Applying Physics to Golf

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 = ------ r

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