The Power of One

December 2004

Strange as it may seem, Intro Physics students often don't appreciate the simplifications we make. Instead they can fixate on the idea that they are dealing with idealizations, simplifications, and arbitrary/convenient choices that mean the subject is unrelated to the real world. I got thinking about this again last month while working on the Mechanics Section of my online Physics problem & solution set. In writing out solution hints for some problems I really wanted to write "just set mass=1" to make it easier. Then I thought, "but what if... this simplification may create more problems than it solves".

Here is the issue. Many students get lost doing algebra, carrying many unknown quantities around in equations. But unless you have a massive number of special case equations available, reducing the subject of Physics to another form of "stamp collecting", this is exactly what you need to do. What generally happens is that any physical quantities not given in the problem or required in the solution could have any non-zero value (maybe even zero). For example, the mass is irrelevant to the speed of a block sliding down an inclined plane. This presents me with some dilemmas:

every formula the student is going to use (normal force, weight, friction) involves mass
however, the mass is irrelevant. Once all formulas are combined, it will cancel out everywhere
if I don't mention a mass value, many students will get lost due an extra unknown that appears all over their formulas
if I do give a mass value this promotes the idea that it matters, or a brighter student might conclude this is a "trick" question because I've included irrelevant numbers. Another student still might stumble on the fact that I gave a number and yet it doesn't seem to affect the solution.
possible solution: tell them "just set mass=1" in the solution notes, and explain that since the mass is irrelevant they can avoid the risk of bad algebra by assigning a convenient value that doesn't introduce any new calculations

The first time you hear a proposal like this it may sound very odd, but it is something I do all the time when solving problems - set irrelevant values to one (or, in some cases, zero). Without thinking about it much we make such convenient/arbitrary choices all the time. For example, in solving a trajectory problem we break up into horizontal and vertical components. Why? Is that a more accurate representation of the problem than some other coordinate system? If I'm told "a bullet is fired at 1000 m/s at an angle of 35% from horizontal", why do I start by drawing an x-y coordinate system with y being vertical and x being along the ground, then breaking the initial velocity up into horizontal and vertical components? Couldn't I draw a coordinate system with the x axis pointing in the direction of the initial velocity, then break the acceleration due to gravity up into components relative to that system? Sure I could, and I would get exactly the same answers, but it would be a lot more math to solve. On the other hand, this second proposal is exactly what we usually do for a problem such as pushing an object against friction on an inclined plane. In that case, the coordinate system with the x-axis parallel to the ground would be much harder to work with. When we solve problems there are often choices to be made which may appear arbitrary to a beginner, but with some experience you learn which choices involve the least work. Being able to make those choices is one of the benchmarks that you are becoming proficient.

In fact, at more advanced levels Physicists often take this one step farther by working in "unit systems" where various constants are simply set equal to one because at the end you know what fudge factors will be required for various results. In that case you're supposed to know things like "if the final answer you're looking for is energy then multiply by xxxxx" or "if the the final answer you're looking for is time then multiply by yyyyy". This ignoring of the constants in formulas can be very convenient for writing and solving problems, but without clear explanation it makes learning the subject more difficult because you can't rely on our old friend dimensional analysis to help you out when looking at the formulas.

So, back to mass=1... In the end I decided that since these problems are intended for beginners, telling them this will probably create at least as many problems as it solves. The beginning Physics student needs to learn by doing many problems that in fact the value of the mass doesn't matter in these particular problems. That is what the problem set is designed to do, provide the student with limitless practice on particular problem types. So, instead of mentioning this in a quick solution hint, I'm putting a link to this page on some problem hints. That way, any beginning students who see the hint get all these qualifications. It lessens the chance of drawing a wrong conclusion, such as "mass never matters with block on a plane problems" or "if they don't mention the mass you can set it equal to one" - conclusions which would lead them to get some problems wrong.

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