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Physical principles for the golf swing

Biology

Dave Tutelman -- February 28, 2023

Limitations

Muscles and joints have their limitations as "engineering elements" of a golf swing. The maximum ability to exert force is an obvious limitation, and is common to a lot of engineering components and materials. But there are some that are more unique or less obvious. Let's look at these limitations, ways people try to overcome them, and what can happen -- or perhaps go wrong -- when you try.

Maximum force

This is the obvious one -- what is the maximum load the muscle can sustain? Put another way, what is the maximum force the muscle can exert? We have a reference that lists this as 9.5 Newtons per square centimeter (13.8 pounds per square inch) of muscle cross-section. But this number is an average over the test subjects; there was considerable variation from individual to individual, as well as a fairly strong bias between male and female subjects.

What can we do to increase the load capabilities of our golfing muscles?

Exercise! Specific resistance training (e.g.- weight training for certain muscle groups) can increase the muscle size, the cross section. Every square centimeter of muscle cross section you can put on should increase the strength of that muscle by almost 10 Newtons.
While the number 9.5 varies with the individual, I haven't found any references that suggest this can be changed with exercies, conditioning, diet, or anything else. I'm not sure how well the number is understood yet.

Here's another question. What if an external force pulls the muscle so that, instead of contracting, it extends? The graph we used earlier has the no-stress length of the muscle as L_o. I have shaded in yellow what is going on for length>L_o, which we have not discussed yet. There are three curves in the shaded area.

The dashed curve, labeled "active force", is the muscle's ability to resist the pull by pulling back, trying to contract even if it does not succeed.
The green curve, which kicks in just beyond L_o and is labeled "passive force", is the result of trying to stretch muscle tissue even if it doesn't try to oppose the pull with its own force.
The red curve is the "total force", the sum of the active and passive forces.

At some point beyond the place that total force goes through a minimum, the muscle assembly fails, and you get injured. It could be a pulled muscle (really a torn muscle) or a damaged tendon, depending on which has the lower failure load of the two.

Range of motion

A joint provides a pivot for two bones, relative to each other. The "range of motion" (ROM) of the joint is the range of angles allowed by the joint. Every joint has some limit for angular motion.

For instance, consider the wrist, and specifically the information in this diagram. In the plane of radial and ulnar deviation, the wrist can swivel 45° towards ulnar deviation or 20° toward radial deviation. That's a total angle of 65°, which is the range of motion of the wrist in deviation.

Only 65° total range of motion? Then how do we get a 90° wrist cock -- ever? Is it only "freaks", people with huge wrist flexibility, that can manage a 90° wrist cock? Let's look at this in a little more detail, because the question is extremely interesting. It shows (a) how you have to look carefully, not just accept vague generalizations, and (b) an important fault in many people's golf swing that stems from a basic biomechanical limitation.

Since radial and ulnar deviation are the fundamental limits of wrist cock, let's look at how much they limit wrist cock. The model in the video is me, and I am measuring the angle between my left arm and a PVC pipe representing a golf club.

The straightest the angle ever gets is in the range of 4° to 10°.
The most acute is in the much tighter range of 79°-80°.

The total range of wrist cock is between 80-4=76° and 79-10=69°. That is a total range larger than the 65° the textbook tells us. But it isn't that much larger. Perhaps my wrist joint is more flexible than average. Or perhaps something else is at work when I grasp a club. Let's consider what happens to the wrist when we make a golf swing.
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We will get into more detail when we look at the forces and torques that make the golf swing work, but let's see what forces and torques might do. Specifically, can they extend the range of motion of a joint.

According to this video, they can indeed. If an external force is applied to the "club" -- other than the force exerted by the muscles attached to the joint -- the radial side can be extended by 9° and the ulnar side by 13°. Bear in mind what the force my right hand is applying to the pipe is actually doing. It is applying a torque to the joint. The torque is the moment of the force I am applying, acting through a moment arm measured from the force to the pivot of the joint.

So what we are saying is that an external torque outside those generated in the joint can extend the range of motion of the joint. There is a limit to how much the range can be extended. This extended range is resisted by the forces due to muscle elongation greater than L_o, which we talked about earlier. But at some point, either the muscle is damaged by overload or, more likely, the joint itself hits mechanical limits. Bones interfere with one another's motion, or ligaments hit their limits.

When we get to the forces and torques exerted during the golf swing, we will see where kinetics external to the joint itself can come from, and the role they play in the swing. But here's a hint: those forces are most likely to be reaction forces, where the club is reacting to our body's forces on it to make it move. Forces exerted back on us by our efforts to make something move are based on that object's inertia (mass or moment of inertia), and are therefore related to how much acceleration we are creating. (That is why I am not a big fan of teaching an athletic motion like a golf swing through slow-motion or position drills. The forces the body exerts for those drills are completely different from what it exerts at speed. It isn't just the size of the force; the real-life forces are often in completely different muscles.)

Before we leave the subject of wrist cock and range of motion, I'd like to mention a "cheat" used by many high-hndicap golfers to increase the wrist cock. I am talking about cupping the wrist, as shown in this video. By cupping my wrist, I can get 20° more wrist cock than radial deviation alone will allow. That's a lot! But it comes at a big price. The extra 20° of wrist cock requires over 40° cup in the wrist. Every degree of cup is a degree more open for the clubface, so a push-slice becomes possible, even likely.

So does the opposite hold true? If I bow my wrist (thus closing the clubface), can I also get in increased wrist cock? The video shows that the answer is "no". There is just about no wrist cock to be gained by bowing the wrist, not even a whole degree.

Exceeding range of motion

What happens if you exceed the ROM? Only bad things. Here are a few examples, which have come to be familiar buzzwords:

Athletes frequently exceed the extension limits on their knee, and sometimes other joints as well, like the elbow, neck, finger, etc. Ever hear the term hyperextended knee? That is exactly what it means. And other joints can be hyperextended as well. The damage is rarely a broken bone, but the ligaments, tendons, and cartilage are damaged.
The knee joint is a hinge; it has only one axis of rotation, that rotation in a fore and aft direction. The sideways range of motion is essentially zero. If there is substantial sideways force on the knee -- a direction it was never intended to flex -- the ligaments holding the knee together are usually the first thing to go. This is called a torn ACL (anterior cruciate ligament) or MCL (medial cruciate ligament), depending on which side of the knee the ligament is on. The knee is particularly vulnerable to this sort of injury, because the trauma can occur while the weight of the entire body is on the leg, which prevents the whole leg from moving sideways to cushion the impact.
A dislocated shoulder is a description of a bone pulled out of the shoulder joint. It could occur from just a pull, but it would have to be a very large pulling force. More likely is that the range of motion of the shoulder joint is exceeded and the force continues in that direction. The bone acts as a lever, which greatly multiplies the force pulling the bone out of the joint. As the diagram shows, the point where the blue bone interferes with the green bone becomes a fulcrum for the lever. The lever arm for the external force is long, typically the distance from the shoulder to the elbow or perhaps even to the hand. The lever arm for the force pulling the ball out of the socket is very short, probably not more than an inch. The result is that the pulling force is much greater than the applied external force, and dislocation is the result.

Dislocations can occur in other joints, but the shoulder is the most common dislocation.

Force-velocity curve

The simplest statement of this limitation is, "If you have to exert a force at speed, then the higher the speed the less force you can exert." Let's start with a YouTube video, which is a really good introduction to the concept.

This is a pretty good introduction, but let's look a little more closely at the curve itself. Most of the presentations I found online, including the one where I got this graph, show the curve approaching zero as velocity increases, but never quite reaching zero -- and certainly never passing it. In the other direction, the curve never quite reaches zero velocity, implying that we could exert infinite force if we do it slowly enough. The curve is clearly not an accurate representation of the variation of force with speed. Consider:

For any way we can exert a force (or, for that matter, a torque), there is a velocity beyond which we not only can't exert a force but we can't even keep up with whatever we are trying to exert a force on.
Even the slowest application of force has a limit to how much force we can apply.

For instance, consider riding a bicycle. If you get to a downhill, the bike will pick up speed, and it may be all you can manage to keep up with the pedals; surely you are not exerting much force. In fact, you will get to a speed where you can't even keep up with the pedals because the bike wheels are turning too fast. (In order to experience this, you may have to do the experiment with the bike in a lower gear; the highest gearing ratios are designed so you can keep up with the pedals even at very high speeds.) So there is indeed a speed at which you can't add any force to the pedals.

But wait! It's even worse. Suppose you were on a track bike and your feet were in toe clips that bound them to the pedals. Track bikes do not have freewheels; you can't just leave your feet still and coast. Your feet -- which we said were bound to the pedals -- had to move at the speed of the pedals. Well, if you can't keep up with the pedals to exert force, that means that the pedals are pulling you along. At that point on the force-speed curve, your feet are being dragged along for the ride. The bike pedals are exerting a force on you. Another way of saying that is, you are exerting a negative force on the pedals. Here is a force-velocity curve that reflects the reality that, above some speed, the force you exert is negative because you can't keep up with the velocity.

There are similar relationships between torque and the angular velocity of the objecty you are trying to twist. That is called a "torque-velocity curve", but it is hard to search for; almost all your results will relate to electric motors, not muscles and joints of the human body.

This limitation, the torque-velocity curve going negative, was the subject of one of the more exciting disputes in the golf biomechanics world around 2015. We will see it later, but if you can't wait I have written a long article about it.

If you are interested in details of force-velocity curves, their history and their actual (empirically determined) shapes, here is a reference paper on the subject.

Antagonistic muscle pairs

The very nature of how joints work requires two opposing muscles to move the joint backwards and forwards. For instance, let's get back to the elbow joint. We know it is a hinge with only one axis. But there are two muscles that move it, the triceps for extension and the biceps for flexion. Biologists refer to pairs like this as "agonist" and "antagonist" muscles. For instance, to extend the arm at the elbow, the triceps contracts; it is the agonist muscle. The biceps is the antagonist for this motion. (If we wanted to flex the arm, the roles of the two muscles would be interchanged.)

Why is the inactive muscle called an antagonist? Because it is not really inactive. It is doing some contraction in opposition to the active agonist muscle. It takes care and training to remove any trace of antagonist tendency from the supposedly slack muscle.

Try this.

Sit in front of a desk; the elbow will be roughly at desktop height.
Place your left hand flat on the desk. Your forearm should be approximately level, and the elbow at a right angle.
Relax the left arm and feel both the biceps and triceps muscles with your right hand. Both muscles should be soft. If they are not, make sure the arm is completely relaxed.
With your right hand on the triceps so you can feel the muscle, push down hard on the desktop. The push should come from the elbow. Feel how the muscle goes from soft to hard and enlarged. That is because the triceps is exerting a force intended to extend the elbow joint. It cannot actually extend because the desktop is preventing it, but the triceps is contracting hard trying to make extension happen.
Repeat, but feeling the biceps with your right hand. Again, be sure both muscles are completely relaxed and soft before you apply the force. Now push down on the desktop with your left arm. Notice that the biceps hardens just a bit. It hardens much less than the triceps, but it is still exerting some force. It is the antagonist to the triceps, which is exerting the main force.

This means that, in order to exert a certain torque at a joint, the agonist muscle has to exert more contraction force, in order for the net force (agonist minus antagonist) to get to that torque.

Why should antagonist muscles have any effect at all when the agonist muscles contract? They can stabilize the joint, holding the bones together. They regulate quick motion, keeping a check on a limb that might be damaged (e.g.- hyperextension) if the full force of the agonist is not checked.

I have not been able to find a reference saying how large the antagonist force is as a fraction of the agonist force. I'm sure that is because it varies, not only from antagonistic pair to pair, but other things like the speed of the motion and any training that the person may have done.

What can we do about the limitations?

Is there some way to eliminate or at least minimize some of these limitations? Yes, and it has to do with training. We can't elminate them entirely, but we can extend our performance if we exercise away some of the limits.

The maximum force you can exert on a load can be increased by strength training. Proper training combined with nutrition can increase the cross-sectional area of the muscles. I have not found a reference that says that you can also train up the nominal 9.5N/cm², the nominal strength of muscle tissue. But that number varies considerably from person to person, so perhaps it is something that can be trained. (Or maybe it is just genetic. Nature or nurture? Don't know.)

The range of motion can be increased by stretching. Flexibility exercise is the essence of ROM training.

The force-velocity curve is interesting. Here's a graph of a linear force-velocity curve with two different types of training, strength training and speed training.

If you do strength training, you can increase the force you can exert right up to the maximum at zero velocity. But you don't get any benefit with maximum velocity.
If you do speed training, you can increase the velocity for every force, including the maximum velocity before the force goes negative. The curve shows no gain in maximum force; but, knowing what speed training looks like, I suspect you'll also get some static strength benefit from it as well.

A similar graph is available from quite a few sites (here's one) with the unrealistic inverse curve. I think the linear graph is easier to understand visually, which is why I am presenting it that way. Moreover, I'm not sure I believe the cited curve, which shows training introducing a loss of strength for middle velocities.

The antagonistic behavior of antagonist muscles, I once read, can be trained away. I can't find any reference to that, and I'm not sure how wise an idea that is anyway, given its function of stabilizing the joints. It may give slightly increased performance at the expense of increased risk of injury.

But a better way to deal with it is to train both sides of the antagonistic pair with "antagonist superset" exercises. (Just look it up; there is a lot of information on the subject.) It consists of back-to-back sets, called "supersets", of exercises for antagonistic pairs. No rest between the back-to-back sets; you rest between supersets. Advantages of doing this include:

Effective strength training.
Time-efficient strength training; you don't have a rest for every set, but rather for every two sets.
"Structural integrity", as one video calls it. The antagonistic balance is maintained, which reduces chance of injury.

Does this minimize the effect on performance of antagonist muscles? In a way it does, but not really. What it does is make the agonist muscle stronger, so it can do a better job of overcoming a mostly relaxed antagonist. If you look at percentage of (a) antagonist activation and (b) increase in strength of both, the math works out to unambiguously better performance.

Last modified -- Apr 3, 2023