🌿 A Framework for Solving Hard Physics Problems

This note only concerns itself with course problems. Solving research problems does have some similarities but is a different kind of hard than course problems.

Studying physics you have to solve a lot of difficult problems, and over time you develop some intuition for how to solve hard problems which I want to share today.

  1. State all the known variables. What information does the problem give you? Problems are usually made such that it gives you all or nearly all the information you need to solve the problem directly, so carefully examining the information given can give valuable insights into how to solve the problem. However, sometimes you have to make some reasonable assumptions to fill in the information the problem doesn't give you directly.
  2. State related concepts. Next, it can be helpful to state all the related concepts. If the problem is a part of a book, the related concepts can often be found in the previous chapters. Otherwise, you have to do some more thinking yourself.
  3. Relate the known variables to the related concepts. Can you use algebra to relate the known variables to the related concepts? Or do you need to do other conversions?
  4. Examine the solution. Do the units match up? Does the answer match up with your intuition if not can you explain why not?


Okay that's a lot. Let us look at an example. Let us look at problem 1.37 of Daniel Schroeder's Introduction to Thermal Physics. The problem goes like this:

In a Diesel engine, atmospheric air is quickly compressed to about 1/20 of its original volume. Estimate the temperature of the air after compression.

1. State all the known variables

We note that we are talking about atmospheric air, and that it is quickly compressed. Finally we note that the final volume $V_1=\frac{1}{20}V_0$.

Moreover, we can make some additional assumptions based on the problem which it didn't exactly give us. We can assume that the initial temperature is a standard temperature such as $300$K, and we can assume that the degrees of freedom of the atmospheric air is $f=5$, since atmospheric air is mostly diatomic, and the temperature is low enough for the vibrations being frozen out. Finally, we can assume that the initial volume is something like $1L$

2. State related concepts:

Since the air is compressed quickly, we can assume that we are talking about adiabatic compression, so $V_1 T_1^{f/2} = V_0 T_0^{f/2}$

We can also note that atmospheric air is well approximated by the ideal gas equation $PV=NkT$. However that doesn't seem too useful for this problem.

3. Relate the known variables to the related concepts

We are interested in the final temperature $T_1$, and we know the initial temperature $T_0$, and we have estimated the initial volume $V_0$. Finally we know the final volume as a function of the initial volume, and the degrees of freedom of atmospheric air, so it seems that the equation is sufficient to solve the problem with just a bit of algebra. We find

$$T_1 = \Big(\frac{V_0}{V_1}\Big)^{2/f} T_0$$

If we insert our values, we find $T_1=994\text{K} \approx 720\text{C}$.

4. Examine the solution

First, we check if the units match our expectations, we expect the temperature to be a unit of temperature which is a sign that we have done the algebra correctly.

We expect the final temperature to be higher than the initial temperature since we are doing adiabatic compression. We see that this is the case. Finally, we expect the temperature to be higher than the autoignition temperature of Diesel as Diesel engines do not need spark plugs. We see that the autoignition temperature of Diesel is $210$C which is well below the final temperature.

After the critical examination of our solution we conclude that the solution is likely a good solution.

What if that doesn't work

One powerful tool if the above doesn't work is to try to relax the problem. Can making additional assumptions help solve a simpler version of the problem? Does that give some insight into how you can solve the more complicated problem?

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