Physical principles -- torque 1

Force-torque 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 (F₁ and F₂) each with its own moment arm (L₁ and L₂). The forces are exactly the same in magnitude and direction, but are applied to the wrench in differeht places. Moment arm L₁ is only half the size of L₂. That means the torque -- the turning "force" -- supplied by F₁ is only half that supplied by F₂.

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.

M₁ = F * L₁
M₂ = F * L₂

But we know that L₂ is twice the size of L₁, so our equations become:

M₁ = F * L₁
M₂ = F * 2L₁

Apply a little high school algebra (or maybe it was junior high school), and we can see that:

M₂ = 2M₁ or M₁ = ½M₂

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 foot-pounds, a length times a force. In most other countries, where the metric system is observed, the basic unit is newton-meters, 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 inch-pounds. It is relatively obvious how to convert that to, say, foot-pounds. Just change the inches to feet by dividing by 12. So the two examples show torques of 20 foot-pounds and 40 foot-pounds.

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 small-fingers 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

Physical principles for the golf swing