Physical principles for the golf swing
Torque
Forcetorque relationship
Dave Tutelman
 July 23, 2022
What is torque?
We know that a force moves a body. A torque is the same in concept, but
instead of moving a body, it rotates
the body. In order to do so, it requires at least one force plus an
axis for the body to rotate around. Often, more than one force is
involved.
That sounds pretty abstract. Let's look at a few examples to make it
more concrete.
Our first example is the knob on a
stove. As pictured here, it includes a bar that the fingers can push
against.
In the figure, there are two fingers, each pushing against the bar but
in opposite directions. In more precise terms, there are two forces
(the red arrows) in opposite directions. Note that the forces do not
have the same "line of action"; they are displaced from one another by
more than an inch.
There is an axis of rotation as well. The green centerline is an
extension of the shaft that the knob rotates on.
Note that the two forces are not only in opposite directions; they are
applied on opposite sides of the axis. That isn't necessary, but it is
frequently the case. As we shall see when we get quantitative, opposed
forces on opposite sides of the axis reinforce one another, and make
the torque larger.

The stove knob could just as easily
have been a doorknob, which is a common example to illustrate
torque. In this example:
 We see two opposed forces. The hand gripping the doorknob
is pressing on it hard enough to be able to apply a frictional force
tangent to the surface.
 We see the green axis, in the obvious place for it to be.
 The forces are not only opposed, they are applied on
opposite sides of the knob.

The next example, a wrench
rotating a hex nut, is a little different.
Here, there is only a single force. The axis here is the axis of the
(unseen but implied) bolt that the nut is attached to, depicted by the
green crossed dotted lines.
We are going to use this example to begin thinking about torques
quantitatively. But how do we even know this is a torque? How do we
know it isn't just a force acting to move the wrench, rather than
rotate it. We know because:
 There is an axis that wants to remain in one place; that is
where motion will be resisted.
 The line of action of the force does not go through the
axis; it is off to one side of the axis.
Intuitively, we should be able to tell by looking that this torque will
cause the nut to rotate in a clockwise direction. If you can't see that
yet, I hope you will by the end of this chapter.

Force required to produce torque
Let's flesh out some detail on that picture of a wrench applying torque
to a nut.
What we added in this picture is a measure of the distance from the
force to the axis of rotation. The distance is called the "moment arm",
and the torque produced can be referred to as a moment
as wall as a torque;
they mean the same thing. The formula for torque is:
Torque = Force * Moment
arm
Let's explore some of the implications of this definition.
Here is the same wrench, but with two forces (F_{1} and F_{2}) each with its
own moment arm (L_{1} and L_{2}). The forces
are exactly the same in magnitude and direction, but are applied to the
wrench in differeht places. Moment arm L_{1} is only half
the size of L_{2}. That means the
torque  the turning "force"  supplied by F_{1} is only half
that supplied by F_{2}.
Let's look at this algebraically. If you are
comfortable with quantitative reasoning, then we should express it in a
form conducive to quantitative manipulation  algebra. Let's designate
torque, or moment, as M_{i}.
Since the forces are equal, we will refer to them without subscripts to
distinguish them.
M_{1} = F * L_{1}
M_{2} = F * L_{2}
But we know that L_{2} is twice the
size of L_{1}, so our
equations become:
M_{1} = F * L_{1}
M_{2} = F * 2L_{1}
Apply a little high school algebra (or maybe it was junior high
school), and we can see that:
M_{2} = 2M_{1} or
M_{1} =
½M_{2}
That is the same as we concluded using words and intuition; now we know
it mathematically as well.
Now we can see why we need a big wrench when we want to make a screw
extremely tight, or to remove an overly tight screw. It isn't that the
wrench needs to be strong. It needs to be long!
Every added inch of length is more torque you can apply to the screw
using the same force.
So... how much more torque?
More to the point, what are the units of torque?
Let's remember that torque is a force times a length, so the units
would be a unit of force times a unit of length. In an American
mechanic's shop, the typical unit of torque would be footpounds,
a length times a force. In most other countries, where the metric
system is observed, the basic unit is newtonmeters,
for the same reason  a force times a length.
Levers
So far, we have been using photos
as the basis for our diagrams. Most textbooks use stylized drawings to
depict torque discussions. They
consist of a rigid beam or lever
sitting on a pivot point called a fulcrum.
Here's an example of that style. The distance from the fulcrum to a
force is called the lever arm
for that force.
For the units shown here, length in inches and force in pounds, the
units of torque are inchpounds. It is relatively obvious how to
convert that to, say, footpounds. Just change the inches to feet by
dividing by 12. So the two examples show torques of 20 footpounds and
40 footpounds.

Now we are in a good position to
see what levers are used for. Mechanical engineering texts call them
"simple machines", and they are used to:
 Amplify a force, at the expense of requiring more movement.
 Amplify a movement, at the expense of requiring more force.
 Redirect a force in a different direction.
Here is how each of those purposes works, along with the formal names
for each kind of lever.
Amplify a force
 In this diagram, we have a substantial load (the red force) that we
need to lift. Obviously, we need an upward force to lift it. We can get
by with a much smaller force (in green) if we give that force a longer
lever arm than the load has. The way it works is:
 The green force generates
a torque at the fulcrum, equal to the force times the lever arm.
 If that torque is greater than the opposite torque
generated by the load (its force times its lever arm), then the load
can be lifted.
 Note, however, that the force has to move farther than the
load. In fact, the force has to move as much farther than the load as
the force is amplified. For instance, if the force's lever arm is 3x
the load's lever arm, then the load will only be moved 1/3 as far as
the force has to move.
If we had already covered the physics of energy, it would be clear why
this has to be. It is simple conservation of energy, for those of you
who already know a little physics.
This explanation works for each of the lever descriptions below. The
only thing that differs is where the force and load are applied, and
the ratio of their sizes.
Note that this kind of lever is referred to in engineering textbooks as
a Class
1 Lever.
It is characterized by the load and force on the same side of the
fulcrum, and a longer lever arm for the force (to allow amplification
of the force).

Amplify a
movement
 This looks very much like the previous diagram, but this time the
load has a longer lever arm than the force. This allows us to move the
load farther than the force moves, but at the expense of requiring more
force. The way it works is comparable to a Class 1 Lever, but with a
few of the details reversed.
 The green force generates
a torque at the fulcrum, equal to the force times the lever arm.
 If that torque is greater than the opposite torque
generated by the
load (its force times its lever arm), then the load can be lifted.
 Note, however, that the force has to be greater than the
load. In
fact, the force has to be as much greater than the load as the movement
is amplified. For instance, if the force's lever arm is 1/3 the load's
lever arm, then the force has to be 3x as large as the load..
This is called a Class 2 Lever,
characterized by the load having a longer lever arm than the force, but
both still on the same side of the fulcrum.
Take special note of the fact that amplifying the movement requires a
proportionally larger force to do it. This is an important part of the
reason The Leverage Myth is a myth and not
a fact.

Redirect a
movement
 In our examples above, we were using an upward force to lift a load.
But suppose we have a load to be lifted, and the force we have
available is not upward. Suppose it is downward, or sideways, or at any
other angle. We can use levers to redirect that force so it provides an
upward push on the load.
In the upper diagram, we have a downward force that we want to use to
support or even lift a load. If the force and the load are on opposite
sides of the fulcrum, then the lever can turn the force around as we
need it. This sort of lever, with force and load on different sides of
the fulcrum, is called a Class 3 Lever.
The lower diagram shows how, if we allow the lever to be something
other than a straight rod, we can accept a force in any direction and
use the lever to redirect it as we need. The only thing that matters is
that the torque generated by the
force opposes the torque generated by the load.
 If the torques are equal, then the load is supported.
 If the torque of the force is greater than the torque of
the load, then the load can be moved. (More technically, we can see
that the load would be accelerated.)
BTW, it should be clear that any of these configurations can be used to
amplify a force or a movement by simply playing with the ratio of the
lever arms of the force and the load. Of course, it doesn't come for
free; whichever you amplify, force or movement, you lose proportionally
from the other.

Now let's
look at two special types of torques that have significance in
analyzing the golf swing: the couple,
and the moment
of a force.
Couple
A couple is a pair of forces on an object, those forces being:
 Equal in magnitude.
 Opposite in direction.
 Not operating in the same line, parallel and at a distance
from each other.
We have already seen an
example of a couple: turning the stove knob. Here we have two equal
forces of magnitude F in opposite
directions and separated by distance d. The torque of
the couple is the magnitude of each force times the distance between
them:
M = F d

Here
is an example of a couple in the context of a golf swing. It is the
"hand couple", a push by the right hand and an equal pull by the left
in the
process of "release".
A few fine points you should know, to prevent jumping to a wrong
conclusion:
 This is only part of the hand couple. Each individual hand
also contributes a couple of its own, consisting of a thumb push and a
smallfingers pull.
 The hand couple is not the dominant moment in the release
of the club. Yes, it is significant, but the moment of the force (which
we'll see next) is even more important.

Moment of a force
We have looked at the moment of a force already. Each of our wrench
examples shows the moment of a force.
In
the context of a golf swing the moment of force is critically
important. In particular (and we will put more precision on this
later), if there is no fixed axis to rotate around, we should assume
that the axis of rotation is through the center of mass of the object.
Here is an approximate image of the golfer and club during the
downswing. The hands are pulling the club's handle along a roughly
circular path. We see in the picture:
 A force F, the resultant
of adding together
 A force tangent to the hand path, accelerating the hands
along that path.
 A centripetal force perpendicular to the hand path,
creating the circular curvature.
 The line of action extended along force F.
 The distance d from the
center of mass of the club to the line of action.
The moment of force is:
M = F d
The moment or torque is in a direction to release the club toward the
ball. (We'll generalize this with a handy
rule of thumb next.)
In many descriptions of how the golf swing works, this is erroneously
identified as "centrifugal force" throwing the club outward. That
includes my own more naive description in the clubfitting tutorial. If
you are up to understanding torque at the level of detail in this
chapter, then you are better off thinking about moment of force than
the fictitious centrifugal force. And when I talk about "inertial
forces releasing the club", this is what I am referring to.
This
is an important concept, and we will return to it before we are done
with torque. It really is where most of the clubhead speed comes from.

Handy rule of thumb
Here's a visual rule of thumb you can use to tell which way a force's
moment will turn an object.
A force applied to an object will turn the
object so that its center of
mass will line up with the line of extension behind the force.
Here's an animation to illustrate the rule.
We have a force (red vector)
pulling at the end of a cylinder. The dotted red line is the line of
action of the force extended backwards. Notice how the cylinder is not
only moved by the force, it is also turned. That is the moment of force
at work. And  the point of this rule of thumb  it turns in the
direction that will make it line up its CoM on the dotted red line.
We encounter this again and again when thinking about what the pull on
a golf club's grip is doing to the motion of the club. A classic
example is the one we just saw, where the release of the club is driven
by the moment of the force pulling the club.

Last
modified  Aug 8, 2022
