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Also see George's essay "What is the Role of an Amateur Scientist?

21 December 2001

What Kinds of Pushes are There?

by George E. Hrabovsky, President of MAST

News from MAST

Hello again. Welcome. From all of us at MAST let me wish all of you a very happy holiday season.

This column will be about the different kinds of forces that exist.

What Forces are There?

Last week we established that you can derive formulas for forces based solely on the variables we use use to express them. We also established that by examining such variables we can determine if a formula is correct, since all terms of a formula must have the same units. Perhaps I should clarify what I mean by a term. It is another of those words that we use without really thinking about. A term represents one part of an expression separated by addition or subtraction. The following expression has two terms.

Another way of writing this is,

This formula has three terms,

This one has four terms,

So, we have now talked about mass and force in general. It is time to get specific. Given Newton's second law of motion,

what forces exist to replace the ?

There are two ways to look at this question. We could ask it and then make an encyclopedia of the different expressions for the forces that exist. This is a very useful thing to do, but it takes a long time and lots of space. The second way is to think about what sorts of forces are possible. How can we do this? We look at Newton's second law and realize that the part of it tells us that force always depends to one degree or another on the mass and acceleration of an object. So, in out first approximation we can say that force is a function of mass and acceleration,

A good start, but we must look more closely. Acceleration itself is a function of velocity and time (it is the time derivative of velocity). Recall from earlier columns ("Pushing the Pedal", and "Free for Fall") how acceleration and velocity are related. So, we now realize that force is actually a function of mass, time, velocity, and acceleration.

Now we think about velocity and realize that it is a function of position and time (see the column, "How Fast?"). This leads us to realize that force is actually a function of mass, time, position, velocity, and acceleration.

That's it! This covers all of the forces that we can imagine in mechanics alone. There are other forces due to charge and properties that only come into play within the nucleus of an atom; and we will eventually talk about these, but those forces way beyond our capabilities at the moment. Indeed there are only three forces that our formula above does not cover, this formula establishes every other force imaginable! I think that is enough of a revelation for this week.

Theory Challenge Answer for Last Week's Column

Given an area of a surface over which a layer of fluid is flowing with velocity , we will see layers of fluid flowing at different velocities from that we will call and that these layers will be separated by a distance . A force will be generated parallel to the surface but opposing the direction of flow, called friction. It turns out that a formula for this force is given by

where is the coefficient of viscosity and is normally given in units of poises (1 poise = 1 gram per centimeter per second). Converting this into dimensions we have,

>From this we can see that the formula is dimensionally correct.

Theory Challenge

Determine what your example force is a function of (what are its independent variables). Do not include constants!

Books That I Like

This is actually a fairly simple idea and it escapes me as to why it only really appears in advanced undergraduate texts on mechanics. Two such books follow (and, incidentally, I know all three authors reasonably well).

Keith R. Symon (1971), Mechanics, Addison-Wesley Publishing Company. One of the best undergraduate-level tests, though it doesn't discuss chaos, it covers just about everything else you could want, it has lots of practice problems (with the answers to odd-numbered problems), and pretty lucid discussions of the concepts.

Vernon Barger, Martin Olsson (1995), Classical Mechanics: A Modern Perspective, McGraw-Hill. I like the order of topics in this book. It uses Lagrange's equations of motion beginning in the third chapter and focuses on the conservation laws. It has chapters on gravitation, Newtonian cosmology, special relativity, and even an introduction to chaos. One word of caution, do not think of this rather short book as being casual, many of the derivations are sketchy with many details left to the reader to sort out.

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Converted by Mathematica      December 19, 2001