Hint: When analyzing an experiment, always try and work with straight lines whenever you can!Hey everyone, thanks for checking out this video about a really handy trick often used by experimental physicists. They use it when analyzing data that they've gathered. It's a really clever way to plot a straight line relationship using logarithms (or logs)!
The key principle is that we should always try to plot straight line relationships if possible, because it makes analysis much easier (and in this case, more powerful too). If we have a theory that tells us that two quantities are related by a power law relationship (i.e. one quantity is proportional to the other to some power) then we can take logs of this relationship.
After taking logs, we can plot the log of the independent variable on the vertical axis of a graph, and the log of the dependent variable on the horizontal axis. This, alongside the mathematical manipulation discussed in the video, allows us to not only find the power in our power law (remember one quantity is thought to be proportional to the other to some power), but we can also find the constant of proportionality. This is because the power is the gradient (slope) of our straight line of best fit, and the vertical axis intercept of our graph is the log of the constant of proportionality in our theory.
(Just as an aside, we can easily use linear regression to plot our line of best fit, because it finds the best line that minimizes the average distance between the line and the data points.)
This log method is powerful for a couple of reasons. Firstly, it allows us to find the power in our hypothesized relationship without having to guess what the power could be in order to make a plot. This easily shows us if our theory is wrong. For example, our theory might tell us that the power we are looking for is 2 (i.e. A is proportional to B^2) but our experimental results tell us the power is 3. In this case we know our theory needs tweaking. The added benefit here is that this log method will also tell us if the power is some non-integer quantity! We could find the power to be 2.53429 or something.
Secondly, there is a huge benefit to finding the constant of proportionality using this method too - if our theory is well developed enough that we know what we think the constant of proportionality should be, then we can check this from our experiment, with no extra data needed other than what we would normally gather anyway.
Lastly, if we cannot plot a straight line using this method, then we immediately know that the two quantities are not related by a power law! Maybe one quantity depends on the sine of the second, or something different - but it cannot depend on the second quantity to some power.
If you're unfamiliar with logarithms, I'd suggest starting here:
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