FORCES

The key of dynamics is on the “free body diagram” or FBD where forces are sketched. But, what are forces? The question seems stupid, everybody knows what forces are. It’s not difficult either to understand why they’re shown with tiny arrows (vectors), everybody can figure that out. But…

Believe me if I tell you that the 80% of mistakes in the resolution of problems that have to do with dynamics come out from the following:

a) Students tend to invent forces that are not playing a part or do not exist.

b) Students usually do not take into account forces that are actually playing a part in the problem, but they did not realise that they were there.

c) Students notice a force but they place it on the wrong body, or in the opposite sense, or they do not realise that it is actually the reaction of a known action and, therefore, it is not an unknown variable.

I can assure you that the quid of dynamics is this one: state the forces with which we deal in each problem and do it right. I will tell you the most usual mistakes and give you some advice. Take it or leave it.

Forces aren’t tangible, visible things: they are inventions of mankind to explain natural phenomena. A guy who is experimenting various forces, for example, is not struck by arrows like John Wayne at the end of the movie. You have to reason, deduce each vector.

To start, you have to make up your mind that, however familiar a force can seem, there HAS to be a body exerting it, producing it; and another one receiving it, “suffering” it. And when I say “body” (don’t laugh for what’s coming now) I mean something touchable, visible, heavy, colourful, hard or soft… an OBJECT, something that actually exists… you know what I mean. Nope, it seems you don’t. Let’s see… is gravity an object? Nooo! Congratulations! Gravity is not an object, therefore, it cannot exert or receive forces. You’re starting to get it.

You have to sketch every force that takes part in the dynamics of a body in a FBD. For example, I have borrowed these two from a problem in which there are two karts: A and B.

Here come the FBD, take a look:

I’m going to start analysing the forces that operate over kart A. There is a horizontal force to the right called F. I ask myself, who applies that force? I don’t know, but the problem says “A horizontal force of 30 kgf is applied over the kart A”. This means that the person who wrote the problem guarrantees the existence of that force. Something or someone has to be doing it: a locomotive, Tarzan, whatever, but something or someone is doing it.

F_BA is the force that B applies on A. In the situation of this problem, the karts are touching each other, that’s why that force exists.

P_A is the weight of A. Who drags A downwards? Earth! That’s the object, round, voluminous, sunny, pretty…

V_A is the force that the railways extert over A. Any other forces acting over A? If I ask that question during class, I’m sure someone courageous will answer “The normal!” OK, I answer, and I draw it (a vertical arrow pointing upwards) and then I ask “who applies that force, who pushes the kart upwards?” That question has no anwer, that means you were about to invent a force that wasn’t there. We erase it.

Someone, trying to rescue his frend, can say “The railways!” Great, but we already named that force V_A… don’t like the name? Wanna call it “normal”? Name it whatever you like (within certain limits), but don’t draw it twice!

I insist on the question: what other force? “The force of velocity”, “And who exerts it, who applies it?” No one? Then it doesn’t exist. That’s it.

Never call different forces the same way. In this problem, for example, a 65.3% of the students who draw a FBD call W the weight of each kart… WRONG. In algebra, which is the language of physics, P = P. They are two different forces, they must have different names. They could have the same value, nevertheless they must be called differently. Use subscripts, that’s what they’re there for. Some insolent kids answer (I’m actually into the character of teacher) “but if I know what body I’m referring to, when I use it I will replace each one with the correct value”. It’s wrong anyway.

You may not know it, but what you have to learn is to produce a message directed to another person, not to you. Nonetheless, you did make a mathematical mistake, and it’s very good of your professor to correct it. Anyway, don’t forget that every force you find is going to become a number that later will travel through an algebraic forest. It’s very likely of them to get confused and lost.

When two bodies interact, and you are interested in the dynamics of both of them (like in this problem about karts), the best way to dominate the forces is with the double subscript, which also emphasizes clearly the action-reaction couple: the length of F_BA is equal to that of F_AB.

I have more pieces of advice, but I have the feeling that you must be kind of tired. Practice will do the rest. Congratulations for having gotten all this way.

• The concept of force was originally introduced by Archimedes, although in purely static terms. Galileum was the first one to give a definition in terms of dynamics. Newton is considered to be the first one who gave a mathematical and modern definition of force.
• Nowadays, there are four fundamendal forces considered as such: gravitational, electromagnetic, and weak and strong nuclear forces. The first one models the universe at a macroscopical scale. The second one shapes the structure of matter from the atomic to the mollecular and particular scale. Nuclear forces govern subatomic scale. And then, there’s will, you know.

TRICKY QUESTIONS:

• What are the units in which forces are measured?