic Rolling Wheel II
EDIT: The SPOKETIP COORDINATES for the contracted ellipse at ground time t=0, are posted here ... post link > Relativistic Rolling Wheel II
EDIT: My analysis approach is presented by JTyesthatJT's prior graphic, posted here ... post link > Relativistic Rolling Wheel II
EDIT: My original rolling wheel analysis which was modified to produce the above plots ... Relativistic Rolling Wheel II
Relativistic Rolling Wheel II
SPREADSHEET DESIGN ... The disk may be considered a spoked wheel. My spreadsheet rolls the wheel over a full rotation, transforming from axle system coordinates to ground frame coordinates at discrete 1 degree intervals. It considers only 1 atom (at radius R') of 1 rotating spoke (N) of a wheel of radius R, per axle. The initial EVENT is when the spoke of interest is in the 6:00 orientation, with its spoketip at ground contact. The spreadsheet calculates the ground frame coordinates for any event defined by "the spoke atom of interest" in the axle frame. The ground frame solutions constitute the x,y,t coordinates of the cycloid of the atom over a rollout of the wheel for any desired number of degrees, I did 1 full rotation here (360 deg).
METHOD OF CALCULATION ... I ran the spreadsheet calculation once for (the spoketip atom of) each of Gron's 16 spokes one spoke at a time, the spoketip atom located at R'=R=1. So SP_{1} > SP_{16} where SP_{n} is the nth spoke, starting the full rotation at its initial ground contact event, which occurs prior to the spoke 1 contact event (so left of the colocated origins). The axle always being at y=y'=1, I set the axle system's ground contact event for each start event as ...
x' = 0
y' = 0
t' = ( 0  2piR(1(SP_n  1)/16))/v )
So that initial groundcontactevent in ground coordinates starts here ...
x = γ( 0  2piR(1(SP_n  1)/16)) )
y = γ*y' = 0
t = γ( 0  2piR(1(SP_n  1)/16))/v )
The zeros above, represent the colocated origins at x,y,z = x',y',t' = 0,0,0 and signifies the final ground contact event to be considered for each spreadsheet run.
In this way, the input coordinates and solns for all runs are also indicative for a single rolling wheel, as though it had begun from ...
x,t = γ2piR, γ2piR/v ... at the prior spoketip 1 contact event per ground, and ending at "the EVENT" of interest (colocated origins x,y,t = x',y',t' = 0,0,0). OK ...
Ground coordinates are calculated for an entire cycloid for each of the 16 spoketips (1 at a time), then those solns are isolated when the current horizontal axle range is 86.6025% the current spoketip range (each wrt the colocated origins as reference), since the emission photons must traverse a distance of s=ct when the axle is at x=vt (for the photon to arrive at the colocated origins at t=t'=0) ... as per JTyesthatJT's prior related Figure 9 illustration. So the x,y coordinates are isolated by the following requirement ...
axle'shorizrange = 86.6025% x spoketip'srange ...
x = 0.866025 * √(x_{SPn}² + y_{SPn}²)
The spacetime solns are selected for that requirement (for the given run, given spoketip), and we then have one of Gron's spoketip solns for his figure 9 part C. Rerun the spreadsheet 16 times, and the solns poduce the above plot.
************************************************** ******
Laurieag, you had asked about the rotation aspects of this scenario. There is an angular rotation wrt the inertial axle and the inertial ground systems, but also the mechanical rotation of the disk as well. Wrt the latter, it is effected by the former per he who translates wrt the rotating disk, which is why the radial elements of Gron's disk are generally curved per the ground observer. Wrt the former ...
γ = 1/cos(θ) where θ=sin^{1}(v/c). θ signifies the frame rotation, ie the angular orientation differential between the 2 inertial spacetime systems in 4space. The photon traverses the slanted path of ct = 4 per ground, but per axle traverses the vertical diameter of the wheel directly along a stationary y'axis across a distance of y' = ct' = 2, so ...
t = γt'
ct = cγt'
ct = γ(ct') ... and since ct' = y'
ct = γy' ... and since y' = y = 2R'
ct = γ(2R') ... where 2R' is the disk's vertical diameter
4 = γ(2) ... where R'=R=1, γ=2, and 2R' is the vertical height of the wheel including at t=0.
ie, the slanted lightpath's length over a duration of t=4 is gamma times the height of the rolling disk at t=0. Geometrically, its the angular orientation differential in 4space (θ) between the 2 inertial systems that allows the above relation to hold true.
************************************************** ******
JTyesthatJT,
These were the solns produced ...
Spoketip emissionevent locations (where N is the Nth spoke)...
N... x...... y....... t
1 0.000 0.000 0.000
2 0.600 1.115 1.266
3 1.877 1.737 2.558
4 2.954 1.970 3.551
5 3.769 1.988 4.261
6 4.316 1.877 4.706
7 4.615 1.695 4.914
8 4.735 1.463 4.957
9 4.655 1.218 4.812
10 4.413 0.971 4.519
11 4.026 0.741 4.091
12 3.535 0.525 3.574
13 2.941 0.344 2.960
14 2.272 0.196 2.280
15 1.546 0.088 1.548
16 0.782 0.022 0.783
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 04152014 at 06:04 AM.
JT,
Hmmm. Looking briefly at some spreadsheet estimates here, it looks like Gron may have run his part C figure for something upwards of v=0.99c, as that would put his spoketip 2 out about where he has it, and the left edge of the part C at ~3.8 or so (consistent with his figure). I haven't run all the numbers to validate all the points though. But, his contracted ellipse appears indicative of 0.83c to 0.866c. I think what happened, is that Gron accidentally superposed a part C figure for a very high v (~0.99c) atop his A/B figures, while his part B was meant for 0.866c (or maybe the 0.8c of the discussion in that paper just prior). Which means, the spoketips in part B (of the contracted ellipse) won't work for the linear extrapolation method you used to verify it, referenced here ...
Relativistic Rolling Wheel II
... because parts B & C are apples to oranges. That is, the spoketips in B won't be where Gron has them placed, if they should instead be indicative of a velocity of the part C figure.
If I get rambunctious, maybe I'll figure the precise velocity to produce Gron's part C precisely, and lay it to rest. I think he superposed a figure with a velocity different from part B, maybe by accident. Or, he never bothered to contract the ellipse to the required size (maybe too thin to place all the spoketips clearly). A v=0.99c produces a contraction of 1/7.
Thank You,
SinceYouAsked
Very nice solution, SYA! I was touching on that idea when I posted this earlier:
Note the hypotenuse (in red) has a v in the denominator, whereas the horizontal axle range (in blue) is the same value, except it does not contain the v in the denominator. Since v=0.866c in this case, that is the same thing as the 86.6% that you are using. My equations in green and orange are only for the special case shown in the drawing. But since you know the x,y coordinates of all points on the cycloids, you can always know what the green and orange equations are for all the spoke tips, not only the special case I have shown above. This lets you solve the intersection points. Very cool.
By the way, your part C diagram looks like it has the 10 point in the same place as above, but instead of corresponding to point 14.5 on part B, yours corresponds to point 15 on part B. Thus the 4.5 in my above equations would have to be changed to 5 for them to apply to your diagram. (Actually I calculate that the 5 is really more like 5.02865).
......
May I ask how you determine where this holds true?
axle'shorizrange = 86.6% x spoketip'srange
Do you have to scan through your spreadsheet visually, or is this found automatically? Thanks.
Well, Gron's part C figure was not to scale either, as it's highest point should be y=2R. So, the emission point from the spoketip of spoke 10 should be a little higher up than the height of R, I figure. And, his event locations in part C are not related to the spoketip locations of his part B figure, since B & C are indicative of completely different (and largely different) relative velocities.
Maybe we should email Dr Gron that his part C was offkilter, just in case no one ever mentioned it to him (which I doubt though)?
For any full rotation of any selected spoke, where a calculation is run for each 1 degree of rotation, my spreadsheet calculates this at each 1 degree interval ...
ratio = x_axle / √(x_spoketip²+y_spoketip²)
I added an adjacent column with a formula that isolates the above solns for this condition ...
IF( (ABS(ratio)0.866025) < 0.005, "EVENT", " ")
and so the word "EVENT" is displayed for any spreadsheet row where the ratio is within 0.005. This was necessary since I run calculations for discrete interval separated by 1 degree of wheel rotation. Also, since I did not feel like searching for those solns by eyeball, although it would not have been bad now that I see it.
I then take the 2 sequential solns, one just above 0.866025 and the other just below 0.866025, and extrapolate the x,y values that correspond to the precise ratio of 0.866025 (which is between the 2 sets of solns) ... however, I do that extrapolation linearly, while in fact it is not. Not a big deal though, since the accuracy error of the soln should be down in the dust given the size of my partc C figure.
I did not run all those computations from the same worksheet. I did the extrapolation in another worksheet, where I also pasted the final set of ground coordinate solns (for each spoke run) into a table. I could take a little time and automate the entire process in the same worksheet, now that I know the calculations are correct.
Thank You,
SinceYouAsked
Good work SYA,
As a final cross check.
Could you also extend your spreadsheet to include the respective x axis axle offsets for the timings to get the emission point axle spacing in actual emission order? The axles x axis positions and the time between them should all show the constant linear velocity of the wheel rolling along the road despite the timing and spacing variations.
Laurieag,
I'm no expert on this, but I have done some reading on it. My understanding is that the galactic rotation curve problem, is unrelated to the relativistic effects noted wrt the Gron rolling disk analysis. The relativistic effects noted in the Gron rolling disk scenario, are the result of BOTH a relativistic spin rate of the wheel (per axle), in conjunction with a relativistic translation rate between axle and ground observer. There are galaxies that are known to NOT move at appreciable relativistic rates wrt the Milky Way's systemcenterofmass, yet their outer stars travel at rates far exceeding what Newton allowed for. All stars at a given radius from the galaxy's systemcenterofmass move with essentially the same rate. The effects in Gron's analysis, such as dynamic spoke rate and radial curvature, would relate only to galaxies whose centers move at appreciable relativistic rates wrt our Milky Way's center. But the galactic rotation curve problem exists even for galaxies that are not of relativistic translation, and stars n a given orbital circumference do not speed up or slow down depending on where they are on their circumference (in the manner that Gron's spoketips do in his rolling wheel analysis). Therefore, and while relativistic effects always apply, I see the galactic rotation curve problem is independent of relativistic effects.
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 04022014 at 04:09 AM.
The point I was trying to make is that my blue line is shorter than my red line by a factor of v=0.866, just like the 86.6% you used in your solution. I didn't really mean to shift the discussion back to Gron's figure C being offkilter. I don't think there is any need to point that out to him at this point, unless you feel that we should.
That's exactly what I was wondering. Thank you!
Here are SYA's data points sorted into chronological order:
But none of those emission events happen at the same place on the wheel, so we cannot see the translational velocity of the wheel from that data alone. If SYA would provide the axle's x coordinate for each of the above events, we could see the translational velocity of the wheel as a linear relationship between the axle's x coordinate and the t coordinates given above.Code:N x y t 8 4.735 1.463 4.957 7 4.615 1.695 4.914 9 4.655 1.218 4.812 6 4.316 1.877 4.706 10 4.413 0.971 4.519 5 3.769 1.988 4.261 11 4.026 0.741 4.091 12 3.535 0.525 3.574 4 2.954 1.970 3.551 13 2.941 0.344 2.960 3 1.877 1.737 2.558 14 2.272 0.196 2.280 15 1.546 0.088 1.548 2 0.600 1.115 1.266 16 0.782 0.022 0.783 1 0.000 0.000 0.000
EDIT: This post has been superceded by the following subsequent post ...
post link > Relativistic Rolling Wheel II
as per JTyesthatJT review, which lead to a need to correct unnoticed truncations (to 3 decimal places) during calculation. The reference above extends the calculations to 6 decimal places, the original intent.
************************************************** ************
Laurieag,
In relation to my Gron figure 9 post here ... Relativistic Rolling Wheel II ...
For the relativistically rolling wheel, with axle at v=0.866025c in the direction of increasing x, rolling toward the reference event (where wonder camera snaps its wonder photo) at the colocated origins at x,y,z = x',y',t' = 0,0,0.
EDIT: In the 2 figures below, the column header "Axle Range wrt Origin" should "Axle Velocity", which is the average axle velocity between the current event and its arrival at the colocated origin.
I. Here are the emission events sorted in order by "spoke number" ...
* ........ v = x_{A}/t_{A} = 0/0 ... which is undefined at that time. However by definition, the velocity is v=0.866c at all times.
** ..... x_{A} : √(x_{SP_N}^{2}+y_{SP_N}^{2}) = 0/0 ... which is undefined "at that time", although the ratio forever remains 0.866.
*** ... gamma = γ = 1/√(1v²/c²) = 1/√(1(undefined)²/c²) ... which is then undefined at that time. By definition, γ = 2 always since v = 0.866c inertial.
II. Here are the emission events sorted in order by "firing time" ...
* ........ v = x_{A}/t_{A} = 0/0 ... which is undefined at that time. However by definition, the velocity is v=0.866c at all times.
** ..... x_{A} : √(x_{SP_N}^{2}+y_{SP_N}^{2}) = 0/0 ... which is undefined "at that time", although the ratio forever remains 0.866.
*** ... gamma = γ = 1/√(1v²/c²) = 1/√(1(undefined)²/c²) ... which is then undefined at that time. By definition, γ = 2 always since v = 0.866c inertial.
Again, my coordinates are off eversoslightly because I linearly extrapolated the location of the firing event from between the 2 closest events in my spreadsheet solns ... since I calculate solns for only every 1 degree of rotation (over a full rotation). The relation is not linear though, so my final solns are off a wee bit, but not worth mentioning far as the size of the figure goes. Note that the gamma values are accurate to within 0.8% (ie 8/1000 wrt spoke 8), so less than 1% error there. The spreadsheet could be modified to calculate (say) each 1/100th a degree of rotation, if ever desired.
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 04032014 at 03:12 AM.
Thanks, SYA. Did you mean to change the time for N=8 from t=4.957 to t=4.963, or was that a typo?
The calculated speed varies a bit from v=0.866c. That's to be expected with any method that involves approximation, but I would have thought your error margin would have been a little less than that, considering your large data base.Code:N x_axle t_axle v_calculated 8 4.298 4.963 7 4.256 4.914 0.8571 9 4.167 4.812 0.8725 6 4.076 4.706 0.8584 10 3.913 4.519 0.8716 5 3.690 4.261 0.8643 11 3.543 4.091 0.8647 12 3.095 3.574 0.8665 4 3.075 3.551 0.8695 13 2.563 2.960 0.8663 3 2.215 2.558 0.8656 14 1.975 2.280 0.8633 15 1.341 1.548 0.8661 2 1.097 1.266 0.8652 16 0.677 0.783 0.8695 1 0.000 0.000 0.8646
I'll recheck that.
EDIT: Indeed, an accidental TypeO there. Good eye, and many thanks. I corrected the prior post with the spreadsheet image. The numbers are as they were prior.
Yes, please recheck your velocity calculations there. My spreadsheet calculates the numbers I posted in the prior post's spreadsheet image. The velocity computation for the axle at any spoke emission point is the same, at 0.866c, except spoke 8 which is 0.867152c, rounded to 0.867c.Originally Posted by JTyesthatJT
Thank You,
SinceYouAsked
Okay, I'll try that:
Actually it just makes that first velocity calculation worse, as (4.2984.256) / (4.9574.914) = 0.042/0.043 = 0.9767cCode:N x_axle t_axle v_calculated 8 4.298 4.957 7 4.256 4.914 0.9767 9 4.167 4.812 0.8725 6 4.076 4.706 0.8584 10 3.913 4.519 0.8716 5 3.690 4.261 0.8643 11 3.543 4.091 0.8647 12 3.095 3.574 0.8665 4 3.075 3.551 0.8695 13 2.563 2.960 0.8663 3 2.215 2.558 0.8656 14 1.975 2.280 0.8633 15 1.341 1.548 0.8661 2 1.097 1.266 0.8652 16 0.677 0.783 0.8695 1 0.000 0.000 0.8646
If I understand your 0.866 calculations correctly, those actually represent the ratio of the axle'shorizrange to the spoketip'srange. By design, your method ensures this will be 0.866, or at least very close.
However, what I am calculating above is dx/dt for the axle itself. For example, between the #6 emission and the #10 emission, I get this:
(4.0763.913) / (4.7064.519) = 0.163/0.187 = 0.8716c
It is still close, and the average dx/dt over all the data is 0.8670c. It is just not as close as we would have thought for each incremental dx/dt. I suppose it has to do with the small details of what you are doing, especially the interpolation.
OK,
I had some horrendous PC problems during my later posts here. I will need to recheck all my solns, make sure the one you just found was not the only inadvertent change. I'll let you know soon.
EDIT: I have good news. I noticed that a procedure I had (between sheets) was truncating to 3 decimal places (I did not realize it), while the LT soln "related decimal places" actually went out much further. The truncations were producing all the errors, which are highest around the spoketip 8 emission event. I redid all the solns while maintaining a 6 decimal place accuracy throughout, and all the errors virtually vanished. I'll repost the data shortly. Thank you for the review JT!
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 04022014 at 08:26 AM.
NOTE: This post supercedes the prior one here ...
post link > http://www.thephysicsforum.com/speci...html#post12695
... in which the calculations were correct, but the results off slightly due to truncation to 3 decimal places during calculation. The prior post was redone here to 6 decimal place accuracy, and the axle velocity EVENTTOEVENT remains virtually fixed at 0.8660c.
************************
Laurieag,
In relation to my Gron figure 9 post here ... see link > Relativistic Rolling Wheel II ...
For the relativistically rolling wheel, with axle at v=0.866025c in the direction of increasing x, rolling toward the reference event (where wonder camera snaps its wonder photo) at the colocated origins at x,y,z = x',y',t' = 0,0,0.
EDIT: The 2 rightmost columns should read ... 1,000,000th resolution, as the calculations below were run to 6 decimal points, versus the prior posted version which had calculations unfortunately truncated to 3 decimal places.
I. Here are the emission events sorted in order by "spoke number" ...
* ......... v = x_{A}/t_{A} = 0/0 ... which is undefined at that time. However by definition, the velocity is v=0.866c at all times.
** ....... x_{A} : √(x_{SP_N}^{2}+y_{SP_N}^{2}) = 0/0 ... which is undefined "at that time", although the ratio forever remains 0.866.
*** ..... gamma = γ = 1/√(1v²/c²) = 1/√(1(undefined)²/c²) ... which is then undefined at that time. By definition, γ = 2 always since v = 0.866025c inertial.
**** ... The "axle velocity is between events" is calculated between the prior and current events. The "axle velocity wrt origin" is calculated as average velocity from event to origin.
II. Here are the emission events sorted in order by "firing time" ...
* ......... v = x_{A}/t_{A} = 0/0 ... which is undefined at that time. However by definition, the velocity is v=0.866c at all times.
** ....... x_{A} : √(x_{SP_N}^{2}+y_{SP_N}^{2}) = 0/0 ... which is undefined "at that time", although the ratio forever remains 0.866.
*** ..... gamma = γ = 1/√(1v²/c²) = 1/√(1(undefined)²/c²) ... which is then undefined at that time. By definition, γ = 2 always since v = 0.866025c inertial.
**** ... The "axle velocity is between events" is calculated between the prior and current events. The "axle velocity wrt origin" is calculated as average velocity from event to origin.
Again, my coordinates are off eversoslightly because I linearly extrapolated the location of the firing event from between the 2 closest events in my spreadsheet solns ... since I calculate solns for only every 1 degree of rotation (over a full rotation). The relation is not linear though, so my final solns are off a wee bit, but not worth mentioning far as the size of the figure goes.
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 04032014 at 03:19 AM.
Indeed. It was due to an unbeknownst truncation during calculations from 6 decimal places to 3. That's what did it
Those were instantaneous axle'shorizrange to the spoketip'srange, not the velocity, however it should equal the translation velocity at all times, in a perfect calculation. Indeed, I cherry picked the LTs solns for when ... axle'shorizrange / spoketip'srange = 0.866025. However, the "axle velocity wrt orgin" column in my data was not related to spoketip locations, but rather direct from the inverse LTs from axle frame to ground frame, which should also always equal 0.866025c in our case.
Yes, even at 3 decimal places, they were loosely close to expected values. The 0.9767c, I had a problem with, which led me to the decimalplacetruncation deficiency. Thanx JTyesthatJT for the verification! Looks much better now. And many thanx to Laurieag, as well !
Thank You,
SinceYouAsked
I figure we got something accomplished wrt Gron's model, because we did not have to instead debate relativity theory with cincirob for weeks I wonder if he's studying, and maybe we'll see him back in town eventually, armed and ready with the right tools? Wouldn't that be somethin?
Thank You,
SinceYouAsked
And so, we have effectively applied the TerrellPenrose analysis to Erhenfest's spinning disk, as Gron did. When opportunity arises, I'll see if I can figure out the velocity that produces Gron's figure 9 part C, just so we know what might have happened there.
Thank You,
SinceYouAsked
My connectivity with (only) this forum has dropped to crawl, or standstill. It's been virtually impossible to post for a couple days now, although occasionally it seems to work as normal. I figure the site is making some system changes, or such. As a verification, is anyone else having problems as such?
Thank You,
SinceYouAsked
I haven't noticed any problems. However, I remember cincirob said he was having a hard time using the other relativistic rolling wheel thread, (which is why he created this one). When he disappeared, I wondered if perhaps he was having trouble with the new thread as well, and gave up out of frustration.
QUICK EDIT:
This is just a test of the quick edit feature.
ADVANCED EDIT:
This is just a test of the advanced edit feature.
Last edited by JTyesthatJT; 04032014 at 04:26 AM.
The problem seems to arise if I go into a post for edit. It then runs like a pig, dragging me to my knees. No matter what I do after that, it runs slow. After hours of that, it started running great, at top speed like usual, about a half an hour ago. I went into a post for EDIT, just to see (based on suspicion), and it allowed a quick edit if I do the QUICK EDIT feature. If I select ADVANCED EDIT, it then takes forever to load the page, drags my connection with www.thephysicsforum.com to its knees from there out ... possibly for hours. Every other web page works fine and fast though. I'm just trying to determine if I have a problem with my PC, versus maybe a forum's designed programming that slows one down once too many EDITs are made (maybe)? This does sound similar to what cincirob had experienced prior.
Thank You,
SinceYouAsked
Just to keep this thread on topic, I would still like to build a spreadsheet for the relativistic rolling wheel based on Excel's builtin iteration feature. I think it should be possible to instantly Lorentztransform the coordinates of any number of spoke tips for any velocity. Then I think it should also be possible to instantly find the resulting TerrellPenrose coordinates. I don't have time right now, but I'd like to do that eventually. When and if I ever do, I will post a link to the spreadsheet.
JTyesthatJT,
I eagerly await.
BTW, I never posted them, but here are the inverse LT transformed coordinates for the spoketips of the contracted ellipse at t=0 ...
N x y t
1 0.000 0.000 0.000
2 0.431 0.493 0.000
3 0.500 0.964 0.000
4 0.474 1.314 0.000
5 0.409 1.576 0.000
6 0.321 1.767 0.000
7 0.220 1.898 0.000
8 0.111 1.975 0.000
9 0.000 2.000 0.000
10 0.111 1.975 0.000
11 0.220 1.898 0.000
12 0.321 1.767 0.000
13 0.409 1.576 0.000
14 0.474 1.314 0.000
15 0.500 0.964 0.000
16 0.431 0.493 0.000
For reference ...
Emission event coordinates for figure 9 part C …
Relativistic Rolling Wheel II
Gron’s figure 9 for v = 0.866025c …
Relativistic Rolling Wheel II
Thank You,
SinceYouAsked
Hey, one day, maybe we might animate sequential camera snapshots per ground, instead of animating how the rolling wheel always presently exists in spacetime. I'm not certain, but I don't think anyone has done that before on the web yet, to my knowledge. Basically, predict how a rollingwheelvideo would look per a ground observer at the t=t'=0 event.
Thank You,
SinceYouAsked
Last edited by SinceYouAsked; 04032014 at 11:54 PM.
I've already given up on the iteration method. It doesn't do what I'd hoped it would.
Thank you, too! Oh, and cincirob would surely argue with the results.
You could just copy your C diagram a horizontal distance of 2piRgamma/N, and then shift all of your spoke numbers by one digit. Repeat for however many frames you want in your animation. However, that method does not really show how the points would move between each pair of frames.
Alright, here is a new spreadsheet!
http://www.filedropper.com/macrowheelcalc_2
This one uses two different macros. One converts points to the Ellipse (ctrl+E) like the part B diagram, and the other converts points to the oblong (ctrl+Q) like the part C diagram. The macros only apply to the top line of the spreadsheet, so once I got a result I wanted, I just copied it down to a lower place on the spreadsheet. The end result is 16 points on the ellipse, and 16 points on the oblong. Everything seems to match the data posted by SYA. Please note that my x axis runs through the axle, so the camera is located at y=R'=1 and all of my y coordinates are less than SYA's by one radius R'=1.
Basically the only cells you should change are the ones underlined in yellow, namely, v, R, and W. W is a wheel point measured in degrees from the 6:00 o'clock point on the wheel. The R value should let you choose inner points of the wheel, rather then edge points, but I haven't tested that yet. I hope you enjoy it, and please do let me know if you find any thing wrong with it. Thanks!
EDIT: When I tested the R feature, I found that I had to change something to make it work properly. So the link above has been updated to "macrowheelcalc_2" which is the most current and correct version.
JTyesthatJT,
Looks good, however it seems you may have accidentally pasted atop the formula in cell J2 for t', with a calculated t' result (of near zero). Yes?
EDIT: Ahh huh. I see. Cool how the macros (using the control key) cranks it out like that. Most excellent ! It may be good to add in the formula for t' in the header cell though.
Thank You,
SinceYouAsked
Thank you, SYA! I should have explained that the macros advance the value of the J2 cell until a desired outcome is reached. The macros will stop on their own, but sometimes they have to run for awhile. For the oblong macro, I start with t'=5 which means it takes a long time to reach something like t'=1. I could probably change it so that is starts at t'=2 or something, , but for speeds very close to c, it might need values closer to t'=5. I haven't checked that yet. EDIT: Now I'm testing starting with t'=2 and so far I haven't found any problems. This makes it much faster.
This method was inspired by your method. It basically does the same thing you did, except it is more automated. You don't have to scroll through lots of rows looking for certain things, because it all happens automatically in the top row.
But t' is not calculated by any formula, it just advances in small increments.
OK, I’ve been away for a while. Death in the family and a vacation. I got a chance to review JT’s numbers and they are correct. They are close to the solution for a small flat spot at the contact area. Because the solution for spoke tip location in the road frame is the location that the point causes the spoke tip to arrive at the some point on the road that it was in the wheel frame. Because of the time differences for the tip locations I don’t believe they exactly match what a flat contact are would be but they are close and are exact as the contact area approached zero length. I was mistaken that it would be significantly different.
One thing that did become clear as I went through this thought process is that the curved spoke calculations are not shape dependent; that is, they do not define the shape nor do they require any particular shape in the road frame. Any solution would require that as the wheel rolls, the same set of points on the rim contact exactly the same set of points on the road during each revolution.
Interestingly, the Gron wheel fails this criterion during acceleration. For instance, if one rolled a wheel at arbitrarily slow speed, one would match up a certain set of wheel rim points with a certain set of road points. But, taking that wheel to .866c, on would find that this onetoone match no longer exists. In most universes, we would call this skidding, not rolling. However if the speed is held constant, at .866c or any other, then the same rim and road points will match as long as the speed is constant. Either building the wheel at speed of stretching the material must be invoked. A rack and pinion solution fails under these conditions.
Another structure that works as the Gron solution, is a static round structure with the points moving around the structure. It is at least strange that a static structure produces the same spoke tip solutions as one that is rotating. The ideas in this paragraph and the one above are what bother me about the Gron solution.
Another wheel that one might consider consists of a strong, arbitrarily thin rim around a foam rubber like disk; or the same rim with elastic spokes. This wheel would allow the necessary relativistic contractions caused by the contraction of the rim. And, since the contraction of the rim in the wheel frame would be in the same proportion of the road contraction in the wheel frame, the point for point match of road and rim would be maintained at all speeds. Further, the rack and pinion would also work.
Perhaps the most interesting thing about this wheel is that it predicts roll out to be 2piR per revolution where R is the radius of the wheel at rest even though the wheel radius at speed is <R at speed. Accordingly, the angular velocity of the wheel w = v/R, again, where R is the rest radius of the wheel and not the contracted radius. And, w increases with speed. The Gron wheel analysis predicts that w decreases with speed which is an artifact of the Gron’s model. One might wonder if the model is too far from reality since it predicts results contrary to a wheel where relativity is allowed to work normally. Again, the static structure with points moving around it is a correct interpretation of Gron’s analysis because it is effectively a clock and would slow down in the view of the road observer and the elliptical shape would be correct.
Spoke tip locations in the road frame for the rimandelastic center would be calculated using ROS just a Gron does. As noted above, the spoke tip locations do not demand any particular shape so the question of determining the shape of the wheel in the road frame is still a question. The ellipse shape bothers me because it is the answer one gets for a nonrotating object. While I can’t make a definitive argument against the ellipse, I will point out that the shape of the wheel is the wheel frame is determined by placing the observer at the center of rotation of the object. It seems to me that rotating objects should always be observed from the rotational center. In the road this would produce a nonelliptical shape. JT has criticized this approach in the past arguing that such a wheel interferes with a close fitting fender. And that is true if one assumes the Gron wheel; however, the rimandelasticcenter wheel may not have the problem because the wheel shrinks and is only close fitting when the wheel is at rest.
For those who subscribe to the Gron analysis I would point out the decreasing speed with increasing velocity. The analysis is identical to a nonrotating clock face where the tips of the hands of the clock match the translational velocity. Rotation of the object itself is not considered in the determining the shape of the wheel in the road frame or in the wheel frame since relativistic contraction is not allowed.
cinci
Very sorry to hear that cinci.
Well, JT will be very happy to hear you have validated his numbers. Good work.
Regarding "flatten wheel at contact area", the question is as to why anyone would want to consider that in the first place?
A radial element (or spoke) that buckles up to support a flattened contact area (as in the case of a rubber tire) would for obvious reasons never exactly match the solns of a spoked wheel that does not flatten at the ground. Nor, would anyone expect those solns to be exactly the same. But the lesser the flattening, the more the scenario tends towards Gron's scenario. It's good that you have run your numbers and obtained that conclusion.
The linear spokes of the axle system are transformed by the LTs to the ground system. The curvature of the spokes as they exist in the ground system, are the result of spoke rotation and the relativistic effects caused by the steady translation between axle and ground. The only thing that is required, is that one accepts the results of the LTs. The LTs require that rotating linear spokes per the axle POV curve per the ground POV.
Incorrect. For classical (non relativistic rates), and assuming the atomic bonds are not overcome by the classical forces of friction and centrifugal force, the wheel's perimeter atoms align 1:1 with road atoms as they make contact, as the wheel rolls out 2piR upon the ground. The Gron scenario is not about a road, but rather a spinning disk from a POV of relative translation. However, one can easily imagine a virtual flat road 1 micron beneath Gron's spinning disk, passing by at a translation rate that allows for the closestpointsofapproach to be momentarily at rest with one another. Gron's scenario assumes a wheel "simply exists at a relativistic rotation rate", and we ask not how or why (ie, its a wonder wheel). Rotating cincumferences measured at 2piR per axle must exist at a proper length of gamma*2piR per the rotating wheel itself. Given such, the wheel must roll out gamma*2piR upon the ground, since wheel and ground (wonder) atoms are at momentary relative rest upon contact, and again (just as in the nonrelativistic roll rate case) there must exist a 1:1 atomic alignment between wheelperimeteratoms and road atoms upon contact as it rolls. The wheel rolls out a length of gamma*2piR. It's this simple ... there are gamma times more atoms along any circumference for a relativistic rate of rotation than a nonrelativistic rate of rotation. While the Gron scenario is an impossible one, it does allow the exploration of SR's relativistic effects under the case of relativisticrotationrates.
EDIT: Ahh, by "taking the wheel to 0.866" I now see you meant "rolling up the wheel's rotation rate to a translation velocity of 0.866c. In that case yes, the wheel slips and rack and pinion fails during rollup to relativistic rates. This is to be expected though, per the relativity. Considering the round wheel (per axle) at a higher steady relativistic rate of 0.866c (per ground), my above statement applies.
Yes, I would say the perimeter atoms or your rotating ringofatoms are transformed to the same solns as Gron's perimeter, given the same perimeter rotation rate. However your spokes would not be curved, they would be linear. There's nothing strange here, however it must be kept in mind that it's an impossible scenario in any reality, just as Gron's is. These are thought experiments using wonder wheels and/or wonder wheel building procedures, to explore the kinematics of the theory under relativistic rotation rates.
What cincirob should consider, is how relativity handles the Gron scenario. Do that, and you will never ask about foam disks, spinning perimeters upon static wheels, marble races, wheels with flattened patch areas at contact, rack and pinion, etc. None of these are possible in the real world, given relativistic rotation rates (rolling or not). Consider the case of Gron, where a wonder disk spins at relativistic rate, and an inertial passerby gazes upon it. Show how the LTs can determine what the passerby beholds ... support the scenario, kinematically, which is what Gron did.
Again, no real wheel can exist at a relativisitic rotation rate. However, kinematically using wonder wheels, the proper cirumferences increase by the gamma factor during the rollup. While the axle's frame holds the wheel forever at 2piR, the wheel itself and the ground hold it at gamma*2piR. The ground observer determines that "by the LTs", or "by the summation of atomic length's upon contact for a full rotation" upon the ground. If the rotation rate is defined wrt full rotations of the wheel, the rotation rate (omega) must slow down with higher roll rate, given there are (gamma) more atoms of a rotating wheelperimeter to roll out. There "are" more atoms, since the wheel was rolled up to higher rate. The more perimeter atoms, the longer it takes a full rotation to complete.
That question was answered a very long time ago. The shape of the wheel, is determined by transforming the locations of wheelperimeteratoms, and considering those transformations for only an instant of ground time t. There is no secret or mystery here.
The wheel is length contracted per anyone who moves wrt the rolling wonderwheel, no matter what their rate and no matter where they are relatively located. For a roll rate of v=0.866025 (per ground), the wheel is a 50% lengthcontractedellipse per anyone of the ground frame, no matter where they are. If a fender cutout hugs the rolling wheel, the wheel is obviously contracted just as the fender cutout is. No rotating wheel atom ever reaches the contracted perimeter of the fender cutout, because no atom of the rotating circumference (per axle) ever departs that circumference (per axle). A stationary round tape measure of the axle frame always overlays the very same rotating wheel circumference (per axle).
See above responses. Well, so much for cincirob's absence being related to the study of relativity.
I must say it again ... We use wonder wheels, because no real wheel can withstand the classical centrifugal forces at relativistic rotation rates. We are exploring kinematics here.
Thank you,
SinceYouAsked
Last edited by SinceYouAsked; 04082014 at 05:58 AM.
Welcome back, cinci.
I'm very sorry for your loss.
Thank you for thinking this through and admitting that you were mistaken on that.
In order for acceleration to be included, I would have to abandon the idea of building the rim "at speed", because that speed would be constantly changing. One way around this is to simply connect each pair of adjacent spoke tips with an elastic band. This forms a rim that is free to stretch. Now I can accelerate the rolling speed from 0.000c to 0.866c without the rim breaking apart due to relativistic contraction. At all times, the velocity of the road is the same as the velocity of the contact point, so there is no skidding. A rack and pinion would still fail in this case, though, so you are correct about that.
Your points are moving inside the static structure, so the points themselves are nonstatic. That is why they produce the same spoke tip solution as the nonstatic spokes. If you want to see a static solution, try leaving all your points at fixed locations inside the static structure. This produces a spoke tip solution that is different from Gron's.
Yes, that is a good approach as well. I agree that the rack and pinion would work in that case.
That is the same thing as saying that the wheel rolls out gamma*2piR where R is the wheel radius at speed.
No, the angular velocity of that wheel in the axle frame would be w=v/R where R is the wheel radius at speed. This is just a rotating circle. It does not affect the angular velocity that the circle had a larger radius when it was at rest.
Even if you meant the road frame, it would not be correct to say that the angular velocity of that wheel would be w=v/R where R is the rest radius of the wheel and not the contracted radius.
The Gron model does not predict that w decreases with speed. In the axle frame, w always increases with speed, everywhere on the wheel. In the road frame, at the contact point, the angular velocity always increases with speed. The only places where the Gron model predicts that w can decrease with speed in the road frame is for points which are not at the contact point. The most obvious such point is at the 12:00 o'clock point. But that point is not in contact with the road, so there is still no skidding happening.
The wheel is similar to a clock, except the angular velocity of the wheel is not constant in the axle frame, as it would be for a clock. The wheel's angular velocity increases linearly with v in the axle frame, whereas a clock would remain at a fixed angular velocity independent of v. Other than that, it is similar to a clock in that it becomes timedilated according to the road frame.
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The ellipse shape is also the answer one gets for points rotating inside a static circular object, as you pointed out above. But as I pointed out above, if the points are not rotating, the spoke tip arrangement is different from Gron's.
That is what I have been calling the axle frame, but it is a nonrotating frame. The wheel frame rotates and is therefore not inertial. That is why it would not be a good frame to choose for the simplest analysis.
Wait, what? What type of wheel are you talking about which does not produce an elliptical shape in the road frame???
Oh, the rimandelasticcenter wheel is still an ellipse. It is identical to the Gron wheel, except the radius is smaller by a factor of gamma, the rim does not have to be built at speed, and the rim does not have to stretch. But everything else is the same.
For the rimandelasticcenter wheel, the closefitting fender must be installed at speed. And if that fender is a circle in the axle frame, then it will be an ellipse in the road frame, and the wheel must not interfere with it.
Which have all been addressed above.
The rotation is considered. If it were not, then the spoke tips would be the same distance apart at the bottom of the wheel as they are at the top. The spoke spacing at the bottom would not be compatible with the wheel rolling out gamma*2piR. The angular velocity in the road frame would not be lower at the 12:00 o'clock point than it is at the contact point.
Relativistic contraction is allowed to happen. If it were not, then the rim would not have to be built at speed, or allowed to stretch. Or the spokes would have to be allowed to compress as with the rimandelasticcenter wheel. If relativistic contraction was not allowed to happen, we could just start with a solid nonrotating disk and then roll it up to relativistic speed without any concern for the length contraction of the rim. You know, like you do in your model.
cinci: I got a chance to review JT’s numbers and they are correct. They are close to the solution for a small flat spot at the contact area. Because the solution for spoke tip location in the road frame is the location that the point causes the spoke tip to arrive at the some point on the road that it was in the wheel frame. Because of the time differences for the tip locations I don’t believe they exactly match what a flat contact are would be but they are close and are exact as the contact area approached zero length. I was mistaken that it would be significantly different.
SYA: Well, JT will be very happy to hear you have validated his numbers. Good work.
Regarding "flatten wheel at contact area", the question is as to why anyone would want to consider that in the first place?
cinci: Apparently you have never looked at the wheels on your car. There’s always a flat spot. Even on steel train wheels there’s a flat spot.
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cinci: One thing that did become clear as I went through this thought process is that the curved spoke calculations are not shape dependent; that is, they do not define the shape nor do they require any particular shape in the road frame. Any solution would require that as the wheel rolls, the same set of points on the rim contact exactly the same set of points on the road during each revolution.
SYA: The linear spokes of the axle system are transformed by the LTs to the ground system. The curvature of the spokes as they exist in the ground system, are the result of spoke rotation and the relativistic effects caused by the steady translation between axle and ground. The only thing that is required, is that one accepts the results of the LTs. The LTs require that rotating linear spokes per the axle POV curve per the ground POV.
cinci: Yes, we all know the LTs are used because we’ve all done the calculations and I assume you’re mentioning it in an attempt to critique the use of the contraction formula. But the shape of the wheel is not determined by the spoke curvature as I say above. Let me explain that to you. Since Gron’s wheel skids on the way up to speed, let’s let his wheel skid all the time; for instance, consider a case where the velocity along the road is not the same as the tangential speed of the spokes. Say v = .4c and v(tip) = .8c. If you can figure out how this works, you’ll find that the wheel shape in the road can be found by contracting all the chords by (1  .4^2)^.5. The spokes will be curved even more than they are in the Gron solution. And while I’d like to see you fumble your way through this, I’ll save you the trouble by pointing out that a wheel skidding at .4c with no rotation will also have the same shape.
If you think the shape is determined by anything other than length contraction, you are kidding yourself. And if you think length contraction is different than using the Lt, you don’t know relativity.
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[I}cinci: Interestingly, the Gron wheel fails this criterion during acceleration. {onetoone correspondence of wheel and road “atoms”} For instance, if one rolled a wheel at arbitrarily slow speed, one would match up a certain set of wheel rim points with a certain set of road points. But, taking that wheel to .866c, on would find that this onetoone match no longer exists. In most universes, we would call this skidding, not rolling. However if the speed is held constant, at .866c or any other, then the same rim and road points will match as long as the speed is constant. Either building the wheel at speed of stretching the material must be invoked. A rack and pinion solution fails under these conditions.
SYA: Incorrect. For classical (non relativistic rates), and assuming the atomic bonds are not overcome by the classical forces of friction and centrifugal force, the wheel's perimeter atoms align 1:1 with road atoms as they make contact, as the wheel rolls out 2piR upon the ground. The Gron scenario is not about a road, but rather a spinning disk from a POV of relative translation.
cinci: Oh dear boy, you just got through explain rollout to me and now you say Gron’s analysis isn’t about the road. What I said above is quite correct and you know it.
Your two principle arguments now are “these wheels can’t be real” and “Oh, it isn’t about that”.
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SYA: However, one can easily imagine a virtual flat road 1 micron beneath Gron's spinning disk, passing by at a translation rate that allows for the closestpointsofapproach to be momentarily at rest with one another.
cinci: Well yes, it’s hard to have a rolling wheel without something to roll on.
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SYA: Gron's scenario assumes a wheel "simply exists at a relativistic rotation rate", and we ask not how or why (ie, its a wonder wheel).
cinci: Scientists always ask how or why and “wonder wheel isn’t the only solution as I have pointed out.
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SYA: Rotating cincumferences measured at 2piR per axle must exist at a proper length of gamma*2piR per the rotating wheel itself. Given such, the wheel must roll out gamma*2piR upon the ground, since wheel and ground (wonder) atoms are at momentary relative rest upon contact, and again (just as in the nonrelativistic roll rate case) there must exist a 1:1 atomic alignment between wheelperimeteratoms and [Uroad[/U] atoms upon contact as it rolls. The wheel [U]rolls out a length[U] of gamma*2piR. It's this simple ... there are gamma times more atoms along any circumference for a relativistic rate of rotation than a nonrelativistic rate of rotation.
cinci: After saying the Gron solution isn’t about the road you mentioned road or ground or rollout 5 times.
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SYA: While the Gron scenario is an impossible one, it does allow the exploration of SR's relativistic effects under the case of relativisticrotationrates.
cinci: The wheel I suggested explores all the same effects and doesn’t deny the phemomenon of contraction in the wheel frame.
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cinci: Another structure that works as the Gron solution, is a static round structure with the points moving around the structure. It is at least strange that a static structure produces the same spoke tip solutions as one that is rotating. The ideas in this paragraph and the one above are what bother me about the Gron solution.
SYA: Yes, I would say the perimeter atoms or your rotating ringofatoms are transformed to the same solns as Gron's perimeter, given the same perimeter rotation rate. However your spokes would not be curved, they would be linear.
There's nothing strange here, however it must be kept in mind that it's an impossible scenario in any reality, just as Gron's is. These are thought experiments using wonder wheels and/or wonder wheel building procedures, to explore the kinematics of the theory under relativistic rotation rates.
cinci: Incorrect. First, there aren’t any spokes but if you built concentric rings and positioned points that would coincide with the spokes, you get the same curvature. Apparently after all JT’s help, you still don’t understand why the curvature occurs.
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cinci: Another wheel that one might consider consists of a strong, arbitrarily thin rim around a foam rubber like disk; or the same rim with elastic spokes. …………
SYA: What cincirob should consider, is how relativity handles the Gron scenario. Do that, and you will never ask about foam disks, spinning perimeters upon static wheels, marble races, wheels with flattened patch areas at contact, rack and pinion, etc. None of these are possible in the real world, given relativistic rotation rates (rolling or not). Consider the case of Gron, where a wonder disk spins at relativistic rate, and an inertial passerby gazes upon it. Show how the LTs can determine what the passerby beholds ... support the scenario, kinematically, which is what Gron did.
cinci: Actually the disk I described is quite possible and will work just as I described; so it the marble race. Both are much more possible than the Gron wheel and will even work at low speeds. Gron’s wheel denies relativistic phenomena in the wheel frame so it doesn’t work at any speed. The “spinning perimeters upon static wheel” is your invention so you explain it.
It’s surprising that you mention the rack and pinion since the gron solution completely fails there. Oh, and it is possible to build a rack and pinion.
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cinci: Perhaps the most interesting thing about this wheel is that it predicts roll out to be 2piR per revolution where R is the radius of the wheel at rest even though the wheel radius at speed is <R at speed. Accordingly, the angular velocity of the wheel w = v/R, again, where R is the rest radius of the wheel and not the contracted radius. And, w increases with speed. The Gron wheel analysis predicts that w decreases with speed which is an artifact of the Gron’s model. One might wonder if the model is too far from reality since it predicts results contrary to a wheel where relativity is allowed to work normally. Again, the static structure with points moving around it is a correct interpretation of Gron’s analysis because it is effectively a clock and would slow down in the view of the road observer and the elliptical shape would be correct.
SYA: Again, no real wheel can exist at a relativisitic rotation rate. However, kinematically using wonder wheels, the proper cirumferences increase by the gamma factor during the rollup.
cinci: Yes, they do. But that is magic and I’d rather deal with science.
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SYA: While the axle's frame holds the wheel forever at 2piR, the wheel itself and the ground hold it at gamma*2piR. The ground observer determines that "by the LTs", or "by the summation of atomic length's upon contact for a full rotation"
upon the ground. The rotation rate (omega) must slow down with higher roll rate, given there are more atoms of a rotating wheelperimeter to roll out. There "are" more atoms, since the wheel was rolled up to higher rate. The more perimeter atoms, the longer it takes a full rotation to complete.
cinci: Yes, it’s that “holding the wheel forever at 2piR” that screws up things like having angular velocity increase with speed and rack and pinions not working.
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cinci: Spoke tip locations in the road frame for the rimandelastic center would be calculated using ROS just a Gron does. As noted above, the spoke tip locations do not demand any particular shape so the question of determining the shape of the wheel in the road frame is still a question.
SYA: That question was answered a very long time ago. The shape of the wheel, is determined by transforming the locations of wheelperimeteratoms, and considering those transformations for only an instant of ground time t. There is no secret or mystery here.
cinci: Mystery? No, but there might be a question.
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cinci: The ellipse shape bothers me because it is the answer one gets for a nonrotating object. While I can’t make a definitive argument against the ellipse, I will point out that the shape of the wheel is the wheel frame is determined by placing the observer at the center of rotation of the object. It seems to me that rotating objects should always be observed from the rotational center. In the road this would produce a nonelliptical shape. JT has criticized this approach in the past arguing that such a wheel interferes with a close fitting fender. And that is true if one assumes the Gron wheel; however, the rimandelasticcenter wheel may not have the problem because the wheel shrinks and is only close fitting when the wheel is at rest.
SYA: The wheel is length contracted per anyone who moves wrt the rolling wonderwheel, no matter what their rate and no matter where they are relatively located. For a roll rate of v=0.866025 (per ground), the wheel is a 50% lengthcontractedellipse per anyone of the ground frame, no matter where they are.
cinci: A length contracted ellipse? Who’d athunk it. You finally woke up.
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SYA: If a fender cutout hugs the rolling wheel, the wheel is obviously contracted just as the fender cutout is. No rotating wheel atom ever reaches the contracted perimeter of the fender cutout, because no atom of the rotating circumference (per axle) ever departs that circumference (per axle). A stationary round tape measure of the axle frame always overlays the very same rotating wheel circumference (per axle).
cinci: Only if you decide to ignore relativity for the wheel in the wheel frame. It’s not necessary to take such a drastic shape. One wonders why anybody considered it a rational thing to do in a relativity study.
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cinci: For those who subscribe to the Gron analysis I would point out the decreasing speed with increasing velocity. The analysis is identical to a nonrotating clock face where the tips of the hands of the clock match the translational velocity.
SYA: Rotation of the object itself is not considered in the determining the shape of the wheel in the road frame or in the wheel frame since relativistic contraction is not allowed.
cinci: Yes, and that’s what makes Gron’s solution fail as a rolling wheel. I don’t have a problem with what he has done except that it has very little in common with a rolling wheel and he, and you, should recognize that when it is plainly pointed out to you. And when better solution is offered.
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SYA: See above responses. Well, so much for cincirob's absence being related to the study of relativity.
cinci: Your responses add nothing of value to the discussion and at least one is wrong. My comments above are spot on.
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SYA: I must say it again ... We use wonder wheels, because no real wheel can withstand the classical centrifugal forces at relativistic rotation rates.
cinci: You’re using the wonder wheel because you don’t have the imagination it takes to get beyond it.
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cincirob,
Your responses continue to show you do not understand the meaning of relativistic effects, let alone how they apply to the case of relativistic rotation. I encourage you to study the related posts made for you, the LT solns for Gron's scenario presented for you, and all the visual aids attached for you, in your 2 threads here. I'm sorry, but I have no interest in discussing your belief system regarding Gron's failure to use length contraction, or rack and pinion, foam wheels, marble races, or wonderous onthefly wheel building procedures. It's a kinematic analysis. Gron's is the simplest. Learn it first. Learn what the relativistic effects mean, for then you will understand nature under that model. Until then, you will eternally ask these same questions, which by the way all stem from lack of understanding. All you have to do is ask, but you're gonna have to start asking the right questions, instead of relentlessly restating the same old ones.
What you need is to learn the meaning of the LTs, because this is where all your confusion stems. You should start with the all inertial scenario, and work your way up to the spinning disk later.
Have a nice day
SinceYouAsked
cincirob,
I'd like to discuss one issue of the Gron analysis, namely, the angular velocity at the contact point. Starting in the road frame, consider two points on the wheel: The axle, where the velocity is v, and the contact point, where the velocity is zero. The distance between those two points is R. Thus the angular velocity at the contact point is w=(v0)/R or w=v/R. Note that this identical to the classical formula, and that w increases as v increases. The wheel is rolling, not skidding.
Also note that since the velocity at the contact point is equal to the velocity of the road, (and all frames agree on this), there is no skidding in any frame. So, if nothing else, you can stop saying that the wheel skids in Gron's analysis. Thanks.
PS:
I agree with SYA that you'd be better off working on an allinertial scenario, but if you must discuss the wheel, then at least you should focus on one issue at a time.
cinci: While I can’t make a definitive argument against the ellipse, I will point out that the shape of the wheel is the wheel frame is determined by placing the observer at the center of rotation of the object. It seems to me that rotating objects should always be observed from the rotational center.
JT: That is what I have been calling the axle frame, but it is a nonrotating frame. The wheel frame rotates and is therefore not inertial. That is why it would not be a good frame to choose for the simplest analysis.
cinci: In the road frame, the contact point is the center of rotation. It makes sense to me that one should work the problem from that point if that’s what makes sense in the wheel frame. Why change strategiy when you change frames?
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cinci: Another structure that works as the Gron solution, is a static round structure with the points moving around the structure. It is at least strange that a static structure produces the same spoke tip solutions as one that is rotating. The ideas in this paragraph and the one above are what bother me about the Gron solution.
JT: Your points are moving inside the static structure, so the points themselves are nonstatic. That is why they produce the same spoke tip solution as the nonstatic spokes. If you want to see a static solution, try leaving all your points at fixed locations inside the static structure. This produces a spoke tip solution that is different from Gron's.
cinci: Why would I use static points? I’m demonstrating that the Gron solution is exact only for a nonrotating structure with points that move. It’s the nonrotating structure that produces the ellipse. I don’t believe that there’s no difference between a rotating and nonrotating structure
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[I]cinci: Perhaps the most interesting thing about this wheel is that it predicts roll out to be 2piR per revolution where R is the radius of the wheel at rest even though the wheel radius at speed is <R at speed.{/I]
JT: That is the same thing as saying that the wheel rolls out gamma*2piR where R is the wheel radius at speed.
cinci: I don’t think so. This wheel always rolls out 2piR because the wheel contact point and the road have no relative velocity. It also means angular velocity increases with speed.
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cinci: Accordingly, the angular velocity of the wheel w = v/R, again, where R is the rest radius of the wheel and not the contracted radius.
JT: No, the angular velocity of that wheel in the axle frame would be w=v/R where R is the wheel radius at speed. This is just a rotating circle. It does not affect the angular velocity that the circle had a larger radius when it was at rest.
Even if you meant the road frame, it would not be correct to say that the angular velocity of that wheel would be w=v/R where R is the rest radius of the wheel and not the contracted radius.
cinci: I did mean the road frame. A revolution of the wheel in the road frame rolls out 2piR, exactly as it would if it were classical; therefore w = v/R where R is the rest dimension. Yes, you can calcuolate a different w by using the contracted R but you never liked to do that before. Either way the angular velocity increases with speed as it should.
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JT: The Gron model does not predict that w decreases with speed.
cinci: When did you change horses? You’ve always said the wheel slows down like a clock. And you were right. As the Gron wheel speed approaches c, the roll out per revolution in the road approaches infinity and you never get a revolution. I was the one who argued that w = v/R gives a different answer.
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cinci: Spoke tip locations in the road frame for the rimandelastic center would be calculated using ROS just a Gron does. As noted above, the spoke tip locations do not demand any particular shape so the question of determining the shape of the wheel in the road frame is still a question. The ellipse shape bothers me because it is the answer one gets for a nonrotating object.
JT: The ellipse shape is also the answer one gets for points rotating inside a static circular object, as you pointed out above. But as I pointed out above, if the points are not rotating, the spoke tip arrangement is different from Gron's.
cinci: And if my uncle’s name was Mary he would probably be my aunt. Why on earth would you bring up nonrotating points?
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cinci: While I can’t make a definitive argument against the ellipse, I will point out that the shape of the wheel in the wheel frame is determined by placing the observer at the center of rotation of the object. It seems to me that rotating objects should always be observed from the rotational center.
JT: That is what I have been calling the axle frame, but it is a nonrotating frame. The wheel frame rotates and is therefore not inertial. That is why it would not be a good frame to choose for the simplest analysis.
cinci: The easiest way to analyze a rotating object is the use polar coordinates and I think that’s what Gron did. To describe the rotation of a spinning wheel with rectangular coordinates, you would really have to define an infinite number of coordinate systems, each at a different angle and assess small portion of the wheel. Then you could put together the story of a rim that wants to shrink, etc. But if you really did that, wouldn’t you have to transform each of the infinite number of frames to the road frame? And you don’t do that, you picked one.
It’s only simple in polar coordinates. Why wouldn’t you use them in both frames?
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cinci: In the road this would produce a nonelliptical shape.
JT: Wait, what? What type of wheel are you talking about which does not produce an elliptical shape in the road frame???
cinci: Any of them. As I have pointed out with two different examples (the clock and the marble race), Gron’s analysis is not for a wheel but for a static structure.
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JT: For the rimandelasticcenter wheel, the closefitting fender must be installed at speed. And if that fender is a circle in the axle frame, then it will be an ellipse in the road frame, and the wheel must not interfere with it.
cinci: You must be kidding. Why would anyone build the fender at speed? I’m not going to use a more rational wheel model than Gron and then do something irrational with the fender. And even if I decided to engage in such silliness, I would build the fender to fit the wheel shape that results, not arbitrarily pick a round one.
No, you argument about fitting the fender made sense; the problem with it is that the Gron wheel model doesn’t make sense and it would run into the fender. That could be taken as another another reason we shouldn’t accept the idea that he is analyzing a wheel.
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cinci: For those who subscribe to the Gron analysis I would point out the decreasing speed with increasing velocity. The analysis is identical to a nonrotating clock face where the tips of the hands of the clock match the translational velocity.
JT: Which have all been addressed above.
cinci: Addressed? I guess so. But the fact remains that The Gron analysis predicts decreasing angular velocity in the road while a more rational (more rational meaning that the length contraction is not ignored) wheel predicts the opposite. If you get opposite answers for what is supposed to be the same problem using relativity, wouldn’t that be a paradox? Well I don’t think it’s a paradox, I think the Gron model is a poor representation of a rolling wheel.
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cinci: Rotation of the object itself is not considered in the determining the shape of the wheel in the road frame or in the wheel frame since relativistic contraction is not allowed.
JT: The rotation is considered. If it were not, then the spoke tips would be the same distance apart at the bottom of the wheel as they are at the top. The spoke spacing at the bottom would not be compatible with the wheel rolling out gamma*2piR. The angular velocity in the road frame would not be lower at the 12:00 o'clock point than it is at the contact point.
cinci: How many times are you going to cop out on this by launching off into a discussion of spokes. The shape of the wheel is determined only by length contraction.
Want proof? Then let your spokes rotate so that their tips are 2v where v is the velocity of the wheel down the road; that is, let the wheel be spinning as if you were burning rubber in a car. Tell me if you get a different shape for the wheel. Or, more simply, let the wheel just skid so that the spokes aren’t rotating at all and tell me what shape you get for the wheel.
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SYA: Your responses continue to show you do not understand the meaning of relativistic effects, let alone how they apply to the case of relativistic rotation. I encourage you to study the related posts made for you, the LT solns for Gron's scenario presented for you, and all the visual aids attached for you, in your 2 threads here. I'm sorry, but I have no interest in discussing your belief system regarding Gron's failure to use length contraction, or rack and pinion, foam wheels, marble races, or wonderous onthefly wheel building procedures. It's a kinematic analysis. Gron's is the simplest. Learn it first. Learn what the relativistic effects mean, for then you will understand nature under that model. Until then, you will eternally ask these same questions, which by the way all stem from lack of understanding. All you have to do is ask, but you're gonna have to start asking the right questions, instead of relentlessly restating the same old ones.
What you need is to learn the meaning of the LTs, because this is where all your confusion stems. You should start with the all inertial scenario, and work your way up to the spinning disk later.
cinci: I’m not confused at all. And this is more than a restatement of old questions so you’re just running for cover.
In fact, I compared the Gron solution to a different, more rational wheel model which is new information. I found that while Grons’s solution predicts such things a decreasing angular velocity with speed in the road frame while the new wheel predicts the opposite. Also, Gron’s model says a rack and pinion won’t work while the new wheel would permit it. If both analysis are consistent with relativity it seems a little odd and perhaps paradoxical that there are these differences. Or, it means that one of them is a lousy model. Did you give an opinion on this? No, you stick to your perennial ad hominem attacks. If you had a cogent argument you wouldn’t have to do that.
I suggested a little analysis for you that would definitively settle the argument as to whether the elliptical shape in the Gron analysis is anything more than simple contraction. No comment there either because you know I’m right and don’t want to admit you’ve been blowing smoke on this issue.
You claim I don’t know relativity which is nothing more than a cheap shot, basically namecalling, which takes no more intellect than one expects from a 5th grader in a schoolyard argument. It’s time for you to grow up and deal with the issues I raise or simply go away. Either would be an improvement.
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Gron doesn't use a nonrotating structure. He just analyses the locations of the points on the wheel. Surely you understand that a point on a rotating, nontranslating wheel would travel a circular path around the axle, correct? The wheel itself is the only "structure", and it is rotating!
Here is the math for your shrinking wheel:
R_speed = R_rest / gamma
Solve for R_rest:
R_rest = gamma * R_speed
You said (correctly) that your wheel rolls out this distance per revolution:
d = 2pi * R_rest
Simply substitute the value of R_rest into the above:
d = gamma * 2pi * R_speed
Which is what I said, to which you replied that you don't think so. Now, if you cannot think clearly enough to work that out yourself, you are not up to the task of criticizing Gron, who is an expert on relativity.
Okay, so let's consider your shrinking wheel with a rest radius of 2.000. Spin it up to 0.866c and it now has a radius of 1.000 at speed. The road frame says the axle has a velocity of 0.866c and the contact point has a velocity of 0.000c. The road frame also says the distance between those two points is 1.000.
I say the angular velocity at the contact point should be this:
w = (0.8660.000) / R_speed
or
w = 0.866 / 1.000
But you say the angular velocity at the contact point should be this:
w = (0.8660.000) / R_rest
or
w = 0.866/2.000
Have you taken complete leave of your senses? I don't even want to finish addressing your post. Please rethink this.
You yourself are the sum of your positions that define you, and your own posts are by your own hand. Your ink is on the paper. I mean, it's not like anyone doesn't realize it after all this time here. I encourage you to restudy the theory, not to ridicule you, but to help you. I would not have spent the time I have with you, if I did not like you. Everyone's just trying to help you here, but you simply refuse to realize it.
I might add, an INSTANT CENTER of rotation is not an AXIS OF ROTATION, unless it continues to be so over duration. Relativity deals with axes of motion, otherwise called propagation paths. You misassume you may apply the Fitzgerald Length Contraction Formula (LCF) to the stringofatoms that constitute a pivotarc about the ground contact point, considered in only the ground instant of time t. Those atoms (1) do not reside on the same path of propagation as they in fact travel along different cycloids, (2) are not all moving at the same speed in that instant and in fact all differ in velocity from each other, and (3) their speed and direction continuously changes (differently) as they go. For these reasons, the LCF fails in your case, and your pivotarc model falls down. The LCF was designed for 2 separated points, always comoving at steady rate.
Also, the atoms that constitute your linear horizontal chord per axle in the axle instant t', are not the same atoms of a horizontal chord of the wheel per ground at that same height y=y'. They are all different atoms, with the exception of the one atom at x'=0. Therefore, when you say the LCF is the same thing as the LT, you are mistaken. The LCF has no time variables, and it assumes all atoms of your horizontal chord per axle are the very same atoms of a horizontal chord at that same height y=y' per ground. The LTs, assert that the same atoms of the horizontal chord per axle, exist as curvilinear in the ground system (Gron's model), and while all those atoms are simultaneous per ground (at time t), they correspond to differing times t' of the axle system. Curved radial elements per ground requires curved chords per ground. Of course, JTyesthatJT has already explained all this, if memory serves.
The radius of a rolling wheel should not shrink during rollup, because there is no lengthcontraction along radial elements, since there is never any velocity along those axes per axle. As such, the wheel's radius cannot change, and therefore its height cannot change. Relativity requires that no lengthcontraction exist wrt axes orthogonal to the axis of motion, such as y and y' (at the axle's location). Given such, the height of the wheel cannot change per ground during rollup, because the radius cannot ever change per axle. Ie, y = y' at the axle. You said you have a competing wheel model that is superior to Gron's, its assumptions completely abiding by relativity theory as Gron's did, that allows for a shrinking radius with rollup. Please provide the reference. Who did it, and in what year? I would like to read the assumptions made, both classical and relativistic.
Thank you,
SinceYouAsked
Last edited by SinceYouAsked; 04092014 at 07:12 PM.
Gron made the following point in his Conclusion.
If the wheel actually has any thickness it will stop many of the light rays from traveling to the camera.4. If the disk is regarded as a 2dimensional surface it can be put into rotation in a Born rigid way, that is without any displacements in the tangential plane of the disk, by bending for example upwards so that it obtains the shape of a cup.
I think he is talking about a solution to the Ehrenfest paradox in which the disk warps out of plane, (for example it could warp into something like a wok shape). The idea is to explain how the circumference of a circle could get smaller while the diameter remains constant.
In that case, you are correct that the camera would only be able to photograph a fraction of the surface of the wok at any given time.
Yes. Just to add to that ... Gron's #4 (an alternative scenario) assumes the flat round disk physically deforms to recipe during rollup (into a woklike shape). The wok's curved radius remains constant at R with increased rotation rate, but rotating circumferences and radii (as measured in an intersecting flat x,yplane) shrink by the factor 1/gamma per the axle system, which means the properlength of circumferences never changes even during rollup. Upon any circumference, whatever the number of consecutively aligned atoms that existed before rollup, exists during and after the rollup. The wok rolls out 2piR upon the (virtual) road over a full rotation. As such, the height of the wheel is shrunken in the axle and ground system, both (of course) recording the same height. This is fundamentally consistent with Gron's own rolling disk analysis, except "the scenario's allowed manner of deformation" differs between the 2 cases. Relativity applies in all the same ways, in either case.
One point though ... Erhenfest assumed a Born rigid body must shatter due to rotating circumferential contraction given unchanging radii, per axle. IOW, he assumed a Born rigid body "could not deform". Once the contraction of rotating circumferences (upon unchanging radii) outweighed the atomic binding forces, the body shatters, or dismembers so to speak. Gron's #4 is actually allowing for a deformation of the atomic configuration, without breakage, and to recipe. Once we allow deformations, we can arrive at 100s of different possible scenarios to analyse, however relativity applies the same for all of them. For example ... we could assume that inner circumferences fatten with increased rotation rate, allowing the all circumferences to remain at the proper length before, during, and after rollup. It's all wonder bodies in wonderous scenarios. Gron's scenario setup is the simplest, using a kinematic analysis, to ascertain the implications of relativity theory as applied to relativistic rotation rates, per inertial nonrotating POVs.
Technically, Gron's own scenario allows for a deformation, although he does not state so. His relativistically rotating disk could never exist in the first place, as defined, unless the atoms were miraculously allowed to deform to a recipe that always allowed for perfectly round disk of unchanging radius per axle ... because more atoms must shift into the circumferences (from adjacent locations along y) during the rollup to maintain the Born rigidity (and not shatter). No wheel can attain such relativistic rotation rates because of the classical forces alone, so all such scenarios are wonder scenarios using wonder bodies. As such, there is no need to consider all the possible scenarios of various desired wonderous deformations, because the relativistic nature of rotation is determined by Gron's simplest scenario. That nature applies the same to all rotatingbody scenarios. And if one does not understand Gron's scenario, there is no good reason whatever to proceed to any other related scenario, because "Gron's is already defined" for us all.
Thank You,
SinceYouAsked
JT: Okay, so let's consider your shrinking wheel with a rest radius of 2.000. Spin it up to 0.866c and it now has a radius of 1.000 at speed. The road frame says the axle has a velocity of 0.866c and the contact point has a velocity of 0.000c. The road frame also says the distance between those two points is 1.000.
I say the angular velocity at the contact point should be this:
w = (0.8660.000) / R_speed
or
w = 0.866 / 1.000
But you say the angular velocity at the contact point should be this:
w = (0.8660.000) / R_rest
or
w = 0.866/2.000
Have you taken complete leave of your senses? I don't even want to finish addressing your post. Please rethink this.
cinci: That’s the angular rotation of the 6:00 spoke which for years I pointed out to you as the answer to CANGAS’s question. And for those same years you used roll out distance to calculate angular velocity; roll out distances per unit time to determine w.
The roll out distance for Gron’s wheel is 2piR/(1 – (v/c)^2)^.5 so at v approaches c, roll out distance approaches infinity. That means you get one revolution in infinity times….the Gron wheel slows down in spite of the fact that the 6:00 spoke angular velocity is always w = v/R.
The contracting wheel always rolls out 2piR so in terms of revolutions per unit time it is always w = v/R where R is the rest dimension. The 6:00 spoke is as you say above and as I have always said.
I haven’t lost my senses and I haven’t lost my memory.
As for finishing the post, don’t bother if you’re going to go this route.
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You mean number of rotations per unit time, like RPM (revolutions per minute). Classically RPM is equivalent to w, but usually in different units. However, in the relativistic scenario, even when v is held constant, w is smallest at the top of the wheel, and greatest at the bottom of the wheel. So there is a whole range of values of w, but there is only one value of RPM for the whole wheel, so it's like apples and oranges.
Yes, that is correct. The reason the RPM's slow down as v approaches c is because of what is happening near the top of the wheel. Meanwhile, at the 6:00 point, w remains its classical value.
Only if R is the rest radius. If R is the vertical radius at speed, it rolls out 2piRgamma, as I proved mathematically. Funny how you skipped that part of my post.
Oh I understand now. You are talking about RPM of the whole wheel, rather than w at the 6:00 o'clock point. Sorry, that was a misunderstanding on my part. In that case, you're right, the RPM is more like 0.866/2.000 in the road frame, rather than the 0.866/1.000 angular velocity at the 6:00 o'clock point. That just shows that your contracting wheel has a timedilated RPM in the road frame. The rest radius in the denominator (0.866/2.000) makes the RPM of the whole wheel slower than the 6:00 o'clock point's angular velocity, which you agree is 0.866/1.000. I'm glad you agree.
So, the contracting wheel behaves exactly the same way as the Gron wheel for any constant value of v where both wheels happen to have the same vertical radius at speed.
Sorry, I didn't realize that you were using RPM and w interchangeably.
You are correct that the shape is the same regardless of whether the wheel is rotating or not. But Gron's solution is not only for the shape, it is also a solution for the location of known points on the wheel. That part of the solution depends on whether the wheel is rotating or not.
Last edited by JTyesthatJT; 04112014 at 12:42 AM.
JT: Of course you have already shown that any wheel which rolls out gamma2piR (where R is the vertical radius at speed) means that as v approaches c, the roll out distance approaches infinity, and therefore the RPM approaches zero. Too bad you skipped over the part where I proved that your contracting wheel also rolls out gamma2piR (where R is the vertical radius at speed) and therefore behaves exactly the same way as the Gron wheel.
cinci: In the wheel frame, the rim and the road contract by the same factor. Therefore, the circumference and roll out in the wheel frame is
2piR(1 – (v/c)^2)^.5 because the road is contracted and so is the wheel.
In the road frame, the road is not contracted and the roll out is 2piR (Rest radius)
The pointbypoint correspondence of wheel and rim do not change with speed as they do with Gron’s wheel. Every rotation of the wheel produces 2piR in the road frame where R is the rest radius.
Your calculation of gamma 2piR (contracted rest frame) is interesting but not applicable; there is no round wheel in the road frame. So you haven’t proved anything.
In the road frame, the curved spokes of the wheel will allow it to roll out 2piR (Rest R) per revolution at all speeds. The angular velocity of the contracted rolling wheel does not slow down; the Gron wheel does. That should be enough to show it is a lousy model.
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Yes, you are correct that the contracted wheel rolls out 2piR for all velocities. I had edited my prior post to remove the statement which you are quoting above. I realized it was incorrect.
I agree that the contracted wheel model is perfectly fine. It works for rack and pinion, which is an advantage over the other type of wheel. So, I would model the contracted wheel like this:
v = translational velocity of axle as measured by road
v = translational velocity of road as measured by axle
γ = 1 / √(1  v²/c²) = gamma
Ro = proper radius of nonrotating wheel
R' = radius of spinning wheel, as measured by axle frame
R' = Ro / γ
Note that the above equation is the one which you say "is interesting but not applicable" because you thought I was referring to the road frame, which I am not.
Note that I am assuming the wheel is rolling, thus the speed of its perimeter is equal to the translational speed v.
W = the angle of some radial line on the Wheel (measured from the 6:00 o'clock radial line) as measured in degrees by axle frame at time t'=0
R = the radius of some concentric circle on the wheel as measured by axle frame
Note that any point on the wheel can be described in the axle frame by (W, R) sort of like polar coordinates.
F = the angle of some radial line on the Fenderdisk (measured from the 6:00 o'clock radial line) as measured in degrees by axle frame
F = W+((360vt')/(2πR'))
Note that the fenderdisk is just like a nonrotating wheel superimposed on the rotating wheel.
x' = Rsin(F)
y' = Rcos(F)
t' = the time in the axle frame when wheel point (W,R) is in the same place as fender point (F,R)
Note that these 'primed' variables refer to the spacetime coordinates of an event in the axle frame.
That event can be described as, "the wheel point (W,R) is in the same place as the fender point (F,R)".
x = γ(x' + vt')
y = y'
t = γ(t' + (vx' / c²))
Note that these unprimed variables refer to the spacetime coordinates of an event in the road frame.
That event can be described as, "the wheel point (W,R) is in the same place as the fender point (F,R)".
Done, and done. Next!
Last edited by JTyesthatJT; 04112014 at 07:18 AM.
Careful, the "physically contracting wheel scenario" is nothing but an alternative impossible wonder scenario using a wonder wheel that deforms to recipe "by definition". No real wheel can survive rollup to such relativistic rates. As such, cincirob's scenario is no improvement at all. It's nothing more or less than another potential wonder scenario, one which Gron himself pointed out. Granted, IF one wanted a rack and pinion to work kinematically, one would want a wheel that always rolls out 2piR over a full rotation no matter what the speed. However, no pinion can withstand such relativistic rotation rates in any reality, let alone deform to recipe in the way one wishes while violating the laws of physics surviving the centrifugal forces. It's all "wonder scenarios". Relativity works the very same way in any case, and that's the part cincirob has not yet understood...
Cincirob has always held the Gron model in error, because it rolls out gamma*2piR upon the road at speed, and as such cinci has no good understanding of what the Gron solns mean ... because he does not understand the meaning of relativistic effects in the very first place. Cinci needs to start afresh with allinertial scenarios, then work his way up to accelerating/rotating bodies.
PS: It's hard to know whether cincirob's wheel scenario is the same one that Gron mentions in his Conclusion #4, versus cinsirob having his own concoction of sort. He says his model makes sense but Gron's does not, so this suggests his notion differs from Gron's #4. Since cincirob can never put his model into mathematical form, and because he changes the subject every 2 seconds, there's no good way to know what he ever means. His posts are generally poorly typed and worded, which of course always results from confusion. One thing is for certain, he does not understand Gron's model, which means he does not understand the meaning of the LT solutions.
Thank You,
SinceYouAsked
JT: t = γ(t' + (vx' / c²))
x = γ(x' + vt')
y = y'
Note that these unprimed variables refer to coordinates in the road frame.
Done, and done.
cinci: Yes, and it is the second equation that tells you that to get the ellipse shape you are simply contracting the chords as if they have no velocity relative to the wheel frame and as if they are not already contracted according to that velocity.
BTW I am having a lot of trouble with the site and don’t see all the messages. I did manage to see the last one from SYA. Please tell him to stop making a fool of himself. Thanks.
That is just a Lorentz transform equation. I always suspected that you distrusted those.
Let's consider the 9:00 o'clock point on the edge of your compressible wheel, (with a rest radius of 2.000 and v=0.866c). Go through my whole list of equations, top to bottom, using t'=0.000, and you should find these values :
v = 0.866025c
γ = 2.000000
Ro = 2.000000
R' = 1.000000
W = 90.00000
R = 1.000000
F = 90.000000
x' = 1.000000
y' = 0.000000
t' = 0.000000
x = 2.000000
y = 0.000000
t = 1.732051
Note how that point transforms from the axle frame (x'=1.000) to the road frame (x=2.000). So clearly the equations are not simply lengthcontracting the circle to an ellipse. You have to consider how the time transformation of the 9:00 o'clock point transforms from the axle frame (t'=0.000) to the road frame (t=1.732). Thus, ROS must be considered.
I'll tell him.
You have to consider that the time transformation of the 9:00 o'clock point transforms from the axle frame as t'=0.000 to the road frame as t=1.732. Thus, ROS must be considered.
cinci: ROS tells you that a particular spoke tip was at a different angular position from the view point of the road observer. Once you know that angular position as determined ROS you go to the wheel frame and contract the chord length for it to find it in the road frame. Substituting the ROS equation into the LT doesn’t make that LT go away; every point on the rim of the wheel has to obey LT equation and that equation gives you the length contraction formula. The shape of your wheel is found by nothing more than length contraction. You already agreed that if the spoke tips were moving at a different tangential speed than v the shape wouldn’t change.
Even if you did the transformation as I suggest for real material, you would go through the same thinking and get curved spokes but rim shape would be different. I’m not saying ROS goes away. In fact I even showed you a calculation for an ROSed chord once for a nonellipse.
Any time you and SYA want to stop pretending that I don’t understand the Gron analysis and that I haven’t done it myself will be soon enough. And if I have to keep explaining to both of you that the wheel shape you get is determined only by length contraction I’m going to start telling you that you don’t understand the Gron analysis.
Your last comment on the roll out was “done and done” which doesn’t say much. My point in all that was that the Gron solution proposes a wheel that slows down with increasing speed. It does that by adding material to the rim as a function of speed. You once called my marble race a “contraption”; if the marble race (which could easily be constructed is a contraption then the Gron wheel is contraption squared. Choosing to use that sort of nonsense to illustrate the phenomena associated with relativity makes no sense whatever. Particularly when more palatable options are available. And that isn’t its only problem.
All the equations I posted above pertain to your contracting wheel. Its shape is a circle in the axle frame, and an ellipse in the road frame. If you think there is something wrong with that, then please show me which equation(s) needs to be changed. Until then, there is no "more palatable option available".
Hey cincirob, do you agree my first few equations describe your contracting wheel? Here they are in detail:
v = translational velocity of axle as measured by road
v = translational velocity of road as measured by axle
γ = 1 / √(1  v²/c²) = gamma
Ro = proper radius of nonrotating circular wheel
R' = radius of spinning wheel, as measured by axle frame, where spinning wheel is still a circle
R' = Ro / γ
Note that I am assuming the wheel is rolling, thus the speed of its perimeter is equal to the translational speed v.
Do you agree to the idea that your contracting wheel is always a circle in the axle frame?
JT: All the equations I posted above pertain to your contracting wheel. Its shape is a circle in the axle frame, and an ellipse in the road frame. If you think there is something wrong with that, then please show me which equation(s) needs to be changed. Until then, there is no "more palatable option available".
cinci: If you don’t really understand that ROS has nothing to do with the simple ellipse shape it will be impossible for you to entertain any other shape. And I’m never sure you understand it because you don’t seem to be able to discuss the ellipse shape without bringing up ROS.
SYA: Please post the equation that you refer to above as "_the ROS equation_", just so we all know what you mean when you say it.
cinci: JT just posted his equations. Don’t ask me to repeat things that are obvious just clutter up the discussion.
I already agreed that the ellipse shape results whether or not the translating wheel is rotating. As you said, it could also be rotating with some tangential velocity other than v, and the resulting ellipse shape would still be the same. I understand that ROS does not affect the overall shape of the wheel, but it does affect the shapes of radial lines on the wheel. I assure you that I understand that.
The only reason I mentioned ROS above is because you claimed this equation...
x = γ(x' + vt')
...produces simple length contraction of the wheel, which it does not. Remember, γ>1, so this equation is not simply length contraction.
To illustrate this, I showed you how the 9:00 o'clock point on the edge of your compressible wheel, (with a rest radius of 2.000 and v=0.866c) transforms from the axle frame (x'=1.000) to the road frame (x=2.000). This looks like the opposite of lengthcontraction, but it isn't. One must consider how the time transformation of the 9:00 o'clock point transforms from the axle frame (t'=0.000) to the road frame (t=1.732). When one considers that the axle would be located at x=vt=1.500 at that time (t=1.732), then one understands how the lengthcontraction of the ellipse is produced. There is no place in my equations where simple length contraction is applied, and ROS is always involved. However, I do agree with you that the overall shape could have been produced by simply applying lengthcontraction, but I did not use that method because that would not tell us the shapes of radial lines on the wheel.
cinci: OK, I’m going to say it. You don’t understand you own analysis. And don’t bother with the new wheel, we’re discussing your analysis.
The LT above is the LT from which the length contraction formula is derived. It is the basis for all your calculations. It is buried in your analysis because you start with it and substitute the ROS business into it. Mathematically, any point you calculate is calculated with that equation and it gives you length contraction. Look at the shape, it is a length contracted circle.
When you include the ROS, you are simply picking a different chord to contract. In the case of the spoke in the wheel frame that is 30 degrees from 6:00, you actually transform the chord for the 70.5 degree spoke and that gives you the curvature.
Why 70.5, because the ROS time calculated for the 70.5 degree spoke Is enough time for the wheel to rotate 40.5 degrees. And since it’s negative time, you back the 30 degree spoke up 40.5 degrees. The chord at 70.5 degrees is contracted by your formula by (1 – (v/c)^2)^.5 and, therefore, falls on the ellipse. This is the reason all your points fall on the ellipse. If you weren’t length contracting whjeel chords by (1 – (v/c)^2)^.5 you wouldn’t get the ellipse.
Instead of arguing about this, work it out in steps instead of substituting everything together.
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