# What is the Benefit of Plotting a Log of Net Force vs. Log of Acceleration Graph?

In preparation for a quiz, I've been going over a set of sample questions.

One question in particular asks that if I am to graph the logarithm of the horizontal net force of a system vs the logarithm of the horizontal acceleration, what does the slope and y-intercept of the graph mean?

This one has me quite puzzled. I can understand why you'd want to plot just net force against acceleration. I'm assuming the slope would be the mass and then your y-intercept is the amount of total force when the acceleration is equal to zero. However, I don't get why taking the log of each and graphing them against each other would really give you anything substantial. Any help would be greatly appreciated.

## 2 Answers

Suppose given an equation of the form $$y = Cx^n.$$ If you plot $y$ vs $x$ then it would be hard to find the value of $C$ or $n$ by looking at it. But if we take log on both sides we get

$$log(y) = n ~log(x) + log(C).$$

Now plotting $log(y)$ vs $log(x)$ would give us a straight line with slope $n$ and intercept $log(C)$. In your question the relation is $F= ma$. So plotting $log(F)$ vs $log(a)$ would give a line of slope 1 and intercept of $log(m)$.

It may seem like a pointless thing to do for this case since we already know the relation. But plotting logarithms are useful when you know that two quantities are related by some power law and you want to find the exact equation from experimental data you have.

You apparently expect the plot of force vs acceleration to be linear, but not everything you will plot (e.g. position versus time) will be linear. Another example: plot the mean orbital radius of the planets vs. the period of revolution the planets. It's not linear. But if you plot the log of the orbital radius versus the log of the period it will be linear, and the slope will be related to the mass of the sun, and you should get a slope of 2/3 or 3/2, depending on which item you put on the ordinate.

Your exercise is a simpler relationship: $$F=ma$$ $$log(F) = log(m)+log(a)$$ Is this a straight line? What are the slope and y-intercept? What does the slope of the logaritmic plot tell you about the exponential power relationship between $F$ and $a$.

What you have is simply an example of one way to investigate data and finding a possible relationship between two variables. If the log-log plot is a straight line, it's a simple power relationship. If the log-log plot is not a straight line, the relationship is more complicated, possibly a polynomial or more complicated function.