# Club Matching using Centripetal Force

Dave Tutelman -- July 28, 2005

This discussion started with a post on the FGI clubmaking forum (July 26,2005, "Discussion: new club matching theory"). The post was by Workshop (he insisted on anonymity, so I just attribute it to his "handle"), and it proposed matching clubs based on the centripetal force that the club exerts on the hands. From the reasoning in his post, that means the maximum centripetal force during the course of the swing. A reply by Brian Mack pointed out that he had suggested the same thing over a year earlier.

Since neither seemed to have the math right, I'm posting this article to show what the physics and math really look like. This does not mean that I endorse matching using centripetal force. While it is an interesting idea, I think that swingweight or MOI still have the inside track by a lot. For reasons I'll detail later, I don't believe that centripetal-force matching is the way to go; I expect it to be a failure.

• An "executive summary" of the results.
• An analysis of centripetal matching. In fact, three analyses with different levels of mathematical sophistication:
• A fairly rigorous derivation, using calculus and basic physical principles.
• A more intuitive derivation, which still results in the same fundamental result.
• Numerical examples based on measurements of actual clubs.
• The design of a workshop instrument, much like a swingweight scale, to aid in matching clubs for centripetal force.
• A description of the fitting and building procedure using centripetal matching.
• Why I don't believe in centripetal matching as an effective way to heft-match golf clubs.

## Results

I'm presenting the results before the analysis, so those of you who don't care about the physics and math don't have to wade through it. Here's what the analysis showed:

First and foremost -- Matching centripetal force is identical to matching clubs by their first moment.

So what is this? Most golf physics books look at three "moments" of the club's mass: the total weight, the first moment of the weight, and the moment of inertia or second moment of the weight.

This would seem like a strange result; why centripetal matching turn out to be first-moment matching? Because the equation for centripetal force comes out to be the first moment of the club times its angular velocity squared. (See the analysis.)  Since the angular velocity (not clubhead velocity, angular velocity) should be the same for all matched clubs swung by the same golfer, matching their first moments will match their centripetal forces.

This is a remarkably good thing for those who believe in matching by centripetal force. That's because it is very easy to measure the first moment of a club. You can do it with a special swingweight scale, where the fulcrum is at the butt of the club. Yes, you have to build the scale, but it's a really simple design. I show the design later in this article.

However, I don't believe that matching clubs by centripetal force is likely to result in a set of clubs that feel the same or swing the same. I started this investigation with an open mind. But the more I learn about centripetal matching, the more it seems to be at odds with most of the rest of what we know about golfers swinging golf clubs. The most damning piece of evidence is that the current trend in heft matching is moving from a swingweight match to a moment-of-inertia (MOI) match. This means going to relatively lighter long clubs and relatively heavier short clubs, compared with a swingweight-matched set. But centripetal matching is a big step in exactly the opposite direction: heavier long clubs and lighter short ones, and by a much bigger jump than the difference between swingweight and MOI. So I am not at all optimistic that centripetal matching has a future.

## Analysis and Discussion

Here's the details, for those who care. Some but not all of those details will involve math.

### Relation to the golf swing

There seems to be some confusion about the radius to use when computing centripetal force. I will argue that it is the length of the golf club. More precisely, the center of rotation for the calculation should be somewhere quite close to the butt of the club.

Here's the reasoning:
• We want to match the maximum centripetal force that is experienced during the swing. That occurs very close to impact, and it occurs at the point of maximum clubhead speed.
• If the arms and club were moving as a single unit, the clubhead would be traveling roughly twice as fast as the hands. That's because the hands would be halfway out on that unit. But that's not the way it happens; the clubhead is traveling more than 10 times as fast as the hands. In fact, strobe photos show the hands actually slow down as the clubhead achieves maximum velocity.
• So, unless we want to analyze a very complicated system (trust me, we don't), we need to consider the club rotating about the hands or wrists. I pick the wrists as the pivot point, because a good swing involves allowing the club to release from the wrists rather than powering it around with the hands. A couple of additional comments about this:
• The wrists are convenient for analysis, because we can view the butt of the club as the pivot point.
• If the hands are powering the swing -- and some golfers do swing this way -- it is not hard to show by physics that they need MOI-matched clubs. Centripetal matching is just wrong for them.

### Centripetal Match is the Same as First Moment Match

First of all, the centripetal force to turn an object of mass m, traveling at velocity v, into a curve of radius r is given by the equation
F = mv²/r
This works fine as long as the object can be approximated by a point mass. That is, its size is very small compared to the radius. For instance, a clubhead (height less than 3") swung about the wrists (radius more than 35") can be safely approximated as a point mass. But the shaft cannot. A point near the tip is going much faster, and with a much larger radius, than a point near the butt. So centripetal force to turn the shaft is more complicated to calculate. Let's see what's involved in the calculation.

We need to add up all the mv²/r components of all the infinitesimal points of mass that make up the whole club: shaft, grip, and clubhead. Yes, I know that the clubhead can be simplified, but let's start out treating the club as a whole. (Hint: it will work out no harder that way.)

The first simplification we can make is to convert the linear velocity to an angular velocity. Since we're dealing with a rotating object here, that turns out to be a better set of coordinates. So we convert to angular velocity ω, which is the velocity divided by the radius. So
v = ωr
Plugging that into our equation for centripetal force, we get
F = mω²r
That is the equation in "polar coordinates". Now we can use calculus to sum up all the infinitesimal points of mass dm. The integral is expressed
F = ω²r dm
Each particle has its own dm and its own r. But, since the whole club is rotating as a unit, every particle of mass has the same angular velocity. So we can take ω out of the integral.
F = ω² r dm
Now, let's see if you remember your college physics. The first moment of a point object is the mass times the radius. If the mass is not a point mass, then you have to add up a bunch of particles; each number added is the mass of the particle times the radius. Expressing that as an integral, it is
FirstMoment M1 = r dm
But that is exactly the same integral we have in the equation for centripetal force. So the centripetal force is just
F = ω² M1
Remember, capital M1 is the first moment, not the mass.

So the centripetal force depends only on the angular velocity and the first moment of the club. If we are matching a set of clubs for a particular golfer, and that golfer has a fairly consistent swing from club to club, then the angular velocity of the club at impact will be the same for all the clubs. (That is not to say the clubhead velocity will be the same from club to club. Each club has its own radius, and v = ωr, so the clubhead velocities will be different.) So, for that golfer, the centripetal force will be matched over the set if the first moment is matched over the set.

### Try again, without the calculus

Workshop had trouble accepting the result that centripetal-force matching is the same as first moment-matching. It's likely that the calculus, while essential for rigor, made the math less accessible. So here it is again, using his somewhat non-standard (but basically correct) original argument. Here it is...

In his original proposal, he defines "total head weight" THW as the weight the bottom of the club would exert on a scale if the club were held horizontally and the butt were supported. He asserts that THW is the mass to twirl on the end of a string to feel the same centripetal force. I agree! Of course, we require that the string is the length of the club and it is twirled at a speed equal to the golfer's clubhead speed at impact; I'm sure he intended that and just didn't say so.

So let's go over the calculations with his intuitively pleasing THW instead of the more precise integral calculus:
F = mv² / r
For the "twirl on a string" analogy, the mass is THW and the radius is club length L.
F = THW v² / L
You could match all the clubs by this formula, measuring the golfer's clubhead speed for each club. I believe that is what he was originally proposing. A bit tedious, but my early experiments with MOI matching involved even more calculation -- so I'm in no position to criticize.

But there is an easier way! Suppose you use a fairly well-accepted approximation: most golfers' clubhead speeds are proportional to club length. This is a good assumption if:
• The clubs are well matched to each other and to the golfer, and
• The clubs are all in a length range that the golfer swings well. (If they are outside this range, the golfer shouldn't be using them anyway. They are too long or short for the golfer.)
Stating the assumption mathematically:
v = K L   or    K = v/L
where K is some constant of proportionality. Every golfer has his/her own K; it isn't the same for all golfers. Let's plug this into the equation for centripetal force:
F = THW  K²  L² / L
=  THW  L  K²
But now look what we have! THW times L is exactly the first moment of the club.The first-moment scale I proposed may have measured it a little differently. But they're equivalent, and give the same number. Therefore:
F = M1 * K²
The centripetal force is equal to the first moment of the club times some number which is a characteristic of the golfer. The characteristic number K doesn't change from club to club for that golfer. Therefore, if you match all that golfer's clubs by first moment, you have also matched them according to the centripetal force that golfer will feel.

### Numerical examples

Lab report time! Luckily, I was one of the few in my graduating class that actually did the work for my lab reports, instead of copying them from the archive of previous year's reports. So I still how.

In order to make this more concrete, I measured a few real clubs and computed their first moments. (You wouldn't believe how many clubs I have around the basement and garage. On second thought, if you're reading this you're probably a clubmaker, so you would definitely believe.)

Workshop asked me for examples of a 37" seven-iron and a 44" driver. I was able to get really close to those numbers. I measured two drivers of approximately 44", because there are different lessons to be learned from them.
• One is a steel-shafted driver whose swingweight is pretty similar to the 7-iron. This gives some insight into the difference between swingweight matching and centripetal matching, and what we have to do to the club to go from swingweight to centripetal.
• The other is a graphite-shafted ladies' driver. It is considerably lighter, with a lower swingweight, and presents an extreme case for matching. Even so, we can make the match happen.
I measured the first moment of each club three different ways, each of which is valid and should give the same moment in inch-pounds:
1. THW times the distance from the butt to the bottom of the clubhead where it rests on the scale. This distance is longer than the nominal club length, because it goes diagonally to the bottom-center of the clubhead.
2. Total club weight times the distance from the butt to the balance point. (This was suggested on the forum by Kyle Walker, "kdubya".)
3. An approach similar to the way the special scale would measure. I supported the club above the butt, and measured the force with a digital fishing scale hooked to the shaft at the bottom of the grip, so the length of the grip was the lever arm. Then I multiplied the force by the lever arm.
Here are the results in tabular form:

 37" Seven-Iron 44.2" Driver 43.7" Driver Shaft Steel (Apollo Shadow) Steel (TT Dynamic Lite) Graphite (Mercury Performance) Swingweight D-1 D-1 C-8 THW 319 g = .703 lb 246 g = .542 lb 235 g = .518 lb Length (Butt to point where face meets scale) 38.2" 45.1" 44.7" Computed First Moment 26.85 inch-pounds 24.44 inch-pounds 23.15 inch-pounds Total club weight 420 g = .926 lb 357 g = .786 lb 320 g = .706 lb Balance point to butt 28.6" 30.9" 32.6" Computed First Moment 26.48 inch-pounds 24.28 inch-pounds 23.02 inch-pounds Force on scale 2.44 lb 2.36 lb 2.22 lb Lever arm 11" 10.3" 10.4" Computed First Moment 26.84 inch-pounds 24.31 inch-pounds 23.09 inch-pounds

So, what does the data tell us?
1. The three methods do give the same result. The differences among the methods were less than 1.5% for the seven-iron, and less than half of that for the drivers. That's about as close as one could expect. I trust the balance-point numbers the best. But if I had a well-made scale of the design in this article, I think I'd trust that even more.
2. The seven-iron is "heavier" in first moment than either driver. It is heavier than the steel-shafted driver by 2.2 inch-pounds, and heavier than the graphite-shafted by 3.5 inch-pounds.This is based on the balance-point method; the differences are not exactly the same for all three methods, but certainly close enough.
If we were doing centripetal matching, how could we get these clubs to have the same first moment? Here are several different approaches, all involving adding weight to the drivers. (We do this because it is much harder to remove weight from existing components. Conceivably, we could shorten a club instead of removing weight, but let's assume that the lengths are correct for the golfer and changing them would introduce a new fitting problem. So we'll add weight to the drivers.) First, we'll match the steel-shafted driver to the 7-iron, since it is a more likely scenario -- the clubs are already similar swingweights. The possibilities are:
• The one involving the least arithmetic is to add weight at the balance point of the driver. We would have to add 32 grams there. That's an awful lot of lead tape to wrap around a shaft. One advantage of this is that it does not change the balance point of the driver.
• We could add weight to the head, by adding a slurry of lead or tungsten powder in mouse glue -- or just add lead tape. This will take 23 grams -- which is a lot to add to a clubhead, but certainly not impossible. It will lower the balance point of the club.
• We could get a heavier shaft. This is probably impractical, since it will require an increase of almost 50 grams. The shaft is already steel, so you're not going to find a shaft that much heavier.
• We could combine the last two: you can probably find a shaft 10-15 grams heavier, and make up the difference adding weight to the head. Let's assume we can add a dozen grams in the shaft. Computation shows that you would need to add 17 grams to the head, which is better than 23 grams but not by enough to be worth it.
So the obvious way to do it is to add 23 grams to the head. A good clubmaker can do this effectively.

Now let's look at matching the graphite-shafted driver's centripetal force to that of the 7-iron:
• Let's add weight at the balance point of the driver. We would have to add 50 grams there. That's an awful lot of lead tape. Not practical.
• Adding weight to the head, we would need 36 grams -- which is an awful lot to add to a clubhead, but probably not impossible. It will lower the balance point of the club.
• We could get a heavier shaft. It will require an increase of 75 grams. Since the driver already has a 70-gram graphite shaft, this would require the new shaft to be 145 grams. So, even though we're starting with a light shaft, this is not practical.
• This time combining a lighter shaft with a heavier head works well. Switch to a steel shaft (adding about 50 grams to the shaft), and make up the difference adding weight to the head. Computation shows that you would need to add 12 grams to the head, which is very manageable. That would raise the balance point of the club a bit, but not nearly as much as adding all the weight in the shaft.

So there are several options for bringing the driver up to the same first moment as the irons. I computed the amount of weight needed. But, if you had a first-moment scale in your shop, you could find the weights experimentally pretty easily -- just as you do today with swingweight.

Here's an observation: A set matched for centripetal force will have a much heavier swingweight in the longer clubs than the shorter ones. Some quick calculations suggest that, in a set of centripetally-matched irons, each longer club would pick up more than a swingweight point on the previous one. So they will certainly feel very different across the set than a swingweight-matched set. And they will feel even more different from an MOI-matched set, where the shorter clubs have the higher swingweights.

## Measuring for Centripetal Matching

Remember that a swingweight scale is just a device for measuring the first moment of the mass of the club, about an axis 14" from the butt of the club. The first moment that we want for centripetal matching is about the axis that the club is swung, which is roughly at the butt. So what we need is a "swingweight" scale, but with a fulcrum at the butt rather than 14" from the butt. Here's a simple way to build one.

Build a simple frame (the black piece in the picture) to hold the club. It looks like the frame of a swingweight scale, because it does the same function. The differences are:
• The pivot, or fulcrum, is directly under the butt of the club rather than 14" from the butt. It is attached to the "real world", in the form of a heavy, solid base.
• There is a point projecting out the bottom of the frame to make contact with the tray of a digital scale. The scale must have a "zero/tare" button, to zero the reading with no club in the frame.
The distance from the point to the center of the pivot is the lever arm of the moment. The reading on the scale is the force. So the moment itself is the product of the lever arm and the force. Place the empty frame in position on the digital scale, and zero the meter. Then place the club in the frame and read the force.

If you are measuring to match a single set, you can just match the force. After all, the lever arm on your scale isn't going to change. But, if centripetal matching becomes popular enough to talk about "the number" of a club (like D-1 for swingweight matching) it is probably a good idea to standardize on a lever arm length. That way, "the number" can be just the force, with no arithmetic necessary.

## A Procedure for Centripetal Clubfitting

By now, we should see a procedure emerging for heft-fitting the golfer for centripetally-matched clubs:
• Find a club with a heft that the golfer hits well. Just as with any other heft-matching scheme (swingweight, MOI), finding that first club is an art. Most clubmakers start with the golfer's "favorite club" and work from there.
• Once you've found that club, measure its first moment on the scale, or compute it by any of the methods above. (The easiest and most accurate is probably total weight times balance point, if you don't have a first-moment scale.) It may be 30 inch-pounds for a strong golfer or it may be 20 inch-pounds for a golfer with less strength. (I think those ranges are about right, but the real experimental work is yet to be done.)
Workshop has suggested an alternative, assuming centripetal "theory" can be as simple in practice as it is in theory. (We scientists have a saying, "The difference between theory and practice is bigger in practice than in theory.")
• Measure the golfer's clubhead speed v with some club, and note the length of the club L. You might want to try it with a few clubs of different lengths, to be confident that this golfer is "normal", in that he/she has a clubhead speed roughly proportional to club length.
• Measure the golfer's strength (probably hand strength) to determine the ideal centripetal force F for that golfer. The existence of such a mapping is part of the speculation in Workshop's proposal. But let's assume the mapping can be found by experiment, and that it proves robust across golfers. (I'm skeptical that there is such a mapping. But I'll stipulate that there is, and see where it leads us.)
• Now you can compute the first moment you will need for all the clubs in the set. The equation (without bothering figuring out the unit conversion factors) is:
FirstMoment M1 = F / { (v/L)² }
Whichever way you determine the first moment for the golfer's clubs, the rest of the procedure goes:
• Build the set so all the clubs have the first moment that you have determined is right for the golfer.
• Now you can be confident that all the clubs will impart the same "ideal" centripetal force to the golfer's hands.

### Special case: non-proportional clubhead speeds

Workshop brought up the point that some golfers may not have clubhead speeds proportional to club length. While that's going to be a remarkably small portion of the golfing population, it is certainly a possibility. By the way, that sort of anomaly would cast the same doubt on things like frequency matching and even the "standard" loft progression. The entire club design paradigm is based on proportional clubhead speeds.

But let's deal with the possibility. We still have a fairly easy way to build a centripetally matched set. Here's the procedure:
• Measure the clubhead speed v and length L with all the clubs in the bag. (Note that, if you don't do this, you'll never know whether the golfer has proportional clubhead speed. And I know of no clubfitter that does this for every club in the bag. But still...) Call the speed and length with each club vi and Li, where i is an index identifying the club.
• As before, decide on the centripetal force F that will be used to match all the clubs.
• Calculate the first moment M1i of each club individually, using the formula (again, without bothering with the unit conversion factor):
M1i  =  F / { (vi/Li)² }
• Build the set to the required first moments, club by club. The first-moment scale described below will ease the job considerably in the shop.

## Why I Don't Believe in Centripetal Matching

Let me finish by explaining why I'm skeptical about the usefulness of matching by centripetal force? As I said in the intro, I started this investigation without preconceptions -- prepared to believe in centripetal matching if the science supports it.

Now that we know that it is the same as first-moment matching, and can be measured by a swingweight scale with a fulcrum at the butt, we understand a little more about what it means to centripetally match a set of clubs. In particular, we have some historical precedents of various ways to match clubs over the years, and know how those worked -- or didn't work:
1. According to Cochran & Stobbs' classic book, "The Search for the Perfect Swing", swingweight is a rather arbitrary compromise between the extremes of MOI matching and first-moment matching. That's certainly a valid way to look at it. And, because Cochran & Stobbs toss it off pretty casually, we may assume that first-moment matching had been tried and found wanting. (Remember, the book is over 30 years old.) Which brings us to...
2. More than 50 years ago, Kenneth Smith -- the premier maker of swingweight scales at the time -- offered two models: a 14" fulcrum and a 12" fulcrum. He was arguing that the 12" fulcrum gave the most generally useful match, and he pushed that model. The market didn't agree; most clubfitters bought the 14" model, and eventually the 12" model went out of production. You might find it interesting that MOI matching, which is currently increasing in popularity, can be approximated by a swingweight scale with a still longer fulcrum than 14".
Why is this relevant? Because centripetal matching corresponds to a 0" model of the swingweight scale (that is, the fulcrum is zero inches from the butt of the club). If going from 14" to 12" was unhelpful, it's unlikely IMHO that going still further to 0" will be helpful. I could be wrong here; matching may be sufficiently nonlinear that going all the way to 0" might do something good. That's why I didn't label the idea as outright hogwash. But my money is on it not being the right answer.
3. A growing segment of the clubfitting community is beginning to favor MOI matching over swingweight matching. That suggests that moving from swingweight in the direction of MOI is probably a good thing. A centripetally-matched set is far far from swingweight-matched, and even further from MOI-matched. Over a standard set of eight irons, 3-PW, with a half-inch length increment, the 3-iron will be about 9 swingweight points heavier than the PW. So if swingweight or MOI matching is anywhere near the right way to do it, then centripetal matching is not. And we have about 100 years experience matching clubs with swingweight and/or MOI, so I doubt it's all that far off.
All these precedents suggest that centripetal matching is the wrong direction to go for heft matching. The current state of the art is swingweight matching. Over the past century, successful deviations from swingweight matching have been in the direction of lighter long clubs (e.g.- MOI matching); unsuccessful deviations have been in the direction of heavier long clubs (e.g.- the 12" fulcrum on a swingweight scale). And centripetal matching is the latter; moreover, it's a pretty extreme case of the latter.

### A speculative follow-up

On August 5, 2005, Workshop followed up with the speculation:
"If the theory presented here has some validity, that validity will be shown by the speed gap between clubs as being greater than what Dave T’s present work shows. This is suggested by the methods described above so that the speed gap is more along the lines of 2.5 – 3.0 miles per inch of club length, and this NOT equally proportional, as indicated in my chart of Dave T’s clubs in an earlier post. In that chart the speed gap would be very proportional (approx .85) This is not the case in this theory and it’s the important point of departure to discuss further. In addition, speed of a particular club is the only factor that significantly helps with distance (approx. 50 more grams of mass at club head needed at the same speed to gain 10 yards—not a practical factor.) So the speed gap between clubs for a range of unique players is at the core of the difference of this theory.

"By hefting properly to improve the speed gap between clubs for a player, the calculations for centripetal force vary from what others have presented here. I believe that variance is the significant factor."
In other words, if you suitably match the clubs then golfers will not exhibit constant angular velocity, but instead will adjust their clubhead speed to equalize the centripetal force.

This speculation appears to be, as H. L. Mencken said, "The new logic: It would be nice if it worked. Ergo, it will work." Workshop's statement would be nice if it worked, but I haven't seen one iota of reasoning nor data to suggest that it will. Still, giving the devil his due, let's see if we can test it with data that's already lying around.

Below I will show that, if centripetal matching could give a 2.89mph per inch speed increment (well within the range
Workshop speculates), then a swingweight-matched set would also be centripetally matched. But we already know that a swingweight-matched set gives a much smaller speed increment, and therefore is not centripetally matched, for the vast majority of golfers. So the speculation has to be wrong.

Consider the following chain of reasoning:
1. Let's consider a 44.2" driver and a 37" 7-iron, both with steel shafts. These are two of the clubs we measured for our "lab report".
2. Workshop suggests that centripetal matching could result in a speed increase of 2.5-3.0 mph per inch of club length.
3. Consider our "lab report" clubs. Suppose we take a 100mph driver speed, and a clubhead speed proportional to club length -- the usual clubfitting assumption that Workshop hopes to obsolete with centripetal fitting. Then the 7-iron would have an 84mph clubhead speed. That's a 16mph difference for a 7.2" length, or a speed increase of 2.2 mph per inch of clubhead length. (For middle-range male golfers, this proportional speed number is about 2.0 to 2.2 mph per inch.)
4. When we made the proportional-speed assumption (like 2.2 mph per inch) and centripetally matched the clubs, we found we needed a driver with a much heavier head weight than would be needed for a swingweight match. (Remember, we started with a swingweight-matched driver and 7-iron, and we had to add a lot of head weight to the driver to get it to centripetally match the 7-iron.)
5. Suppose we could achieve the 2.5-3.0 mph per inch in the speculation. Let's take an exmple from the upper end of the suggested range -- 2.89 mph per inch -- as a target number. (No, that was not a random selection; I worked backwards from an interesting result to get it.)
6. Let's ask ourselves, "What sort of weight modification would we have to do to the measured clubs, to centripetally match them under Workshop's speculation?"
• 2.89 mph per inch times 7.2" gives a speed difference of 20.8 mph.
• Let's make the (most favorable) assumption that we haven't hurt the 7-iron performance, but rather improved the driver performance. We'll justify that by keeping the same 7-iron and making whatever changes we need to make in the driver.
• So the driver speed is now 104.8mph.
• From our earlier measurements, the first moment of the 7-iron is  26.48 inch-pounds. What first moment do we need to make a 44" driver, so that at 104.8mph it has the same centripetal force that the 7-iron had at 84mph?
• I did the calculations, and found that the driver would need a moment of 24.28 inch-pounds.
7. This is a rather remarkable result. Why? Because that's exactly the same moment as the original driver we measured, before we started adding weight to the head. And that driver had the same swingweight as the 7-iron.
8. Therefore, if the speculation is correct, a swingweight-matched set should give a speed increment of 2.89 mph per inch. (Why? Because if the speed increment were 2.89 mph per inch, the fairly typical swingweight-matched clubs that we measured would give a centripetal match.)
9. But we already know that a swingweight-matched set gives only 2.2 mph per inch, in the speed range we're talking about. (Why? Because pretty much everybody already uses clubs that are swingweight matched or close to it, and the 85% rule holds for the vast majority of that population.)
10. So Workshop's speculation is wrong, because it leads to a conclusion contrary to observed fact!
When I pointed this out on the FGI forum, Workshop responded that he would go ahead with experiments anyway, to test his speculation. I asked a golf buddy about the experimental design necessary to test such an assertion. (He has a PhD in experimental psychology and decades of experience in subjective experiments; he is a legitimate subject-matter expert.) He said that, in order to get statistically significant results, he would have to measure far more swings -- preferably with more golfers -- than Workshop intends to. Among the problems involved are:
• Variability for a given human subject.
• Measurement tolerances.
• The rather small velocity difference involved. (Workshop and I agree on the velocity differences involved. The test must reliably distinguish a 4mph clubhead speed, in order to distinguish constant-angular-velocity from centripetally-adjusted.)
Given these difficulties with subjective experimentation, I "ran the experiment" on a computer program, which has no human variation. This program, SwingPerfect, starts with the characteristics of a club and the forces imposed by the swing of the golfer, and comes up with clubhead speed, wrist angle at impact, and a few other output parameters. Think of it as a "virtual robot" for club testing. Here's what I learned:
• If speed were proportional to length, then the 7I speed would be 84% of the driver speed. (I knew that before.)
• With the characteristics of the test clubs (the same clubs we have been talking about) and the same swing forces for both clubs, the ratio was 86%.
• But, for the clubs to match centripetally under Workshop's assumption, you would need an 80% ratio -- that's in the other direction altogether.
These results were not very sensitive to assumptions. I varied a bunch of things over a reasonable range, and the ratio remained between 85.5% and 86.5%. This indicates that the golfer would have to make some pretty substantial swing changes in order to force the clubs to give a centripetal match.

So the speculation is very hard to believe. More and more like Mencken's "new logic".

Workshop
did some informal experiments anyway, and came up with results closer to constant-angular-velocity clubhead speeds than to speeds that the golfer was automatically adjusting to give a centripetal-force match. He seems undeterred by the results of his own experiments, nor by my analysis which predicted those results. He continues to search for a way to rationalize centripetal-force matching.

I have dropped out of the discussion.