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 centripetalforce matching is the way to
go; I expect it to be a failure.
This article covers:
 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 heftmatch 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 firstmoment 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
momentofinertia (MOI) match. This means going to relatively lighter
long clubs and relatively heavier short clubs, compared with a
swingweightmatched 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
MOImatched 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 M_{1}
= ∫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
M_{1}
Remember, capital M_{1} 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 centripetalforce matching is the same as
first momentmatching. It's likely that the calculus, while essential
for rigor, made the math less accessible. So here it is again, using
his somewhat
nonstandard (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 wellaccepted
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
firstmoment scale I proposed may have measured it a little
differently. But they're equivalent, and give the same number.
Therefore: F = M_{1}
* 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" seveniron 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 steelshafted driver whose swingweight
is pretty similar
to the 7iron. 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 graphiteshafted 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 inchpounds:
 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 bottomcenter of
the clubhead.
 Total
club weight times the distance from the butt to the balance point.
(This was suggested on the forum by Kyle Walker, "kdubya".)
 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"
SevenIron  44.2"
Driver 
43.7"
Driver 
Shaft
 Steel
(Apollo Shadow)  Steel
(TT Dynamic Lite)  Graphite
(Mercury Performance)  Swingweight
 D1  D1
 C8 
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 inchpounds
 24.44
inchpounds 
23.15
inchpounds 
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 inchpounds
 24.28
inchpounds 
23.02
inchpounds 
Force on scale  2.44 lb  2.36 lb 
2.22 lb 
Lever arm 
11"  10.3" 
10.4" 
Computed
First Moment  26.84 inchpounds
 24.31
inchpounds 
23.09
inchpounds 
So, what does the data tell us?
 The three methods do give the same
result. The differences
among the methods were less than 1.5% for the seveniron, and less than
half of that for the drivers. That's about as close as one could
expect.
I trust the balancepoint numbers the best. But if I had a wellmade
scale of the design in this article, I think I'd trust that even more.
 The seveniron is "heavier" in first moment than
either driver. It is heavier than the steelshafted driver by
2.2 inchpounds, and heavier than the graphiteshafted by 3.5
inchpounds.This
is based on the balancepoint 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 steelshafted driver to the 7iron, 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
1015 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 graphiteshafted driver's
centripetal force to that of the 7iron:
 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
70gram 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 firstmoment 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 centripetallymatched
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 swingweightmatched set. And they will feel even more
different from an MOImatched 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 D1 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 heftfitting
the golfer for centripetallymatched clubs:

Find a club with a heft that the golfer hits well. Just as with any
other
heftmatching 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
firstmoment scale.) It may be 30 inchpounds for a strong golfer
or it may be 20 inchpounds 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 M_{1}
= 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: nonproportional 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 v_{i}
and L_{i}, 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 M_{1i} of each club
individually, using the formula (again, without bothering with the unit
conversion factor):
M_{1i
= }F / { (v_{i}/L_{i})²
}
 Build the set to the required
first moments, club by club. The
firstmoment 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
firstmoment 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:
 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 firstmoment matching. That's certainly a valid way to
look at it. And, because Cochran & Stobbs toss it off pretty
casually, we may assume that firstmoment matching had been tried and
found wanting. (Remember, the book is over 30 years old.) Which brings
us to...
 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.  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
centripetallymatched set is far far from
swingweightmatched, and even
further from MOImatched. Over a standard set of eight irons,
3PW,
with a halfinch length increment, the 3iron 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 followup
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 swingweightmatched set would also be centripetally matched. But
we already know that a swingweightmatched 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:
 Let's consider a 44.2"
driver and a 37" 7iron, both with
steel shafts. These are two of the clubs we measured for our "lab
report".
 Workshop
suggests that centripetal matching could result in a speed increase of
2.53.0 mph per inch of club length.
 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 7iron 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 middlerange male golfers,
this proportional speed number is about 2.0 to 2.2 mph per inch.)
 When we made the proportionalspeed 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 swingweightmatched driver and 7iron, and we had to add
a lot of head weight to the driver to get it to centripetally match the
7iron.)
 Suppose we could achieve the 2.53.0 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.)
 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 7iron performance, but rather improved the driver performance.
We'll justify that by keeping the same 7iron 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 7iron
is 26.48 inchpounds. What first moment do we need to make a
44"
driver, so that at 104.8mph it has the same centripetal force that the
7iron had at 84mph?
 I did the calculations, and
found that the driver would need a moment of 24.28 inchpounds.
 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 7iron.
 Therefore, if the
speculation
is correct, a swingweightmatched 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 swingweightmatched clubs that we measured
would give a centripetal match.)
 But we already
know that a swingweightmatched 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.)
 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 subjectmatter 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
constantangularvelocity from centripetallyadjusted.)
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 constantangularvelocity clubhead speeds than
to speeds that the golfer was automatically adjusting to give a
centripetalforce 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 centripetalforce matching.
I have dropped out of the discussion.
Last
modified August 20, 2005
