The five minute university bringin in the sheaves...

Applying Physic to Go

Now let's start to apply some of the physics we just learned. We'll look at what happens during the golf swing, during impact, and during the ball's flight. Nowhere near everything, of course. We will focus on some issues which are important for clubfitting, especially:

Where does the power for the swing come from? The answer will probably surprise you.

What happens during impact, and how does it turn clubhead design and path into launch conditions? This is the bottom line, turning a swing into a golf shot.

What determines the ball's flight? How do the launch conditions (ball speed, launch angle, and spin) translate into distance and direction.

Once we have these fundamentals under our belts, let's look critically at a few myths that have perpetuated themselves in golf lore.

What powers the swing?

Many golfers, especially in the United States, grew up with sports in which a ball is thrown or hit. Baseball, American football, basketball, or hockey are usually mastered -- well, at least played sort of competently -- before golf is attempted. And those sports generally involve powering the ball with the hands and arms. Yes, the body can play a part in adding to the power. But that part is generally getting the whole mass of the body moving in the direction you want the ball to travel.

The golf swing is different, not just in degree but in principle. Golfers who grew up hitting things with a bat or a hockey stick have developed swing habits that are counterproductive in golf. (Well, the very best at those sports may have incorporated the important elements of a golf swing. But the average Sunday athlete has not.)

Most of the power in a golf swing comes from centrifugal force, generated by the muscles that rotate the body through the swing. Before explaining it further, let's look at the physics of the golf swing. After we see the forces at work in a good golf swing, we'll go back and see what sort of bad habits most golfers carry over from their other sports. Finally, since you're probably going to be skeptical about this -- it really is counterintuitive, and you should be skeptical -- we'll review how we know this to be true.

The double pendulum

When an engineer sets about analyzing a real-world system -- like a golf swing -- he creates a physical "model" of the system. This is a set of elements that are simple enough to yield to calculations, yet complex enough to represent what is actually going on. Finding the right model -- the right balance between simplicity and complexity -- is the first and often the hardest step in engineering analysis.

The simplest model that makes any sense at all for the golf swing is a double pendulum. The two members of the pendulum are:

The golfer's shoulders and arms, taken as a single rigid unit. That's the green triangle in the diagram. We'll call that "the triangle" in the discussion that follows.

The golf club, also taken as a single rigid unit.

The triangle is hinged to the golfer's body (the tan elements in the diagram) so it can turn. Similarly, the golf club is hinged to the other end of the triangle.

This is a very simple model, having only two moving elements hinged together. To see just how simple, let's re-draw it the way an engineer would: as a collection of free, hinged bodies. Now we can see why the model is a double pendulum; it is a black pendulum (the club) hanging from the end of a green pendulum (representing the triangle). While the diagram looks different from the golfer above, it works exactly the same when it comes to physics.

Given the simplicity of the model, it's pretty amazing how close it can get to the actual measured performance of a golfer's swing. True, there are a lot of nuances of the swing that it doesn't capture. But experience has shown it is rich enough to explain where the clubhead speed comes from in a good swing.

Let's look at the next question about using the model. We have two hinges, and we can apply a torque at each of those hinges. Those two torques -- plus gravity -- are the only forces in this model that will cause the golfer to swing the club.

So the engineering model has to say what kind of torque:

The body applies to the shoulders to turn the triangle.

The hands and wrists apply to the club to uncock it and bring it to impact.

It turns out that the torque the body applies to the triangle is considerable, but a good swing applies almost no torque to the grip of the clubs by the hands. Yet more than half of the clubhead speed comes from the club turning about the hands at the bottom of the swing -- much more than could be explained simply by shoulder turn. What is creating that very strong rotation of the club about the hands, if the hands are not being used to supply a "hit" force?

The answer is centrifugal force.[1] Remember that a body in motion wants to keep moving in a straight line. But the golfer is pulling the club around in a circle. According to Newton, the club wants to fly outward from the circle; the force that is trying to pull it out straight with the arms is centrifugal force. That centrifugal force is generated by pulling the club in a circle around the shoulder hinge, and the force wants to pull the club straight out along a radius from that hinge.

How big is that centrifugal force? Let's look again at the formula:

m v2

F = ------

The mass m is a property of the golf club, and the radius r is a combination of the extended arms and the wrist-cock angle. The more acute the wrist-cock angle, the closer the club is to the shoulder hinge -- and thus the smaller the radius. As for velocity v, it increases as the body torque accelerates the triangle.

So, what does the golfer have to do to get maximum centrifugal force in order to get maximum clubhead speed? His job is to "hold the lag" -- keep the club cocked at a right angle to the arms -- until fairly late in the downswing. This keeps from releasing the club until v is nearly as large as it's going to get, which allows a large F to accelerate the club outward and downward just before impact. This, and not torque applied by the hands, is the way to reliable high clubhead speed.

(It is worth noting there is criticism of centrifugal force as the mechanism of the golf swing. More on that here.)

If you spent years swinging a baseball bat before you started golf, this probably defies your understanding of power. A baseball slugger has strong forearms and wrists, the better to whip the bat through the ball. Yes, body rotation is also important, but the hands are active, while for this model of the golf swing the hands are passive.

On top of that, you never hear this from the pros. Instead, you hear stuff about clearing the hips, or keeping your right elbow "tucked" instead of "flying". If this is true, why don't the pros teach it?

So you're probably skeptical that this is how the golf swing works. And you should be! But it really does work this way. First, let me address why the pros don't teach it; then I'll spend some time on how we know it's true.

OK, why don't the pros teach it?

They don't know it! Very few teaching pros or TV commentators have a clue about physics. Terms like "centrifugal force" and "moment of inertia" are buzzwords they throw around without a clue what they really mean. Please don't get me started on this; it's a pet peeve and I could go on all day.

Actually, there are a few pros who do understand, but they are indeed few and far between -- and often skeptical. Jack Nicklaus writes about an incident in 1972, when an expert in golf physics told him about this. He wasn't at all sure he believed it...

"But his theory seems to explain a shot I hit at the par-3 fifteenth in the second round at Firestone. The choice of club lay between a two-iron and a three-iron, and I decided to go with an easy two-iron. Coming into the ball I was deliberately 'soft' with my hands. I've never hit a better two-iron in my life! The ball finished over the green.

"Maybe this explains what happens on those good drives where I have a 'soft' feeling in my hands through the ball... My hands merely went along for the ride."

In a sense, they do teach it -- without knowing or believing it. The two examples above are not contradictory to the way the swing works:

"Clear the hips" creates body rotation with the large muscles, which causes rotation of "the triangle". Anything that increases torque on the triangle will contribute to power.

Tucking the right elbow has little to do with power. That move's purpose is to control the swing plane. When done right, it assures that the ball will go in the direction of the target -- and none of the analysis above deals with accuracy at all, just power.

Now let's look at the reasoning why we should believe it works this way.

Jorgensen's study

Theodore Jorgensen set about to find a physical model that would match the behavior of a golfer with a good, "classic" swing. Here is how he went about it:

He found a golfer with the sort of swing he needed.

He outfitted the golfer with reflective dots on his joints, as well as on his golf club's grip, shaft, and head.

He took a sequence of strobe pictures of the swing, with the reflective dots indicating exactly where all the important parts of body and club were at every moment. At this point, he had a completely instrumented swing, and could compute velocities and accelerations of club parts, body parts, wrist cock angles, etc.

He started his mathematical modeling with a simple double pendulum, and fiddled with the torques until the model gave the same swing as the golfer did.

He couldn't do it with the simple double pendulum, so he added complexity a little at a time until he had an exact match between the mathematical model and the golfer's swing.

So Jorgensen's model isn't quite a simple double pendulum. The figure shows the changes he had to make to the simple model in order to get it to behave exactly like the real golfer. He has to insert a right-angle "stop" so that the wrist-cock never exceeds 90º. And he had to put a little "sway" into the golfer -- a small forward motion of the shoulder hinge during the downswing.

But the important change he did not have to make was to add any wrist torque to release the club at the bottom of the swing. That is accomplished completely by centrifugal force. In fact, once he had a mathematical model that behaved like the golf swing, he ran some "what if" analyses to see whether application of wrist torque could add to power. He found that there is a critical time about 70-100 milliseconds before impact (where the arms are 60º before the impact position) where torque changes from hurting clubhead speed to helping it. That is, any uncocking wrist torque before the critical time will reduce clubhead speed at impact. You can indeed increase clubhead speed a bit by applying wrist torque, but only if you can do it for just the last 70 milliseconds before impact, and not before. It takes a very well-coordinated athlete to get away with this. (If you are interested in more detail about this, I have worked it to death in another article. In particular, the difficulty of applying torque during those last 70 milliseconds is discussed here.)

Interestingly, Jorgensen found that that the same critical time works the other way as well. If you use negative torque (that is, use strength in the wrist to prevent uncocking) early in the swing and then release it 100 milliseconds before impact, you will increase the clubhead speed. In fact, you'll get as much increase in clubhead speed as that well-coordinated athlete would have gotten by a late application of positive torque. And it's much easier to hold off release than to apply a release-aiding torque at exactly the right time.

So Jorgensen's study confirms the notion that power in a golf swing -- clubhead speed -- is a product of centrifugal force and not wrist torque. He adds a lot of detail, but nothing that denies that basic truth.

Muscle energy

If classical scientific-method physics (Jorgensen's approach) doesn't satisfy you, how about biology and physiology?

Coming into impact, a golf club's kinetic energy is based on its mass and speed. It gets there from zero kinetic energy during the time of the downswing, less than half a second. This implies that the muscles have to put out a certain amount of power for half a second. Physiologists know how much power a muscle can provide for a short burst (say, half a second).

When this fairly simple calculation is cranked through, the answer is that over 30 pounds of muscle mass is needed to impart that energy to the golf club. This is muscle that is engaged in generating motion, and does not include muscle used to stabilize the body in the golf swing posture. The 30-pound number has come up consistently in quite a few separate studies aimed at this question.

There isn't anywhere near that much muscle in the forearms, hands, and wrists, so they can't be the major driving force of the swing. You need the big muscles -- the legs, thighs, torso, and shoulders -- to create that much power. That verifies that the clubhead's energy comes from body rotation, not hand torque. But it doesn't unambiguously point to centrifugal force as the enabler.

But we should be able to compute the clubhead speed that would result if we only used body rotation and not centrifugal force. Without any velocity at impact from uncocking the wrist, just from body rotation, we get only about two thirds the clubhead speed that a good swing actually accomplishes. So we need centrifugal force because:

We know the bulk of the power comes from body rotation.

We know that body rotation without wrist-uncock velocity gives a third less clubhead speed.

In order for body rotation to generate wrist-uncock velocity, we need centrifugal force -- because the small muscles in the hands and forearms can't generate that much power.

Trebuchet

Still skeptical? Don't believe physics or biology? How about history...

A few years ago (probably 2004), I was watching a show about Siege Engines on The History Channel, and had a "Eureka" moment. They were talking about the Trebuchet, a rock-hurling device that was invented about 1200AD. It replaced the catapult over only a few decades, because it had more range for a smaller and lighter device. (Still big and heavy to be sure, but definitely more efficient than what came before it.)

My Eureka was because, watching it, I saw an upside-down golf swing. The principle of a double pendulum driven by centrifugal force was right there, and history has proven it very effective. For a description of how a trebuchet works, see the page and the animation I clipped from The Trebuchet Store. (They sell trebuchet kits and the like, in case you find this stuff interesting for its own sake, not just what it teaches about the golf swing.) In short, the inner arm of the pendulum (corresponding to the triangle) is a rigid, pivoting structure, but the outer arm (corresponding to the golf club) is literally a string. You couldn't apply "wrist torque" to it if you wanted to -- it must operate by centrifugal force.

Now one of the most interesting thing about this design is that it has never been significantly improved upon! It has been around for eight hundred years, and it is still the most efficient catapult known. Of course, catapults are no longer used for sieges; cannons and gunpowder took over a few hundred years after the trebuchet's introduction. But:

For those few hundred years, the trebuchet remained king of the siege engines.

Even today, there are catapult-engineering contests; they call them "punkin chunkin'" and obviously they hurl pumpkins -- and bowling balls and other large objects up to and including major kitchen appliances. But the design that still dominates this "sport" -- even in our engineering-knowledgeable age -- is the trebuchet.

Since the equations of motion for the trebuchet are basically the same as the zero-wrist-torque golf swing, we can rest assured that the centrifugally-driven golf swing is very effective indeed.

Hitters and swingers

I'd like to take this opportunity to state very specifically what I mean later in these notes by the terms hitter and swinger. Most clubfitters and many instructors make this distinction, but it tends to be intuitive and imprecise. I believe that:

A swinger is a golfer who depends exclusively on centrifugal force for clubhead speed, and adds no wrist torque during the downswing except that needed to hold a 90º wrist cock.

A hitter is a golfer who depends to some extent on torque applied to the club's grip via the hands and the wrists.

Of course, there are few pure swingers and no pure hitters. But, comparing two golfers, we now have a way to say which one is more of a hitter and which more of a swinger. And, in fact, we can tell from this whether a golfer is primarily a hitter or a swinger.

Notes:

There will be criticism of this simplified explanation, and it is justified. A more complete description would be that the clubhead gets its speed from inertial forces. In particular, most clubhead speed comes from the rotation of the club about the hands, which is powered by forces from the hands that don't go through the CG of the club. There are books and lots of scientific papers that discuss this in more detail. I hope to write an article in the future that explains it for the non-physicist. In the meantime, I feel justified in invoking centrifugal force in this explanation for the physics beginner, even though it is oversimplified and incomplete.

Last modified May 7, 2020

Heft of the club: Swingweight & MOI

What Is It, And Why Do We Care?

Swingweight is an attempt to reflect a couple of important properties of the club:

Feel

Performance

Together, these constitute the "heft" of the club.

Q. Now that doesn't sound so hard. Why not just make all the clubs the same weight?
A. Well, you can do that and still have major differences in the balance point. Total weight alone doesn't reflect what the golfer feels nor what the club does.
Q. OK, then. Make all the clubs with the same head weight, and the same shaft weight, and the same grip weight. That should do it.
A. Well, it would if all clubs were the same length. The real problem is that there is a length progression across the set. As you make the club longer, it will feel more head-heavy and will release differently.
Q. Oh! Well then, how about making the head lighter as the club gets longer.
A. Exactly! But how much lighter? That is the subject of this section...

Swingweight is an attempt to quantify the heft of a golf club, as it affects both the feel and the physics. Let me emphasize the word attempt! Swingweight is not a magic quantity that can be shown by physical laws to have anything whatsoever to do with the things it would like to measure. It is, rather, an empirical approximation with an interesting history over a century old. Because it is so important to understand that swingweight doesn't represent anything really fundamental, let's start by reviewing that history.

Once upon a time...

History

First, a big thank you to D.B. Miko of Mac Shack Golf, for providing me with a number of references on this subject.

By the early 1900s, clubmakers for professional golfers were already using mathematical formulas for matching their pros' clubs for heft across the set. The formula they used was to match the product of the head weight and the square of the length. Thus the longer the club, the lighter the head had to be, by about twice the percentage increase in length.

Let's take an example. Consider a 5-iron of the time: 37.5" long with a 255g head. If we made the club 1" longer (that's 2.7% longer), we would need to make the head 14g lighter (that's twice 2.7% of 255g). So, for each inch longer or shorter in the vicinity of a 5-iron, we'd need to vary the head weight by 14g. Since, for a normal set of irons, the club spacing is a half inch, the progression of head weight from each club to the next is a half of 14g, or 7 grams. Does that sound familiar? Now you know where it comes from.

At that time, clubs were made with hickory shafts and wound leather grips. There wasn't much you could do to change their weights, once the length and stiffness were determined. So none of the matching formulas included shaft or grip weight, because for all practical purposes they were fixed.

Now, think about the physics of such matching. Back in the chapter on Physics: moment of inertia, we saw that "the moment of inertia of each grain of mass is its mass times the square of the distance to the axis." If the variation in shafts and grips could be neglected, then this formula made sure that the set matched for all clubs' moments of inertia about their butts. The clubmakers of a century ago were building moment-of-inertia matched sets. (Well, almost. See note [1] below.)

Now, at that point, there was no such thing as swingweight; neither the measure nor the word was invented yet. But something was needed because the math was a little tedious. Remember, no electronic calculators or computers back then.

In the early 1930s, a clubmaker named Robert Adams invented the swingweight scale. It was a balance that measured the amount of torque the weight of the club exerted about a pivoting fulcrum. The diagram shows a modern swingweight scale taken from the 2006 Golfsmith catalog, but it is basically the same instrument that Adams used alnost 80 years earlier. The weight of the club exerts a counterclockwise torque on the beam, because the center of gravity of the club is to the left of the fulcrum. The clubfitter moves the sliding weight until its clockwise torque balances the torque from the weight of the club. The position of the sliding weight then gives the "swingweight" of the club. Notice from the picture below (a scan from Adams' original patent), how little the design has changed over three quarters of a century.

After much experimenting, Adams concluded that a fulcrum 14" from the butt seemed to give the "best" match, in a subjective sense, for the pros he worked for. Why 14"? Did that correspond to some sort of "pivot point" in the golfer's swing? No, it was just a number that seemed to work; it yielded a set of clubs that Adams' clients felt were well matched. (As we shall see later, this is not a perfect match to moment of inertia, but it's not a bad match at all. So Adams-matched clubs would be a little different from MOI-matched clubs, but not hugely so.) Adams' scale was used to match Francis Ouimet's and Bobby Jones' clubs, with obvious success.

Adams used an arbitrary letter-number scale (e.g.- "D-1") to measure swingweight. That scale, which he called the "Lorythmic" scale, remains the most popular swingweight measure right up to the present.

Around 1945, Kenneth Smith bought Adams' rights to the swingweight scale, and began experimenting with it himself. He came to the conclusion that the 14" fulcrum gave a good match for professional golfers, but a 12" fulcrum would produce a better set for the average amateur, which he called the "Official" scale -- even though the industry has never adopted it as official anything. He was soon producing both kinds of scales.^[2]

So, by the mid-1900s, we have three approaches to heft-matching a set of golf clubs:

The 12" so-called Official scale.
The 14" Lorythmic scale (still the most popular).
Moment of inertia (not much used by then, because it was so tedious compared with a swingweight scale).

The major difference among them is the amount by which the clubhead gets lighter as the club gets longer. Smith believed that the average golfer couldn't handle light long irons and woods, hence his proposed (and never really accepted) change in fulcrum placement.

For example, consider a heft-matched set in each of the systems, using the standard club lengths from the late 1900s (35.5" for a 9-iron and 43" for a driver). Let's choose a common weight for the 9-iron head, and see what the driver head would weigh in a matched set.

System of Measure	9-iron head	Driver head
12" swingweight	284 grams	201 grams
14" swingweight	284 grams	195 grams
MOI	284 grams	180 grams

Thus there is some difference across the set between and 12" and a 14" scale, and a lot more difference between either and an MOI-matched set. In particular, the 12" scale allows the longer clubs to be heavier-headed than an MOI-matched set would. Smith felt that the long clubs had to feel heavier-headed for the Sunday golfer to swing them, though the lighter heads would give better performance to the accomplished golfer. Put another way, the longer the fulcrum, the faster the clubheads get lighter as the club gets longer.

So, you ask, which one is "Right", in some absolute sense? Obviously, "it depends". To understand how to heft-match clubs, we'll have to look at some mechanics of how the club is released during the swing.

Heft and Release

I'd like to thank Bernie Baymiller for the strobe pictures of Bobby Jones. Bernie's father was the director of R&D for Spalding Golf in the 1940s, which is where and when the pictures were taken.

Here is a strobe picture of Bobby Jones swinging a driver. It was taken by Dr. Harold Edgerton of MIT, inventor of the strobe flash, and captures Jones and his club's position at intervals of about 0.007 second. I have taken the liberty of marking three positions of the swing with the "double pendulum", as follows:

The red position, about 70 milliseconds before impact. Note that the wrist is still fully cocked at an acute angle. This is about where Jorgensen has identified the beginning of release, and Jones' swing corroborates that.
The green position, about 20 milliseconds before impact. The wrist angle is much smaller, indicating that the wrists are well into the uncocking process.
The blue position -- impact. The wrist is almost straight here, with just a little wrist cock left. This is a very good impact position; the club is just about fully released.

So "release" is the process of the loss of wrist cock angle, from fully cocked during the downswing to almost straight at impact. And a club that is well-fit to the golfer will release so that the two arms of the pendulum are aligned at impact, with only a degree or two of wrist cock left.

In order to understand how to design the club for proper release, let's review the physics of the club's release. "Release" means rotation of the club about the wrist hinge. According to Newtonian mechanics, such rotation can only occur by the imposition of a torque on the club. The torque comes from two sources:

Centrifugal force rotates the club about the wrist hinge.
In addition, the golfer may apply some torque via through the wrist hinge, using the muscles of the wrists, hands, and forearms. This is a relatively small component of the release torque for most golfers, especially good golfers.

Resisting this torque -- retarding the club from turning -- is the club's own moment of inertia around an axis at the wrist hinge. So, if we assume that the golfer makes the same swing -- applies all the same forces at the same times -- regardless of which club he is swinging, then it would appear that the way to match a set of golf clubs is to match their moment of inertia. That way, identical swings would result in identical release. So, if you find the correct moment of inertia for the golfer for some favorite club, you should build every club in the set to that same MOI.

This is idealistic rather than ideal. Or, as my science teachers used to tell me, "The difference between theory and practice is bigger in practice than in theory." Here are some reasons that our argument for MOI matching may be too simple:

Some golfers don't need or want identical release on all clubs.
Golfers may not apply the same swing to all their clubs.
The analytical model isn't quite right; the centrifugal force working to release the club depends not just on the golfer's swing, but also on some of the same properties of the club that affect the MOI.

Let's address each of these points:

The need for identical release

Most instruction today teaches to place the ball in the same place in the stance, regardless of what club is being used. Usually, that recommended ball position is just inside the heel of the front foot (the left heel for right-handed golfers). For instance, see Butch Harmon's lesson article endorsing constant ball position. (Note: the links here worked at the time this was written. If it doesn't work for you, please contact me so I can find another page.)

With the ball in a constant position with respect to the golfer's stance, an MOI-matched set of clubs should be ideal. All other things being equal, it will result in complete release occurring at the same position in the swing. If you can find an MOI such that the release position corresponds to the ball position, build all the clubs for that golfer to that MOI.

Simple!

But constant ball position was not always the way golf was taught. In fact, it's a fairly recent development. Only a few decades ago, most golfers were taught to play the short clubs back in the middle of the stance, and move the ball forward as the clubs get longer. How do I know? I was taught that way in the early 1950s. And, since it works for me, I have not bothered to change. I'm sure there are many old fogeys like me, who still play that way. Not only that, there are a few instructors who still teach that even today.

So what does that say about heft matching. The first thing we should notice is that we want an earlier release in the short clubs and a later release in the long clubs. We can accomplish this by making the MOI progress across the set, so it is lower in the short clubs and higher in the long ones.

Another way of saying this is: the heads still get lighter as the clubs get longer, but they don't get lighter as fast. Now look at the table above, where we compared the three ways of measuring heft. Swingweight has the property we just described: as the club gets longer, the heads don't get as light for swingweight as they do for MOI. So swingweight matching may be good for a golfer who uses a variable ball position.

Indeed, history also seems to support this. The popularity of swingweight scales with clubfitters dates back to the middle 1900s. And, at that time, variable ball position was the way almost all golfers were taught. So swingweight might have been exactly the right way to match clubs at the time. And it might still be the right way to match clubs for dinosaurs (like me) who still play that way. (Actually, I discovered this the hard way in my early experiments with MOI-matching in 1995. I am continuing to experiment with MOI matching and constant ball position; some day that may be my usual game.)

Different swings for different clubs

Everything up to this point is about physics, not the physiology or psychology of the golfer. Face it, many golfers do not use the same swing for all their clubs. There are some good reasons for this, as well as some bad ones. But, good or bad, we have to fit golfers who may use different swings for different clubs.

Here are a few of the reasons, and what we can do about it:

A lot of modern instruction says to "hit down" through the irons, but to "sweep" the woods. This necessarily produces slightly different swings for irons and woods. In particular, release should not be complete for the irons; there should be more wrist cock remaining at impact than for the woods. Remedy: Match the irons to one MOI and the woods to another. Logic would say that the irons' MOI should be greater than the woods' (to retard the irons' release so there is still significant wrist cock at impact), but you have to determine this by experiment with the individual golfer.
Different clubs obviously have different lengths. This necessarily results in different swing planes. This may result in different application of the muscles to produce the forces. Remedy: This is likely to be a smooth progression across the set, because the swing plane itself is a smooth progression. If so, some sort of "slope" on the swingweight or MOI may solve the problem.
If the clubs don't feel the same (what ever that means, for that particular golfer), then he may change his swing a little in response. (We'll get back to this later in the section, when we talk about backweighting.) Remedy: The simplest way to deal with this is to work hard at making the clubs feel the same. Experience shows that MOI matching provides the most "same feel" across the set for most golfers. But not all; some golfers will be problems in this regard, and will have to be fitted on a club-by-club basis -- or perhaps a more drastic solution like a constant-length set.
Golfers are inclined to think of different clubs in different ways, and swing them differently. Many will apply a different swing to the driver, because they are trying to kill the ball. Many will try to lift the ball with the wedges, rather than hitting down through it. Remedy: If you are trying to band-aid this problem by clubfitting (rather than fixing it properly, with lessons and lots of practice), you'll have to recognize it first, then experiment with clubs to find what works. But let me suggest the first experiment be increasing the MOI for the trouble clubs -- for both these faults.

Centrifugal force and MOI

The analysis so far treats the torque due to centrifugal force as a constant, independent of the design of the club. But it isn't. In fact, most things that will increase the MOI of the club will also increase the torque due to centrifugal force. For instance:

Making the club longer will increase the MOI. It will also move the balance point further from the wrist hinge, increasing the "moment arm" of the centrifugal force -- so the torque will increase.
Making the head heavier will increase the MOI. It will also increase the centrifugal force, as well as move the balance point closer to the clubhead (hence further from the wrist hinge). Both of these effects increase the torque on the club.

The torque changes in the same direction as the MOI, but not nearly as much as the MOI. Experience has shown that MOI matching still produces good results, both in release and feel, despite this flaw in the analytical model. Here's a tentative conclusion I have drawn:

To the extent that the golfer "hits" (applies torque via the hands, wrists, and forearms), using MOI to match the clubs is the right answer for heft matching.
To the extent that the golfer "swings" (depends on centrifugal force to produce clubhead speed), MOI matching does not hurt -- but the golfer is relatively immune to heft errors in the set anyway.

In other words, a pure hitter needs to have the set matched by MOI (either constant MOI for a constant ball position or a sloped MOI for a variable ball position). A pure swinger probably won't be hurt by errors in the heft match; it almost doesn't matter how you match the clubs. A caveat for that last statement: it won't matter for release performance, but that's not all you need to worry about. He may still feel the difference and change his swing accordingly. That is an argument for not being careless about the match, even if you know you are fitting a swinger.

While we're questioning centrifugal force in the model...

This page has drawn considerable criticism on the basis that centrifugal force is phony. True, some centrifugal forces are fictitious, but not all. The centrifugal force in this analysis is an example of the fictitious force, so the criticism is at least partly true. So why do I use it?

Just because it is fictitious does not mean it gives wrong answers. There are plenty of fictitious constructs in physics, that we use because they behave analytically as if they were real, and centrifugal force is one of them.
From a tutorial point of view, it is much easier for the non-physicist, non-engineer reader to understand. Here is a good explanation of the more classical physics. Not many people -- actually not even a majority of physicists and engineers -- would be able to visualize how release works from the diagrams and equations in the reference. You have to be able to mentally step through the Digital Differential Analyzer and see what the output would be.

So, if I were actually writing a program to analyze the swing, I'd use d'Alembert's principle from classical physics (as in the link above) and never use centrifugal force at all. (Not that CF would give a wrong answer if programmed correctly, but it's much harder to program right.) But I still maintain that it's easier for most people to understand using centrifugal force -- so that's the way I'm explaining things for this tutorial.

Notes:

In 2008, André Cantin pointed out that the shafts are different lengths, and so will have different weights. As we can imagine, this increases the moment of inertia of the longer clubs relative to the shorter ones, because there is additional shaft weight at the tip in the longer clubs. I didn't think it would matter much, but did the calculations to see. As it turns out, there is enough of a difference to measure. In fact, it appears that the match is just about halfway between a moment-of-inertia match and a 14" swingweight match. Basic information for the calculations:

The specific gravity of hickory is about 0.7. (That's a density of 0.7 grams per cc.)
The hickory shafts of the time had a diameter of about 0.45" near the tip. (See Wishon & Summitt.)

Among the information that Dave Miko shared with me was some interesting correspondence between Kenneth Smith and Lloyd Rittenhouse, an engineer who got sucked into a discussion of how to convert between Smith's "Official" swingweight and the more common Lorythmic swingweight. The discussion, which went on for a year in the 1976-77 period, centered on Rittenhouse's unpopular assertion that there is no one-to-one correspondence; it is a function of the swingweight and the total weight. Eventually Smith came to see it the same way as Rittenhouse, and Lloyd published a technical paper (I don't know where it was published -- might have been just a private correspondence) on the conversion. Here is the bottom line, the conversion chart from Rittenhouse's paper.

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