So what happens when a ball burns up?

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Re: So what happens when a ball burns up?

Post by LabRat »

Paul, I apologise if my tone offended you, that was not my intention. The best definitions are, in my opinion, those that are accurate, informative and useful. The existence of the transition points in the ball path is important knowledge for bowlers to have if they want to better understand ball roll. The problem is that I don't feel that ar-at is a particularly useful or informative definition of the second transition point, partly because I think it is inaccurate, and mainly because even if it is accurate it doesn't describe to the bowler what is physically happening. It's also not universal - you can't tella guy who throws a full roller that the second transition is where ar=at, because for him it never happens. At the second transition point, the ball stops hooking and starts rolling. At this point, absent external factors, it will also have the maximum friction between the lane and ball. I don't believe that the condition ar=at is either necessary or sufficient for this to happen.
The 'accepted' mechanics espoused by Mo and others is that the at the second transition point, the ball begins and will continue to roll in a straight line, ar=at, and both ar and at will decay to zero as the ball continues to roll. I agree with the ball travelling in a straight line from the second transition point, other than deviation caused by statics and topography. However, for ar and at to both decay to zero, the ball has to back up! Since ar is measured from the fixed reference point of the lane, ar=0 corresponds to a ball travelling straight down the lane. Remember, this is a ball that has finished hooking and is entering the pin deck at an angle to the lane. Whilst there IS a force that would cause this behaviour, the ball motion study demonstrated that the roll path is linear, so it does not act measurably on the scale of a bowling lane.
The requirement that ar=at is not consistent with the physical requirement that the ball continues in a straight line from the second transition point. It is as simple as that. Regardless of tilt, a ball stops hooking and starts to roll when it has zero axis rotation relative to the direction it is travelling, and the ball surface in contact with the lane is not moving relative to the lane. Any ball that has non-zero ar relative to the direction it is travelling in will continue to hook, again regardless of tilt. We already know that the ar=at mantra can't apply to full rollers and inverted tracks, and it is obvious that high tilt bowlers don't bleed off enough tilt to fit the definition either, easily confirmed by measuring the diameter of the dry flares.
There is no theoretical physical foundation or justification for the 'ar=at at the roll point' dogma, and it seems to be one of those urban legends that everyone accepts without thinking about what it means. Apparently there is even a rotation/tilt overlay chart in the USBC Coaching manual (hopefully not the current one) set up to imply at cannot be higher than ar, which is the equivalent of arguing that longitude can't be bigger than latitude.
Now to the flare/friction/burnup issue. I gave the analogy of a car doing burnouts wearing out tyres faster than doing laps. Paul, I'm sorry but I have never heard of 'molecular connectivity energy', and have no idea why it or the heat capacity of lane oil would be relevant anyway. I think you have (understandably) misunderstood what I meant by slow vs fast transitions - I was talking about a ball released into a smoothly changing friction environment (blended pattern) vs one dumped straight into the desert at the edge, and should have made that clearer. A bowling ball has a total kinetic energy at release in the vicinity of 200 joules, give or take around 20% for most people and 14-15lb gear. That's not a lot of energy in heat terms - enough to heat a cup of water by 0.2 degrees or so. However, anyone who has bowled a ball on carpet will see the result when a skidding ball abruptly encounters a high friction surface - carpet burns.
Too abrupt a change in friction generates enough heat to burn dry flares into a ball and damage the friction coat on the lane surface. Mo has said that bowling balls should start to transition in the pattern. Coming off the pattern in a full skid, with most of the axis rotation imparted to the ball at delivery still there, will result in an abrupt grabbing of the lane by the ball and generate lots of heat - from the balls' kinetic energy which is then no longer available to help the ball to hook and hit. Alternately, a ball that 'reads' the pattern properly will see a more gradual and staged increase in friction, and will more efficiently transfer energy into revolutions instead of heat. Throw it into the dry too hard and too early, and the ball literally 'burns up' - and takes the lane surface with it.
How much difference does it make? That may well be a question for a throwbot study, but anyone who has fallen over on carpet knows the difference a bit of speed makes to how bad the carpet burns are.
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Re: So what happens when a ball burns up?

Post by larim »

I believe the Persimmon Driver was a popular choice for a driver in 1980.

Personally, I enjoy a little debate and this reading was a nice refresher coming off a long summer. The nice thing is there is a lot more data and information gathered today than in 1980. Sometimes things evolve and the way you used to think (or currently think) is proved not to be the most accurate... it is important to have an open mind and embrace different information.

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Re: So what happens when a ball burns up?

Post by larim »

I struggle to enter into this debate for fear of time commitment to debate.

Nevertheless, I dive, but I will keep it short.

Please understand that ar = at was derived from current and historical data that was achieved in a controlled environment where variables were understood and controlled.

There is always going to be a situation where a specific variable(s) can be modified or set to extreme limits to offer counter arguments. Certainly, I could oil a lane 60 feet with 50mLs and how many phases would I see if I threw a highly polished pearl ball at 22mph?

I understand the issue with explaining the at = ar to 'average' bowlers. It is not a sexy term.

One quick question or two to LabRat: You understand that there is axis migration as the ball goes down the lane, correct?

I am most certain Paul will have a more formal response to you, so I will let him continue that formal discussion.
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Re: So what happens when a ball burns up?

Post by MWhite »

purduepaul wrote:

What flawed thesis from 1980 do you feel I'm drawing my conclusions from?

I can understand if you limit your observations to people trying to score well while throwing resin equipment, you could come to the conclusion that the AR = AT.

An effective release with resin equipment has the ball lose both AR, and AT at a very rapid rate.

What people call stand up, and face the pocket.

If you observe a well thrown polished urethane ball, or even plastic ball, you will notice the decay rate of AR, and AT are independent of each other.

If you have access to a ball thrower, why not perform a small experiment to show that a ball released with AR = AT = 30 degrees will not hook.

To be fair, have the rev rate high enough that the ball doesn't go instantly from 30 degrees to 0 degrees the moment in encounters friction.
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Re: So what happens when a ball burns up?

Post by LabRat »

purduepaul wrote: Now we are going to have an issue. In my previous post, I stated due to the current measurement technology on the market, neither side can generate measured data to confirm or deny AT=AR.
...
As I stated earlier, if you had a bowler with thirty degrees of tilt and rotation, the ball would not hook. In your theory, it would. So before we go around throwing bad science moniker around in the future, we should be aware that this is all theory anyways.
Now we are getting somewhere. From what Paul is saying here, it seems that the AT=AR relationship has not been experimentally confirmed, due to the lack of measurement technology, and it is a theoretically calculated result. So, where is the maths that supports the AT=AR view?
A ball with 30* of axis rotation will hook regardless of tilt, as even a cursory examination of a force diagram will confirm.
Further, it is trivial to measure axis tilt at the pin deck (oil the deck and measure the resulting ring), and CATS data gives the entry angle fairly accurately, so I would suggest that it would be quite easy to confirm the relationship with a robot.
AT and AR should decay independently, btw. IIRC, Blueprint did a simulation where balls were thrown with identical speed and AR, and with varying tilt and rpm to keep the same track speed. They found the balls reacted exactly the same, and I think it was confirmed via Throwbot. Physically, there is no reason for the decay of rotation and tilt to be linked AFAIK.

edit: the study is here:
http://blueprintbowling.com/Blog/Posts/ ... rcats.aspx" onclick="window.open(this.href);return false;

To larim: obviously there is axis migration (due to flare) as the ball travels down the lane. In the ideal case, this migration does no work on the ball. I do believe that axis migration can, depending on layout, affect AR and AT even in the absence of friction in cases where the migration path is not a great circle, and that highly curved migration pathways have a profound affect on ball roll - long pins on asyms, for instance.

Also, with respect to this: "Please understand that ar = at was derived from current and historical data that was achieved in a controlled environment where variables were understood and controlled."
Could you expand on this? I'm trying to understand the basis that the relationship was first conceived via.

BTW, the AR/AT diagram I referred to earlier is in the Bronze manual with a 2005 copyright, so maybe it's still in use. (hopefully not).
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Re: So what happens when a ball burns up?

Post by MWhite »

LabRat wrote:BTW, the AR/AT diagram I referred to earlier is in the Bronze manual with a 2005 copyright, so maybe it's still in use. (hopefully not).
I just found the diagram you are referring to.

If that graph is taken as gospel, it would explain some of the unique theories some people have about
AR vs AT.

AR = AT = 30 on that graph, would be AR = 0, AT = 30 using a proper graph.

And yes, a ball released with AR = 0, AT = 30 would not hook.
Bad Rot Tilt Graph.png
Whoever created that graph should be dug up, shot, then reburied... with the reburial part being optional.
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Re: So what happens when a ball burns up?

Post by LabRat »

That also explains why some people insist that tilt cannot exceed rotation.
Plenty of other clangers in that material, too. Hopefully it has been updated.
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Re: So what happens when a ball burns up?

Post by larim »

I misspoke regarding AT/AR. I thought some of the Ball Motion study 'setup' was used to gather this info. I have learned this was not the case.


Somehow I think I got confused with this information being derived from the ball motion study, etc.
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Re: So what happens when a ball burns up?

Post by purduepaul »

First of all, thank you for the apology on your tonality. Sometimes when we are passionate about something, we seem a little too harsh. I agree that transition points are important for bowlers to know, however most bowlers read front to back transition poorly. Whenever you state, I feel or I think those are opinion statements, its your opinion that you do not agree with the definition provided. I think where part of the confusion is on the subject is the reference point to define AR. The reference point to determine AT is universal, while the reference point for AR is not.



So let’s pick one reference point and that is in relation to the bowling lane. So AR will never be zero after the second transition point on the bowling lane. Also for regular tracked bowlers (no Negative tilt and full rollers for the moment) AR will be greater than or equal to AT. For AT> AR, the PAP would have to be off of the ball which can not occur. For the average bowler (Starting tilt between 10 and 17 degrees), your ending tilt will be between 3-10 degrees. Bowling balls lose Axis rotation much easier than axis tilt. This is also why bowlers who have high axis rotation and low tilt (Pete Weber) have more violent hook phases of ball motion than a Walter Ray Willams Jr who has lower axis rotation with low tilt. It is my contention that most bowler’s will not see their AT reach 0. The difference per degree of tilt on a bowling ball is 0.075” on the high side of a regulation bowling ball, and we can get into the measuring accuracy discussion at that time if you think you can measure that. The interesting point you make is concerning full rollers. Since AR will always be greater than AT=0 and we know that full rollers still produce entry angle, the hook phase may end because of a different reason in relation to the core’s properties of gyroscopic inertia. The gyroscopic inertia of a bowling ball is very important, and is solely determined by the core of the bowling ball itself. I have a theory of why this would occur, however it would be going into proprietary information that I am not at liberty to discuss.



As for the fourth paragraph, the analogy you gave does not compare because the frictional coefficient of the surface is the same while the frictional coefficient of an oiled lane and dry lane are different. In addition, heat capacity of lane oil is important if you think it is losing energy to heat. You said it lost a significant amount of energy to heat, I disagreed. Any efficiency calculation I have done is total conversion/possible conversion or time. The faster you transfer energy from translational to rotational is the most efficient way. You are thinking in terms of energy still available not lost to friction and heat.
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Re: So what happens when a ball burns up?

Post by purduepaul »

In reference to MWhite’s post #44, the flawed thesis most ball motion simulators are based on is the Smalc thesis from Case Western Reserve from 1980. Yes, I have a copy of it. No, it is not electronic. It’s 464 some odd pages. I know of the existence of seven copies, that’s pretty much it. People in the modern environment throw mostly resin bowling balls. If you want to discuss polished urethane, the second transition point does not occur unless you are on short oil and then maybe. AR is lost more rapidly with resin equipment than AT. I used to have access to a ball thrower, however not anymore. I have a regular full time position outside of bowling.



In reference to LabRat’s post #45, While I appreciate your passion regarding this subject, I would like to see your force diagram on the subject. I’m willing to bet you are missing two to four possible other forces acting on the bowling ball. Some of which I would discuss further but the confidentiality agreement prevents me from doing so. CATS data has significant flaws. Again, I could write a thesis on the effect of a parabolic ultrasonic stream and its effect on data, however I will leave it at my statement above. I will reiterate myself once again, Blueprint is a simulation based on math formulas that were derived. I’ve said it in the past that I believe its in the neighborhood of 85-90% accurate. My feeling at this point is I’m not going to change your mind, and you have not provided any data to back up your claim. If I have time, I will come up with some of the math regarding the situation, however at this point I feel we are beating the proverbial dead horse.
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Re: So what happens when a ball burns up?

Post by MWhite »

purduepaul wrote:For AT> AR, the PAP would have to be off of the ball which can not occur.

This is the equivalent of saying there is no place on earth where the latitude is greater than the longitude.

If you really think what you said is true, there must be a problem with how you measure AR and AT.
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Re: So what happens when a ball burns up?

Post by Mo Pinel »

MWhite wrote:

This is the equivalent of saying there is no place on earth where the latitude is greater than the longitude.

If you really think what you said is true, there must be a problem with how you measure AR and AT.
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Re: So what happens when a ball burns up?

Post by purduepaul »

MWhite wrote:

This is the equivalent of saying there is no place on earth where the latitude is greater than the longitude.

If you really think what you said is true, there must be a problem with how you measure AR and AT.
In the modern game of bowling, AT > AR does not exist.
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Re: So what happens when a ball burns up?

Post by LabRat »

Comments embedded, and below. If something sounds snarkey or disrespectful please blame it on my communication skills rather than demeanour. I don't think disagreeing is disrespectful, btw.
purduepaul wrote:First of all, thank you for the apology on your tonality. Sometimes when we are passionate about something, we seem a little too harsh. I agree that transition points are important for bowlers to know, however most bowlers read front to back transition poorly. Whenever you state, I feel or I think those are opinion statements, its your opinion that you do not agree with the definition provided. I think where part of the confusion is on the subject is the reference point to define AR. The reference point to determine AT is universal, while the reference point for AR is not.

I think the reference frame is the source of the issue, but not in the way you think. I will expand on that later.

So let’s pick one reference point and that is in relation to the bowling lane. So AR will never be zero after the second transition point on the bowling lane.
Unusual, but not impossible. Low rotation bowlers shooting tenpins with the strike ball, fallback shots with forward rolling layouts from deep lines perhaps. This is one of the problems with using the lane as the reference, as it changes the final rolling AR figures if the bowler simply throws the same short from outside or inside.
Also for regular tracked bowlers (no Negative tilt and full rollers for the moment) AR will be greater than or equal to AT.At release, generally yes - it's what happens as the ball goes down the lane that is in dispute. I believe AR can easily decay to be lower than AT for lower tracks. For AT> AR, the PAP would have to be off of the ball which can not occur. This statement is simply wrong. (sorry, but it is!) Any combination of tilt and rotation is possible in theory - the two axes are orthogonal and can be set independently of each other. For AT>AR, the PAP simply has to be tilted further 'up' than rotated 'around'. For the average bowler (Starting tilt between 10 and 17 degrees), your ending tilt will be between 3-10 degrees. That seems a reasonable range. What about higher tilt bowlers, in the 20-30* range? Bowling balls lose Axis rotation much easier than axis tilt.Absolutely correct. This is also why bowlers who have high axis rotation and low tilt (Pete Weber) have more violent hook phases of ball motion than a Walter Ray Willams Jr who has lower axis rotation with low tilt. It is my contention that most bowler’s will not see their AT reach 0.I agree. The difference per degree of tilt on a bowling ball is 0.075” on the high side of a regulation bowling ball, and we can get into the measuring accuracy discussion at that time if you think you can measure that. It's about 2mm per degree in terms of track diameter.I think that is certainly measurable to within a couple of degrees. I'm sure an electronic sensor net embedded in the ball could measure this to arbitrary precision, but that would be an engineers' problem. :)
The interesting point you make is concerning full rollers. Since AR will always be greater than AT=0 and we know that full rollers still produce entry angle, the hook phase may end because of a different reason in relation to the core’s properties of gyroscopic inertia. ?? The hook phase ends for exactly the same reason it does for any track - the balls' side rotation decreases to zero relative to the direction of travel. Using the lane as reference, this is when the AR decays to the entry angle and it has no 'side spin' left. The gyroscopic inertia of a bowling ball is very important, and is solely determined by the core of the bowling ball itself. No, gyroscopic inertia (angular momentum) is determined by the moment of inertia about the PAP and the rev rate. MOI is a property of the whole ball, not just the core, although the core will contribute the most to the total. It is gyroscopic inertia that allows the ball to retain tilt while losing rotation as lane friction applies a torque to the ball. I have a theory of why this would occur, however it would be going into proprietary information that I am not at liberty to discuss.



As for the fourth paragraph, the analogy you gave does not compare because the frictional coefficient of the surface is the same while the frictional coefficient of an oiled lane and dry lane are different. Ok, so you think it's a bad analogy. ;) Here's a better one - the lane surface lasts a lot longer with oil on it than without. Kegel have studied the effect of resin balls on dry lane surface. They wreck it through friction. Properly designed patterns allow the ball to hook as much as they do on a tunnel block without destroying the lane surface, so it is possible to allow efficient energy transfer without generating enough heat to damage the lane surface. In addition, heat capacity of lane oil is important if you think it is losing energy to heat. Obviously it must lose some energy to heat no matter what - you have said as much yourself when you said the ball slows down after the second transition. That energy has to be converted to heat (by friction, both internal and external). You said it lost a significant amount of energy to heat, I disagreed. Actually, I don't think I said this at all - in fact I concluded by wondering how much difference it would make. The heat capacity of lane oil is irrelevant due to the low volume present, and most of the heat generated by friction will be in areas of the lane where the is little to no oil anyway, so any heat generated will virtually all be absorbed by the lane surface and ball surface, not oil.Any efficiency calculation I have done is total conversion/possible conversion or time. Obviously there is a typo here but I don't understand what you mean. The faster you transfer energy from translational to rotational is the most efficient way. You are thinking in terms of energy still available not lost to friction and heat.No, I'm thinking of total kinetic energy at the second transition compared to total kinetic energy at the release point. This will be maximised if friction losses to heat are minimised. I have bowled on lanes with no oil on them, having enough friction to get the ball rolling at the dots certainly looks a very inefficient way to transfer energy. But how bad is it, that is does the ball really lose enough energy to make a difference at the pocket (burn up) or is it simply an entry angle (and possibly no tilt) issue?
With respect to the force diagram, with my paint skills it will take a while so I might draw and scan it instead if needed. In the meantime, I offer this thought experiment:
Take a ball on an axle and spin it such that the axle is parallel to the foul line, (0*AR, 0* AT) . If we release the ball, it will roll straight down the lane, agreed? The track would be that of a full roller, 13.5"in diameter. Now, if we tilt the axis whilst keeping AR at zero, any track we tilt the ball onto will be parallel with the original, ie at 0 AR, and the ball will still, if released, travel straight down the lane. True?
If we now rotate that ball such that AR > 0, the initial full roller track is pointing somewhat to the left. If released with 0 tilt, it would roll to the left as it now has side spin. Now let us first tilt the ball to say 30*, then rotate the ball to 30* AR. Both the initial full roller track and the new 30* tilt track are parallel, so both are at a 30* angle to the lane. The only difference in the velocity vector of the contact point of the ball on the lane is that the 30* tilted track is moving slower than the full roller track - the direction is still at 30* to the lane. Why will this ball not move to the left if released?

I said earlier that I felt the use of the lane as the reference for AR contributed to this issue. The reason is simple - if you use the lane as the reference, the ball almost always starts rolling with (as you noted in your post) a non-zero AR. This is counterintuitive - how can a ball go straight if it still has axis rotation? People noticed that the tilt decays close to the AR for most bowlers, so to 'fix' this, the 'rolls straight when AR=AT' myth was born In fact, if we use the instantaneous direction of the ball as the reference, the ball simply rolls when the AR goes to zero, which makes sense intuitively, works for all tracks regardless of tilt, and is easy to explain physically.

If you still feel that a ball with both 30* of tilt and 30* of axis rotation will not hook if it's rolled straight down the lane, I'll take the time to draw up a force diagram. In the meantime, whole I have outlined the basis of my objection to the AT=AR several times, with respect nobody has offered anything in response other than 'everyone knows this and you're wrong'. In fact the only substantive objection has been 'if the AT>AR the PAP can't be on the ball', which is obviously untrue. You have already acknowledged it's not an experimental result, but the math/physics behind it has never been forthcoming for analysis. There has to be a reason why people think this is true but I still have no idea why.

Your comments about Blueprint and CATS are intriguing. Since their results match up, it appears they suffer from identical error bias which is difficult to believe. What do you feel is the accuracy of the CATS system, and why (or in which specific measurements) do you think it's flawed? Was this system used for the USBC ball studies?

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Re: So what happens when a ball burns up?

Post by J_w73 »

MWhite wrote:

This is the equivalent of saying there is no place on earth where the latitude is greater than the longitude.

If you really think what you said is true, there must be a problem with how you measure AR and AT.
I really think the disagreement is coming from how people are looking at AT and AR relation. I may be completely wrong or out of the conversation loop but I will try to explain something. If you have a person that releases the ball with 90 degrees of AT the ball will be spinning like a top. Does this ball have 0 deg of AR or 90 deg of AR? I think Paul's view is that at that point this is equivalent to 90 degrees of AR. And by that definition all AT angles apply the same amount of AR to the ball.

So Paul.. Is it possible to have a ball with 10 deg of AT and 0 degrees of AR? I think by your definition a ball with 10 degrees of AT has 10 degrees of AR as well. Am I correct in what I am saying?
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Re: So what happens when a ball burns up?

Post by Mo Pinel »

J_w73 wrote:
I really think the disagreement is coming from how people are looking at AT and AR relation. I may be completely wrong or out of the conversation loop but I will try to explain something. I think the way Paul and others see it is that if you have a person that releases the ball with 90 degrees of AT the ball will be spinning like a top. At that point this is also equivalent to 90 degrees of AR. And by that definition all AT angles apply the same amount of AR to the ball.

So Paul.. Is it possible to have a ball with 10 deg of AT and 0 degrees of AR? I think by your definition a ball with 10 degrees of AT has 10 degrees of AR as well. Am I correct in what I am saying?
Thanks for coming back in this form. Yes, we feel a ball with 10* of AT has the have at least 10* AR or you can't represent the PAP as being on the surface of the ball. For the PAP to be represented as being on the surface of the ball in the diagram, AR must be > or = to AT.
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Re: So what happens when a ball burns up?

Post by MWhite »

Mo Pinel wrote:
Thanks for coming back in this form. Yes, we feel a ball with 10* of AT has the have at least 10* AR or you can't represent the PAP as being on the surface of the ball. For the PAP to be represented as being on the surface of the ball in the diagram, AR must be > or = to AT.
The first graph is from the USBC Bronze level manual.

The second graph is from BowlingChat wiki.

Either only one of those graphs are correct, or we are trying to communicate using a completely different basis.
Bad Rot Tilt Graph.png
Rotationtiltball.JPG
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Re: So what happens when a ball burns up?

Post by MWhite »

J_w73 wrote:
I really think the disagreement is coming from how people are looking at AT and AR relation. I may be completely wrong or out of the conversation loop but I will try to explain something. If you have a person that releases the ball with 90 degrees of AT the ball will be spinning like a top. Does this ball have 0 deg of AR or 90 deg of AR? I think Paul's view is that at that point this is equivalent to 90 degrees of AR. And by that definition all AT angles apply the same amount of AR to the ball.

So Paul.. Is it possible to have a ball with 10 deg of AT and 0 degrees of AR? I think by your definition a ball with 10 degrees of AT has 10 degrees of AR as well. Am I correct in what I am saying?
If you observe the PAP on a ball that has an AT of 90 degrees, from that alone, you don't have information about what the AR could be. All of the different possible AR would look the same. That doesn't mean that AR has only one possible measurement.

Reverse the process. Instead of trying to figure the AR and AT from the PAP, select a AR and AT, then determine where that PAP would be.

Again, you will find that if AT = 90, all values of AR place the PAP at the top of the ball.

But if the AT is 89.9 degrees instead, all of the different AR values map to different locations on the ball.
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Re: So what happens when a ball burns up?

Post by Arkansas »

Which one of these pictures are correct? In the one on the wiki, Mo and Paul's position is correct. In the one from Blueprint, tilt can very clearly exceed rotation and still have the PAP on the ball. For example, a ball with 30* tilt and 20* rotation would have the PAP in the red circle. For the wiki to be correct, that PAP could not be represented.


MWhite,

I have a copy of the Bronze manual revised in 2010 and the chart in that manual matches the wiki. What's the date on the manual you pulled that from? I'm guessing it was an older one.
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Re: So what happens when a ball burns up?

Post by J_w73 »

MWhite wrote:
If you observe the PAP on a ball that has an AT of 90 degrees, from that alone, you don't have information about what the AR could be. All of the different possible AR would look the same. That doesn't mean that AR has only one possible measurement.

Reverse the process. Instead of trying to figure the AR and AT from the PAP, select a AR and AT, then determine where that PAP would be.

Again, you will find that if AT = 90, all values of AR place the PAP at the top of the ball.

But if the AT is 89.9 degrees instead, all of the different AR values map to different locations on the ball.
I agree with your example. I just wanted to make sure everyone understood where Paul and Mo were coming up with their argument. I agree only one argument is probably right but I didn't feel that anyone was understanding why Paul and Mo were saying what they were saying. I don't really have a side on this.. Well I do, I'm on the side of physics.
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