Design Notes - Physical Principles p2

Torque

Torque is the amount of "twisting force" applied to a body, to turn it around some axis or fulcrum.

You may remember it from high school physics as the "moment" of a force on a lever. In the lever example (which we'll use for a bit here), the fixed axis is called the lever's "fulcrum". The torque (twisting force, remember?) produced by any force that acts on the lever is the force times the distance from the force to the fulcrum.

For instance...

Suppose I put a 200-gram weight on a lever, at a point 12 inches to the right of the fulcrum. Then the weight will try to turn the lever clockwise (right side down) with a torque of 2400 inch-grams (200 * 12). Putting more weight on the lever will certainly increase the torque, but so will moving the original 200 grams further from the fulcrum. If we moved it out another 12 inches, its clockwise torque would be 4800 inch-grams. Coloquially, we've increase the "leverage" of the force.

How about a little exercise to really feel what we're talking about here.

Lay one of your clubs on (or preferably across) a table or workbench, with the entire grip (and nothing else) hanging over the edge.
Put your finger on the grip near the butt, and press down until you lift the clubhead free of the bench. Note what the force you apply feels like.
Now repeat, but press on the grip about an inch or two from the edge of the bench. Note how much harder you have to press.

What's going on here is that it takes a certain amount of torque to turn the club about the fulcrum (the edge of the bench) and lift the clubhead. Remember that torque is force times distance from the fulcrum. You can apply that torque by either a small force at the butt or by a much larger force much closer to the fulcrum.

Torque has a number of interesting applications in golf club design:

Of course, we're used to reading about the "torque" of a shaft, which is a complete misnomer. Actually, that rating is an angle (say, 3.5 degrees), not a torque. It is the amount a shaft twists when a given amount of torque is applied to it, trying to twist the shaft around its long axis. The smaller the number, the stiffer the shaft in resisting torque.
The Center of Gravity (CG) of any body is informally defined as the "balance point" of the body. A more quantitative definition is that axis about which the clockwise and counterclockwise torques (due to the body's own weight) balance exactly. The informal definition is easier to measure in a finished club, but the formal definition is the way you compute the CG of something you're designing.
Swingweight is a torque, too. It's the torque provided by the weight of the whole club, about an axis 14 inches from the grip. As such, it measures how much pressure (counter-torque) you have to apply to the grip to swing (turn) the club about the axis. Note that, since the head is further from the grip than the shaft, a gram of head weight contributes more to swingweight (torque) than a gram of shaft weight.

Ironic, isn't it, that the only example that calls itself "torque" isn't really torque at all, but rather the twisting motion resulting from torque. (Scientists and engineers call that "torsion", not torque).

Moment of Inertia

When you hit something on its CG, you move it straight away from the point where you hit it. But if you hit it off its CG, it will twist. Let's verify that with another little experiment:

Hang a wire hanger from a finger of your left hand. Its CG will be right below the finger from which it's hanging.
Tap it with a finger of your right hand, on the middle of the long horizontal wire (i.e.- right beneath the CG). Note that it swings back and forth, but it doesn't turn about its vertical axis.
Stop it from swinging. Now tap it with the same finger, but at the end of the hanger. Note how most of the energy from the tap goes into turning the hanger rather than swinging it.

The reason for the behavior stems from Newton's original observation that "a mass at rest tends to remain at rest." In particular, the center of mass (another name for the CG) tends to remain in one place unless prodded. If you hit it with a force that doesn't go through the CG, the body will turn, allowing it to respond to the force without moving the CG any more than it has to.

Remember torque? Think of the wire hanger as a lever. The CG, which wants to remain in one place, is the fulcrum for that brief dynamic moment before anything moves. And therefore the force applied when you tap the hanger is a torque around that fulcrum. The farther from the fulcrum (CG) you tap it, the more it wants to twist (rather than swing).

OK, so how much will the hanger twist in response to an off-center tap? We know mass has an "inertia" that makes it resist a force that wants to move it. (The famous equation "F=ma" means that the higher the mass the more force it will take to accelerate it a certain amount.) Well mass has a "rotational inertia" as well; the higher this rotational inertia, the more torque it will take to produce a certain amount of rotational (or "angular") acceleration. The rotational inertia is called the "Moment of Inertia". (For mnemonic purposes, it is called "MOI" in these notes, rather than the "I" that is its usual designation in physics textbooks.)

As you might expect, the further we place the mass from the CG the more effective it is in resisting torque. If your intuition doesn't tell you this, it's time for another experiment.

Get two identical wire hangers and four identical "weights" (small objects considerably heavier than the hangers, but still light enough so you can attach them to the hangers with, say, masking tape).
On hanger #1, tape two weights to the horizontal wire as close to the center as possible.
On hanger #2, tape two weights to the horizontal wire, as far from the center as possible. One weight should be near each tip.
Repeat the previous experiment with both hangers, but note how "enthusiastically" each hanger swings or turns in response to the same strength of tap.

The videos here repeat that experiment, but on steroids. (Click the thumbnail images to see the videos.) We have made a short barbell with steel weight plates and PVC pipe, and hung it so it can turn. If you don't want to do the experiment yourself, you can see what happened when I did it with the short barbell.

The two barbells (like the two hangers) are exactly the same mass, and they should swing identically in response to tapping the center of the barbell. But, in response to an off-center tap, barbell #1 will turn much faster than barbell #2. That's because the weights far from the axis of rotation contribute a lot more to the moment of inertia than do weights near the axis. Moreover, #1's motion seems to be all rotation, while #2 swings a little as well, indicating that the object's mass is getting into the act, not just its moment of inertia.

Quantitatively, the distance between weight and the axis of rotation is even more significant in computing moment of inertia than it was in computing torque. Whereas the torque of each force is the force times the distance to the axis, the moment of inertia of each grain of mass is its mass times the square of the distance to the axis.

Application to golf

OK, time for a few golf applications of Moment of Inertia. We will take a look at how moment of inertia can make a clubhead more forgiving, and how it can control the timing of the club's release during the swing.

To evaluate the forgivingness of a clubhead to off-center hits, we use the Clubhead MOI. Specifically, we use the moment of inertia of the clubhead as we try to rotate it around its center of gravity.
The other picture shows the entire club trying to rotate around the butt of the club. This Whole-club MOI is the moment of inertia that resists turning the club at our wrist hinge.

Forgiving clubheads:
Consider two putter heads with identical weights and blade lengths. Head #1 is a simple uniform blade, while head #2 has all its weight at the heel and toe.

Head #2 has a much higher moment of inertia than head #1, because all its weight is as far as it can get from the CG. In fact, the moment of inertia of a pure heel-toe weighted putter is three times that of a blade, even though their total weight is the same.
Roughly speaking, the CG is the "sweet spot" of the head. Hit the ball there and you get a nice straight putt.
If you hit the ball off-center, the putter will twist and your putt will go off-line; it also won't go as far, because the clubhead is twisting away from the ball. The higher the moment of inertia of the putter head, the less it will twist so the closer to the intended line your putt will go, and the more nearly it will achieve the distance you intended.

Conclusion: a heel-toe weighted putter head with its higher moment of inertia is more forgiving of off-center hits than a blade. Another way of saying this is that head #2 has a "larger sweet spot".

Now let's talk about cavity-back irons. Clubhead manufacturers talk about the cavity making the sweet spot bigger. Now we're in a position to understand what's really going on. It's not that the cavity is magic, but that it allows all the weight of the clubhead to be moved to the edge of the face. This is called "peripheral weighting", and (just as we saw with the putters) it dramatically increases the moment of inertia of the head. And just as with the putters, this means that the head twists less when you don't meet the ball right on the sweet spot. That's why the club is more forgiving of off-center hits. It ain't the cavity at all; it's all the steel they dug out of the cavity and moved to the periphery.

This also explains why metalwoods are more forgiving than wooden woods of the same weight. The woods have their mass distributed more or less uniformly throughout the head, while the metalwoods are hollow shells (because steel is much heavier than wood). Can you say "peripheral weighting"? I knew you could.

Release of the club: MOI Matching:

The goals of a good golf swing are to have the clubhead traveling at its maximum speed at impact, with full extension to the ball so that impact occurs at the middle of the clubface. At impact, the hands have slowed considerably, so most of that clubhead speed comes from the club rotating around the hands. (For most golfers this is somewhere near the wrist hinge, but I have seen estimates that put the center of rotation anywhere from 5 inches below the butt to 8 inches above. Our assumption will be rotation around the butt.)

This rotation is created by the swing. Resisting the rotation is the Whole-club MOI. One goal of clubfitting is to find the right whole-club moment of inertia so that the club is fully extended when the head reaches the ball. Too much MOI, and the club will not be fully released. Too little MOI, and the clubhead will have passed the hands, causing all sorts of ugly things to happen.

Selecting this whole-club MOI to control release is what is meant by the term "MOI matching". We'll see it in much more detail in the chapter on heft matching.

Last modified Dec 13, 2006