Physical principles for the golf swing
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 screwtop 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 realworld 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 seesaw, 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
counterclockwise.
 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 seesaw. Suppose tenyearold Sam wants to ride the
seesaw with fiveyearold 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 seesaw 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 40momentarm lever will move the payload twice as far as the
20momentarm 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 40momentarm lever:
 Force = torque divided by moment arm.
 F = 120/40 = 3
 F=ma
or 3=3a
 Acceleration a
is 1.
 For the 20momentarm 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
longershafted 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 longshafted
drivers
is hardly universally true. For instance, Jamie Sadlowski (a threetime
winner of the world longdrive 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 counterintuitive 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 wellknown speedtraining exercise: swinging a shaft with a grip
but no clubhead, with the goal of generating as highpitched 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
torquevelocity 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 200300g 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. 
Last
modified  June 11, 2015
