Friction is defined as the force resisting the relative motion of solid surfaces in contact with each other. Friction in scenic motion appears all over the place: in obvious places, such as between the bottom of a pallet and the floor or deck or between the casters on a wagon and the floor or deck; and in less obvious places, such as between ropes and pulleys (and between sheaves and axles or cheek plates within pulley blocks). The friction forces in a system can be insidious: individually, they may be relatively small, but if unaccounted for, they can be the difference between an effect coming off as planned, or a moving unit sitting in place while technicians scramble to find out what went wrong.

Fortunately, the calculations that can help machine designers determine how much friction is resisting the movement of a particular unit are relatively easy. Unfortunately, these calculations depend on a piece of data called the coefficient of friction, which is different for every combination of surfaces in contact with each other; not tabulated in any useful or consistent way; and dependent on dry or lubricated applications. Think about it: sliding two blocks of unpainted wood against each other takes a different amount of force than sliding two blocks of painted wood against each other; it is also different than sliding two blocks of wet wood against each other, and different than sliding two blocks of wood which have a layer of butchers wax applied between them. Type of material, treatment of material, climatic effects (like humidity in your space), and all sorts of other effects alter the coefficient of friction which governs the amount of friction that exists between two surfaces.

With the inter-webs infiltrating every part of our lives, it would only seem logical that there would be a friction fan site (a "fan-friction site?") dedicated to tabulating various friction coefficients. I haven't found it yet. (If you have, or are so juiced up about friction you plan to create one, let me know, and I'll link it here; it'll probably have a pretty high hit count relatively quickly!) What this means is that we either have approximations or must develop empirical data to depend upon. Developing the empirical data isn't terribly difficult, but it does require time, which not many of us have in abundance. Alan Hendrickson describes two easy methods of determining coefficients of friction in his text, Mechanical Design for the Stage. He also provides some rule-of-thumb coefficients to use in a pinch.

Once you've determined a coefficient of friction that's accurate for the conditions you're modeling, the equation is relatively simple: for each instance of friction in the system, you simply multiply the coefficient of friction by the normal force in that instance. The result will be the force of friction resisting the movement in that part of the system. Sum all of these individual friction forces, and you've got the total friction force in the system.

Wait. Normal force? What?

You're right: I tried to pull a fast one there! There are a couple of terms we need to discuss: plane of contact and normal force. The plane of contact is that plane defined by the two surfaces in contact with each other. For a skid or pallet on the floor, the plane of contact would be the horizontal plane between the floor and the pallet. For a wagon on a rake, the plane of contact would be the plane between the top of the rake and the wheels on the wagon.

The normal force is defined as that force acting perpendicular to the plane of contact which causes the two surfaces to press against each other (which in turn generates the friction). For a unit sliding across the floor, the normal force is the weight of the object (or the force of gravity pulling down on the mass of the object). In the case of an object sliding at any angle other than parallel to the ground, only a portion of the weight of that object is acting to generate friction; that portion is the normal force.

Consider a solid cube dropped from a 100' height. If we could ensure that that cube could drop straight down, and dropped it so that it "slid" along a perfectly straight, smooth, wall, what would the friction between the wall and the cube be? Intuitively, we know there wouldn't be any; this is because there's no normal force acting between the cube and the wall--all of the cube's weight is acting to pull it down, not into the wall. Now angle that wall 45 degrees to the floor, and allow the cube to slide down it. We know the cube will travel more slowly because it experiences friction with the wall. This is because some part of the weight of the cube pushes it into the now-inclined wall. In the case of an incline at 45 degrees, we know, intuitively, that half the weight is pushing the cube into the wall, and half the weight is pulling it along the wall toward the floor. If the cube weighs 100 pounds, we can guess that the normal force is about 50 pounds.

The mathematical representation of this is pretty straightforward (though it takes some trigonometry and vector math to derive): the normal force (due to the weight of an object) is equal to the weight of that object times the cosine of the angle of the plane of contact (as referenced to horizontal), or Fn = Fw cos x, where Fn is the normal force, Fw is the weight of an object, and x is the angle between the plane of contact and horizontal. For example, our 100 pound cube sliding along the top of a rake whose angle is 5 degrees from the floor would experience a normal force of 100 pounds multiplied by the cosine of 5 degrees, or about 99.6 pounds.

Of course, weight isn't the only thing that can contribute to the normal force in part of a system; consider a stage hand pushing a wagon: unless they are pushing exactly parallel to the plane of contact (and once they've squat-walked around once during their freshmen year they'll never do it again), some portion of their efforts will actually contribute to the friction in the system, making their jobs a little bit harder. Consider a stage hand pushing a wagon at an angle of 45 degrees with a force of about 40 pounds; half of that force will contribute to lateral motion, while the rest is pushing the wagon down into the floor. This adds to the total normal force, increasing the amount of friction in the system.

If a typical coefficient of friction for a caster is, say, 0.04, that 20 pounds doesn't seem like much. Multiply 0.04 by 20 pounds, and you get about 0.8 pounds of friction force. This isn't a whole lot. But start looking at 1000 pound wagons pulled by cables which are not perfectly parallel to the floor, or elevators whose lift cables have to run over multiple pulleys, and you can suddenly see yourself adding hundreds of pounds of friction to a system in no time at all. This is especially true in theatre where--let's be honest about it, here--we're lucky to get accuracy to within a degree and an eighth of an inch. Those wide tolerances can add up fast.

All of which is to say, it's easy to assume because were using nice pulleys and castors, that friction isn't going to be an issue in our machines. That's a very...ahem...weighty...assumption to make. Better to take the time to include potential friction forces in our calculations.

(NB: I deliberately neglected to discuss the differences between static, kinetic, and rolling friction to keep from muddying the issue. These are important distinctions, and worth reading further about. Check Alan Hendrickson's book, or even the dreaded Wikipedia.)