Working with Ratios

In this class, most of the numerical problems you will get can be treated as a ratio problem. Anytime you see words like "how much bigger", or smaller, or further away, or fainter, or hotter, or any sort of comparison, you can be pretty sure that we have in mind you will make the comparison by taking the ratio with something else (that will be given). Your task is then rather simple. Essentially, you just write out the relevant equation and replace each variable with a ratio, remembering to invert the subscripts if the variable appears in the denominator. Then plug in what you  know, and solve for the value you were asked for.

If that didn't tell you everything you need to know, we now supply a more detailed story. You must

  1. identify the variable that is to be compared,
  2. identify the relation (equation) that tells you how to make the comparison,
  3. find the needed given data,
  4. re-write the equation so that the variable to be compared is on the left, and the equation is in a ratio form, and
  5. solve for the answer (basically just plugging in a few numbers and dividing).
Let's see how this works!

An Example

Suppose the question is "How much faster will a feather accelerate than a hammer if they are pushed by the same force". Let's go through the steps above to solve this problem. You recognize it is a ratio problem because it starts "How much faster..."

1) identify the variable to be compared. In this case, you are asked about "acceleration".
2) we need an equation that relates acceleration, force, and something about feathers and hammers. Newton's force law will do; the mass of the feather and hammer are the other relevant variables. So the equation is F=ma.
3) There's not much data here, the main thing the problem says is that the force should be the same. Let's call the force on the feather F1 and the force on the hammer F2. Then we have that F1 = F2. Since the problem does not say what the masses are, we'll just call them m1 and m2 . Presumably m1 is smaller than m2.
4) We want to re-write the equation to solve for a ratio of accelerations. First, we just re-write it so that "a" is on the left, namely a=F/m . Then we invoke the rule that if we divide one valid equation by another, the ratio must remain valid. In this case, the two equations are just a=F/m for the feather, and the hammer. Thus we can write:
a1 /a2 = F1 /F2  m2 / m1 .
5) the answer we are looking for is the ratio a1 /a2 itself. We know that F1 = F2 , so they cancel out of the equation. We are left with a1 / a2m2 / m1 . That is the answer! The feather's acceleration is faster than the hammer's by the fractional amount that the hammer's mass exceeds the feather's *. If we had given you the masses involved, you would plug in those numbers, and get out a numerical ratio for the accelerations. Note that all ratio answers should end up without units (a ratio is a pure number). Thus, you can check what you are doing if you carry units along with each number you plug in. You can cancel units in ratio just like you do numbers. If at the end you haven't gotten rid of all the units, you have done something wrong.

Using Units as a Check

For example, suppose we had told you that the hammer has a mass of 2 kg, and the feather has a mass of 10 grams.
You would then write a1 / a2m2 / m1 = 2 kg / 10 gm = 1/5 kg/gm. You can see that something is wrong, because the units didn't cancel, although it is good that at least they are the same kind of unit (mass). To get rid of the units you will have to multiply by something with the units gm/kg ; that will cancel what you have there now. Basically what you need is the conversion factor between gm and kg: 1000gm=1kg. So if you multiply by 1000gm and divide by 1kg, you won't have changed anything (like multiplying by 1), but the units will get fixed.
a1 / a2 =  1/5 kg/gm x 1000gm/1kg = 200 . The feather will accelerate 200 times faster if it is 200 times less massive.

Although this may seem like a lot of work, and very complicated, once you get used to what is happening here it will seem very simple, and is very easy to apply. Comparisons like this are quite useful in life, you probably do a version of this all the time without really thinking about what you are doing. The difference here is that we have made explicit the rules that are being followed (the equation), and must be careful in their application.

* You may be surprised that this is the answer, since when you saw the astronaut drop a feather and a hammer, they hit at the same time. That is because in that case, the force being applied to the feather and hammer are both due to the Moon's gravity. Since the gravitational force is proportional to the mass, the two forces are not equal! In fact, the force on the hammer is greater by just the right amount to cancel the tendancy of the feather to accelerate faster if the forces had been equal. The inertial mass of the hammer is exactly equal to its gravitational mass. This is what Einstein had in mind when he stated his "equivalence principle"; it didn't have to be that way, but it is!