# Statics: force and torque

Dave Tutelman  --  August 30, 2014

This article is the start of a larger one on some of the physical principles involved in analyzing the golf swing. I am posting the partial article now because of a discussion in the Golf Sports Science group of Facebook. My position in that discussion is larger and more complex than I have any intention of committing to Facebook. Facebook is a horrible medium for serious technical discussion! So I'm putting it in this article on my web site. It's an article I was planning to write eventually, but I'm writing this part now in answer to Rob Houlding's question.

## Force

For now, let's just say it's a push or a pull. It has a strength (magnitude) and a direction in which it wants to push or pull what it is acting on.

This section will be expanded.
• Most important is to show resolution of forces.
• Introduce "equal and opposite reaction".
• Demonstrate what a force is not, such as a "grab and push".

## Torque

For now, let's just say that it is the rotational analogue of a force. Instead of a push or a pull, it is a twisting effort. I say "effort" because it may not accomplish a rotational movement -- but it tries to. For instance, you apply a torque to remove the top from a screw-top jar. But it may be so securely closed that you can't budge it. You are indeed applying a torque (twisting effort), but without accomplishing twisting motion. The jar (usually your other hand holding the jar) is applying just as strong a torque in the opposite direction, and no motion takes place.

This section will be expanded.
• Define the axis, sense, and magnitude of the torque.
• Torques also have reactions.

## Moment Arm

The question that set me to writing this article was, "On Moment Arms in the golf swing- How would you explain to a golf teacher what a moment arm is as it applies to biomechanics and golf?" asked by Rob Houlding on August 29, 2014.

A moment arm is what transforms a force to a torque, or a torque to a force. This is very important in understanding a mechanical system, including the golf swing. Here's how it works.

 The diagram shows an arm that can rotate around a fixed pivot. A force is applied at the end of the arm, which causes the arm to want to rotate clockwise around the pivot. In other words, there is a clockwise torque that has been applied to the arm. How big a torque? The torque is found by multiplying the force by the distance labeled "moment arm" in the diagram. The moment arm is the distance from the force to the axis of rotation. What turns the force into a torque is that the force is not being applied at the pivot, but rather at some distance from the pivot. That distance is the moment arm. It works in reverse, too. In this case, the pivoting arm has a torque applied to it, twisting it clockwise at the pivot. That will cause the arm to exert a downward force on the blue block at the tip of the arm. How big a force? Well, since torque equals force times the moment arm, the force must equal the torque divided by the moment arm. Simple algebra. So far, all our examples had a horizontal arm and a vertical force. More to the point, the force was perpendicular to the arm. Suppose the angle between the arm and the force were not 90°? As the diagram shows, the moment arm is always measured perpendicular to the force. It does not have to be measured at the point where the force is applied; the important thing is that it is measured: From the pivot... ... To the line along which the force is acting, perpendicular to that line. When you are trying to analyze a real-world force/torque problem, it is often inconvenient (read that as "mathematically complicated") to measure the distance to an extended line of force. You often know the distance to the point where the force is applied, and would rather use that distance. There is a way to handle this without much difficulty. What we need to do is resolve the force, as we saw earlier in the section on forces. We want to resolve the force into a force toward the pivot and another at right angles to that. Think about it; we have a pivot, which defines the arc of a circle. At the point where we are resolving the forces: There is a direction straight at the pivot, which is a radius of the arc. The component of force in this direction is called the "radial force". There is a direction at right angles to the radius, which is tangent to the arc. The component of force in this direction is called the "tangential force". Having resolved the forces into a radial and a tangential component, the torque is equal to: The tangential force, times... ... The distance from pivot to the point of application for the force, along the radius. It is worth noting that both methods -- the moment arm measured to the extension line of the force, and resolution of the force -- give exactly the same result when computing the torque. Everything is scaled to the sine of the angle. (If you are a trigonometriphobe, you need to get over it to understand the science of the swing.) The sine of 90° is 1.0 exactly. So a force at right angles to the radius is just multiplied by the length of the arm. If the angle A is something other than 90°, then multiply the radial distance times the size of the total force times sin(A). If you are trying to scale the moment arm to the extension line of the force, the moment arm is the radial distance times sin(A). If you are trying to resolve the force into components, the tangential force is the total force times sin(A). So it works either way. Either way, the torque is equal to distance * force * sin(A)

## Leverage

I often read the word "leverage" in swing instruction books. And the word is usually used in a way that suggests the author does not really know what leverage is. So let's first review what an engineer or physicist means by the word leverage.

### What an engineer means

It's obvious that the word refers to a lever. Let's look at what a lever is. The diagram shows a long rigid arm with a fixed pivot along its length; that is a classic simple lever.

Note that the pivot is not in the middle of the arm; it is 2/3 of the way to one side. Think of it as a lopsided playground see-saw, with one seat further from the pivot point than the other. (We'll get back to that analogy.) The length of the arm on the right side of the pivot is twice that on the left, 40 vs 20. (Don't worry about units in this exercise; we're keeping things simple so the arithmetic is easy to do in your head.)

Now in order for the lever not to simply spin at increasing speeds -- in order for it to be in equilibrium -- we need zero net torque on the assembly. So let's look at the torque at the pivot.
• On the left side, the torque around the pivot (force times moment arm, remember?) is 10 times 20 = 200. It wants to turn the lever counter-clockwise.
• To keep the lever in equilibrium, we need to have 200 units of clockwise torque. The diagram has already labeled the force on the right side, but let's compute it to make sure it's right. We have a moment arm of 40, so the force must be 200 divided by 40 = 5. And that is indeed the force in the diagram... so the lever is in equilibrium.
Think back to that see-saw. Suppose ten-year-old Sam wants to ride the see-saw with five-year-old Amy. But Sam is a lot heavier than Amy. How can this be made to work. You've all seen this done successfully. Sam doesn't sit on the end of the plank, but moves in toward the pivot until the see-saw balances. He may not know it, but he is adjusting the moment arm so his weight's torque balances Amy's.
Bottom line: a lever is a force multiplier/divider. It converts one force into a torque at the pivot, then generates another force at the other end that is either bigger or smaller than the original force. The "bigger or smaller" is in inverse proportion to the ratio of the moment arms.

An engineer looks upon a lever as a force amplifier. It is a way to generate large forces where you only have a smaller force to work with. Construction contractors and furniture movers view it the same way. That is what they all mean when they say "leverage". And it is quite different from what a golf instructor usually means. Which brings us to...

### What a golf instructor means

While an engineer thinks of a lever as a force amplifier, most golf instructors view a lever as a motion amplifier. In geometrical terms, with the same angular velocity, a longer radius gives more speed. True enough, but that is pure kinematics -- a description of motion without considering forces. Let's look at the kinetics, both motion and forces.

When a golf instructor talks about leverage, it is often in the context of the assumption: "larger radius means more speed." Let's take a look at the same lever we used before, but this time instead of forces we will consider the motion of the ends of the lever.

This diagram shows how far the ends of the lever move for a given rotation of the lever around its pivot. And the motion would appear to support the golf instruction meaning of "leverage". The motion is in direct proportion to the length of the moment arm, so a longer arm should give more speed at the tip.

But is this really what happens? The first hint (to a scientist) that there is a fallacy at work is a consideration of the energy involved. One way of measuring energy expended is the force multiplied by the distance the force moves. The longer end of the lever in our example moves twice the distance, true enough -- but with only half the force. So the energy is the same at either end. Therefore, we should be suspicious that a longer arm would be able to accomlish any more work -- generate more clubhead speed, for example -- than a shorter arm.

Let's look at this notion in more detail -- including numbers.

We are trying to move a payload as fast as possible. The payload is at the end of a lever. For example, think of a clubhead at the end of the shaft, or the hands (holding the club) at the end of the arms. The "engine" driving this machine is a torque at the pivot.

The naive view is, "Of course the payload will move faster on the longer lever. The torque will turn both the levers the same amount, but the 40-moment-arm lever will move the payload twice as far as the 20-moment-arm lever. So that's twice the speed, because speed = distance divided by time."

The fallacy is the assumption that both levers will turn the same amount. To see this, we need to move from the realm of kinematics (motion studies only) to kinetics (motion created by forces and torques).

In the real world, the payload has a mass, and that mass must be accelerated to achieve speed. And we know from F=ma that acceleration requires force. So here is the diagram again, relabeled with the same torque and the same payload mass for two different lever arms. Let's do the simple arithmetic to see what the acceleration is for each.
• For the 40-moment-arm lever:
• Force = torque divided by moment arm.
• F = 120/40 = 3
• F=ma or 3=3a
• Acceleration a is 1.
• For the 20-moment-arm lever:
• Force = torque divided by moment arm.
• F = 120/20 = 6
• F=ma or 6=3a
• Acceleration a is 2.
What does this mean? It means that acceleration is twice as great with the shorter lever arm. If we let both systems run the same amount of time, the payload on the shorter arm will be moving faster than the payload on the longer arm. That is seriously counter to our intuitive expectations. Moreover, it shows there is no logic in the argument that a longer lever arm will generate a faster speed. The difference is the difference between kinematic thinking and kinetic thinking.

But it is a known fact that many long hitters are tall, and that a longer-shafted driver will produce more clubhead speed for a good fraction of golfers. So how can the colloquial (golf instruction) use of "leverage" be wrong? Well, there are other factors at work here, none of them being any intuitive application of leverage. A few:
• The generalization of tall golfers and long-shafted drivers is hardly universally true. For instance, Jamie Sadlowski (a three-time winner of the world long-drive championship) is only 5'10". He gets superior clubhead speed from factors other than his height.
• The longer lever arm may have less acceleration, but it therefore travels a longer time before impact. On top of that, the longer arc radius means it goes farther before impact, making it take even longer. Acceleration accumulates into velocity over time. So, lower acceleration means it takes longer. Ultimately we get the same payload speed at impact. (The detailed calculation is on another page.) This is actually counter-intuitive if your frame of reference is the colloquial interpretation of leverage.
• The torque may not be the same for both cases. For instance, the taller golfer is often bigger proportionally overall. That means bigger (and potentially stronger) muscles, which can apply a larger torque than a shorter golfer would.
So arguing from an assumption that a bigger golfer or club automatically generates more clubhead speed due to "leverage" is a flawed argument. The conclusion may be true or false, but leverage is not what makes it so. For a more complete discussion of tall golfers hitting it further, see my article on the subject.

But I think I can venture a guess as to why people keep insisting that longer levers create more speed. It's because longer levers do create more speed... if there is no load on the system. Again, it's a kinematic rather than a kinetic impression -- motion alone, rather than the forces required to produce the motion.

There is a well-known speed-training exercise: swinging a shaft with a grip but no clubhead, with the goal of generating as high-pitched a whooshing sound as you can. And that model will reinforce the notion that a longer lever creates more speed -- because it does in that case

Try the exercise while moving the shaft only with your hands, not with a golf swing. (The double lever of a golf swing greatly complicates the model, and really doesn't reflect what happens if the longer lever is the arms and not the club.) You can still get more tip speed by a longer shaft. The system, in this case, is limited in speed by the torque-velocity curve. At this speed, you don't lose angular velocity as you increase lever length, so a longer lever does indeed mean more speed.

Now hang a 200-300g clubhead on the end of the shaft. It's a completely different story. The increased moment of inertia of the loaded lever will require more torque to get back to the angular velocity. Turns out you lose angular velocity fast enough to exactly counteract the increased length; the clubhead speed stays the same. And that is only true if the added shaft weight is negligible; if not, the added length will slow down the clubhead.