Physical principles for the golf swing

Forces

Dave Tutelman  --  March 5, 2020


Let's start with what a force is.

It is a simple push or pull. What it ultimately does -- why we are interested in it -- is cause the object to change its motion. But we will get to that shortly.
 
In order to sharpen the idea what a force is, let's say a bit about what it is not.

In the first place, it is not a grab-and-push nor grab-and-pull. As this diagram shows, such an action can result in more than just one force. The forces this hand exerts include:
  • In red, the push or pull we have in mind.
  • A force we have to exert to support the weight of the object. That is the blue force the hands apply. It is upwards to prevent the object from going downward due to gravity.
  • A pair of forces -- a "torque" or "couple" -- the hands apply to keep the object oriented vertically. That is the green pair of forces.
So we have to be careful when we itemize the forces involved in any analysis, making sure that all the forces are individually accounted for. We shall see later that they can be combined for the analysis, but first they must be quantified individually.

Moreover, a force is not a mass. These are often confused.

The diagram shows a pair of one kilogram masses. (You might have called them "weights", but they are really masses.) In particular:
  • The one in space is operating without gravity. So it exerts no force if it just sits there (or really floats there).
  • The one in a gravity field -- say, on earth -- does have a force exerted on it. We call that force the "weight" of the object. Once again, weight is a force, and it is not the same as mass.

OTOH, force and mass have an intimate relationship, which is completely described in Newton's second law, F=ma. In a moment, we will go into detail on that.

Yes, I know that there is a gravitational force on masses in space. It's an unnecessary complication to include in teaching golf, a sport seldom played in space. So let's not confuse the student here.
Finally, before we dig into what forces do, let's mention what a force is mathematically.

A force is a vector.
The magnitude of the vector is the amount of force.
The direction of the vector is the direction in which the force is acting.

Since force is a vector, we can add forces together (but we have to add them as vectors) and we can resolve them into components. Both will happen a lot as we examine how a golf swing works.
 

Kinetics

Isaac Newton's greatest contribution to physics is his three laws about forces on objects. We will first look at the kinetic implications of those three laws -- that is, how forces between objects (masses) interact to move the objects. That is the substance of Newton's three laws of motion.

Later we will look at the kinematics -- just the motions themselves. We do that because it is one of the biggest problems I see when golfers or golf instructors or TV commentators discuss motion. Their confusion of acceleration, velocity, and position is serious. If you want to understand what is going on, you need a more proper understanding. Which brings us to Isaac Newton's greatest contribution to mathematics. Once Newton realized that his laws demanded a rigorous treatment of position, velocity, and acceleration, he invented a new kind of mathematics to deal with it: calculus. We will not be learning calculus here. But we will go through a bunch of examples of motion and see some patterns. If you can retain some of those notions, you will be at least partially immunized from silliness. And if you really understand the examples, you will be very well positioned to learn calculus if you choose to do so.

Newton's laws

So... Newton had three laws relating forces and motion. Let's take each in turn and look at what it really means.

Newton's first law of motion: A body in motion stays in motion at the same speed and direction (same velocity) unless acted upon by some force outside the object itself.

This is pretty simple.
  • If a body is sitting there not moving, it will continue to sit there and not move. No big surprise.
  • If a body is moving with some velocity V (remember, velocity is a vector -- both a speed and a direction), it will continue to move with that same velocity V unless a force is applied to the body.

    What!? You mean it won't slow down over time? That is exactly what it means. If it does slow down, that means some force was slowing it. There are lots of frictional forces in the real world, so most things do slow down. But remember, that was because of a force, a friction force, not because that is just the way physics works. In space, where there is almost no friction, things go on moving for millions and even billions of years.


Let's use Stimp as a golf example. Newton's first law law says that if you roll a frictionless ball along a flat, frictionless surface, it should roll forever. Stimp is a measure of how frictionless the ball and surface really are. In the diagram, we show the conditions that must attain -- the friction that must be eliminated -- to approach a true "body in motion" with no force on it.
  • Stimp = 5: Kind of a shaggy green. Some fairways probably stimp here if cut for an elite tournament.
  • Stimp = 15: This is a very fast green, close to the fastest you will see even on the PGA tour. The grass is trimmed so close, you are almost putting on packed, dry soil.
  • Stimp = 50: This just plain can't be done on dirt. It has to be a smooth, hard surface. Concrete or blacktop might conceivably work, but even that may be too rough; you may need to go to metal plate or something like that.
  • Stimp = 150: Now we've done as much as we can with the surface; we have to look at the ball itself. To get here, we have to lose the dimples (which are a source of friction). We may need to make the ball out of steel in order to (a) get the lossy flex out where the ball's weight slightly compresses the cover, and (b) increase the mass of the ball so it takes more frictional force to slow it down. (That's part of Newton's second law, which we are just a few paragraphs away from.)
Before we go any further, Newton's laws refer to a "body", which I might also call an "object". It is a real physical thing, with a mass, a size, a shape -- everything that we think of a real-world object as having. In real physics, it may be as small as an atom or as large as a planet. But in golf, we are not going to encounter objects much smaller than a golf club or human body part, nor much larger than a person.

Well, I wasn't completely honest about that. Golf only deals with "Newtonian" physics. But a real physicist deals with two other kinds of physics that we never encounter in golf, and probably not anywhere in biomechanics:
  • "Quantum physics" deals with the world of tiny particles. When I said "as small as an atom", that would be into the world of quantum mechanics.
  • "Relativity" deals with huge sizes and very high speeds. The sizes can be planets and stars or larger. The smallest things where relativity explains behavior are probably black holes. Relativistic speeds are comparable to the speed of light. If we think about some twenty-fifth century golf ball going at 500mph, that is still very slow in relativity terms. The error we would incur by ignoring relativity would be only 0.00000000005%. (That's ten zeros.)
If someone tells you quantum physics or relativity is important to golf analysis, they are probably deluded.
So we are studying only Newtonian physics. Therefore, Newton's laws are a good place to start.
 
Newton's second law of motion: If a force is applied to a body, the body accelerates according to the relationship F=ma.

The first law said that things don't change their motion without a force being applied. This second law says exactly how the motion changes for a given force. For purposes of discussion, we will apply basic algebra and F=ma becomes:
a  = F

m
This way, we see what happens to acceleration when we change force of mass. So...
  • Acceleration is defined as the rate of change of velocity. It isn't how fast something is moving, a mistake that so many people make. It is how fast we are changing how fast the thing is moving. Two "how fasts" in that description, and we'll see why there are two when we get to the kinematics of Newton's laws. So far, we are still on the kinetics.
  • Acceleration is a vector. It has a magnitude and a direction. The direction is the same as the direction of the force vector. As for its magnitude...
  • Acceleration's magnitude is proportional to force (we know from a=F/m). Increase the force and you increase the acceleration by the same percentage as you increased the force. For instance, double the force gives double the acceleration.
  • Acceleration's magnitude is inversely proportional to mass (we know from a=F/m; m is in the denominator). Increase the mass and you decrease the acceleration by the same percentage as you increased the mass. For instance, double the mass gives half the acceleration.
For our example of Newton's second law, we'll do something that involves numbers -- only fair since the law itself is a formula, F=ma. Let's figure out what the magnitude is of the force the driver face applies to the ball during impact. Let's assume that impact produces a ball speed of 150mph, and that the force is constant during impact. (The latter is a false assumption, but we'll make it to keep the calculation simple. At the end of the chapter, we will revisit the assumption and see how we could have done it more realistically.)


Here's a diagram of what we need to know in order to solve the problem. (The photos of the ball and the clubface are frames from an excellent Titleist video.) The facts:
  • We know the force is zero everywhere except during the .0004 seconds the ball and clubface are in contact. Then it is some number -- and that number is what we are trying to find.
  • We know that the acceleration is proportional to the force because of F=ma. So the acceleration curve is the same shape as the force curve -- a line that is zero everywhere except during contact, and a flat top during contact.
  • We are given the ball's velocity is zero before impact and 150mph after separation. We will use the international standard for solving this and other numerical problems, so that is 67 meters per second.
  • We know that the golf ball has 45 grams of mass; that's the law! So, if we can find the acceleration, we can just apply F=ma to find the force.
Let's get started! The constant force (assumed) requires a constant acceleration, because mass doesn't change and F=ma. Now remember the definition of acceleration; it is the rate of change of velocity. Since the acceleration is constant, the velocity changes at the same rate -- the same slope -- for the duration of contact. That's why the green curve is a sloped straight line during impact. We need to find the acceleration during impact; if we can find that, we just apply F=ma to find the force.

Impact lasts about 0.0004 seconds, a little less than a half millisecond; we all knew that, right? During impact, the velocity goes from 0 to 67m/s.

Brief digression. In case you didn't know, "per" means "divided by". For instance you can find miles per hour by dividing miles traveled by hours spent traveling. That's the way everything in real-world measurements works. So we write "meters per second" as "m/s", meters divided by seconds. Now we are going to use that fact.

The rate of change of velocity -- the acceleration by definition -- is 67m/s per 0.0004 seconds. So we divide 67 by 0.0004 and get a rate of 167,500 meters per second per second (m/s2). That is the acceleration we were looking for! Note that acceleration is always units of distance per time squared.

Now we just plug 45 grams (which is .045 Kg for international units) and 167,500 into F=ma, and we get:
F = ma = .045 * 167500 = 7537 Newtons.
Well of course the international unit of force is named after Newton. What else would it be?

Let's convert 7537N back into a unit of force we are familiar with, pounds. That gives us 1694 pounds, or roughly 1700 pounds of force to accelerate the golf ball.
 
Newton's third law of motion: For every force applied by one body to another, there is an equal and opposite reaction force.

This final law says that, if you push on something, it will push back with a force of the same amount and exactly opposite direction. If you pull on something, it will pull back with a force of the same amount and exactly opposite direction.

Try it. Push against a wall. Feel it pushing back? That is real. It is a genuine force, not your imagination. But if you don't trust your feel, here is a more tangible demonstration of the third law.

I am sitting in a chair, solidly on the ground. With my foot, I push a plastic crate full of golf balls (for weight and friction). It slides along the ground, following Newton's first law, until the force of friction stops it. While I am pushing the box with my foot (the red arrow while I push and acceleration the box), I can feel the box push back at me. But I don't move, so there is no visible F=ma on me -- even though I can feel the F.

Then I exchange the kitchen chair for something with a little less friction. Well, a lot less friction -- a wheelchair. Now I do exactly the same thing. This time, the heavy crate has more friction than my wheelchair, and you can see the reaction force (the red arrow) at work, performing F=ma on me.


If these explanations leave you cold, there are lots of other tutorials on Newton's laws on the Internet. For instance:


Last modified -- May 29, 2022