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 200gram 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 inchgrams
(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
inchgrams. 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 (countertorque) 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 offcenter 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 offcenter 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 offcenter 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 Wholeclub 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 heeltoe 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 offcenter, the putter will twist and your putt
will
go offline; 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 heeltoe weighted putter head with its higher moment of inertia
is more forgiving of offcenter hits than a blade. Another way of saying
this is that head #2 has a "larger sweet spot".
Now let's talk about cavityback 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 offcenter 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 Wholeclub MOI. One goal of clubfitting is to find the right
wholeclub 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 wholeclub 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
