What Does "Scale As" Mean?

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Chapter 1


Physics is the study of how everything works. We observe how variables behave and interact with each other, and from those observations we attempt to uncover general rules that govern the interactions.

Say you're planting flowers in a row along a sidewalk. Each sidewalk section is 4 feet long, and each of your flowers needs 1 foot of space in which to grow. You need to know how many flowers to buy, per section of sidewalk. Let's set up a mathematical equation to figure this out.

(1 sidewalk section) divided by (1 flower)
= (4 feet) / (1 foot)
= 4 flowers per sidewalk section
1 flower per (1/4) sidewalk section.
If flowers are our X variable and sidewalk sections are our Y variable, then we can say:

X = (1/4)Y.

While X and Y are our variables, the number (1/4) is a constant. This name indicates that while we are free to change X and Y, the number (1/4) does not change.

In an introduction to a physical law, we tend to be more interested in the way variables relate to each other, than we are in constants. Constants are necessary in calculating exact amounts, but if what we want to do is merely uncover the general relationship between two variables, it is often simpler to forget about constant and focus only on the variables.

For example, in the flower-sidewalk calculation, if all we want to know is the general relationship between flowers (X) and sidewalk sections (Y), then for the moment we may ignore the (1/4) and just say

X α Y, which reads:
"X scales as Y"
or "X goes as Y"
or "X is directly proportional to Y."
Now, I won't be surprised if you disagree with this statement. After all, if you want to know how many flowers to buy, you certainly do need to remember that factor of (1/4) -- otherwise you're likely to buy far too few (or too many) flowers! This practical application is one in which an exact answer is important. And as I just mentioned, if you want an exact answer, then you must keep all your constants.

But if all you want is a grasp of what is going on, it is useful to ignore constants. Your first encounter with the Inverse Square Law will be an instructive example:

Experiments indicate that the gravitational attraction between two bodies is related to the mass of each body and the distance between the two bodies, by the following law:

Force of Gravity == F = G(M1)(M2)/(R2),
  where M1 = mass of first body
        M2 = mass of second body
        R = distance between the centers
            of each body
        G = an experimentally determined
This equation says that the force of gravity increases as the product of the bodies' masses increases. That is, the force is directly proportional to the product: (M1)(M2.) And the force decreases as the square of the distance increases, or: the force is inversely proportional to the square of the distance.

Generally, then, the force scales as the product of the masses divided by the square of the distance:

F scales as (M1)(M2)/(R2), or
F α (M1)(M2)/(R2)   (Equation 1.)
If we want to make any exact calculation using this relation, we include the constant G:
force equals G(M1)(M2)/(R2), or
F = G(M1)(M2)/(R2)   (Equation 2.)
Which is more useful, Equation 1 or Equation 2? Well, it depends. If you have just been introduced to them, equation 1 is better. More important than being able to make calculations is to solidly grasp what is happening. You need to understand how the variables interact with each other. Until you do, then adding constants into the equation might complicate it and make the fundamental relationships more difficult to see.

If you are already familiar with the concept, and you want to use it in a calculation, then you need G. The numerical value of G depends on the units in which you measure the other variables.

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