[Physics] Why is the relationship between centripetal force and velocity straight, in circular motion

Question

I don't understand why the graph of force vs velocity^2 is a straight line in uniform circular motion. The equation for force in UCM is $F=m\frac{v^2}{r}$, I thought the graphs of such equations would be quadratic as the highest degree is 2. What am I missing to fully grasp the relationship between the two?

Answer 1

This is simpler than you think. You are quite correct that:

$$ F = \frac{m}{r}v^2 $$

So:

$$ F \propto v^2 $$

and a graph of $F$ against $v$ is a parabola. So far so good.

But now let's define a new variable $u = v^2$ so our equation becomes:

$$ F = \frac{m}{r}u $$

Now $F \propto u$ and if we draw a graph of $F$ against $u$ it will obviously be a straight line. You say:

I don't understand why the graph of force vs velocity$^2$ is a straight line in uniform circular motion.

and it's because when you plot $v^2$ on the $x$ axis, instead of just $v$, you are drawing the graph of $F$ against $u$ that I described above and it's a straight line.