homework and exercises - For circular motion in a vertical plane, why does Net Force = Centripetal Force?

I'm struggling with some of the concepts pertaining to the forces and acceleration associated with circular motion in a vertical plane (only concerned with what happens at the 'top' and 'bottom' of the loop for now though).

Let's say that we have a roller-coaster going around a circular (vertical) track. I understand that at the bottom of the loop, the $F_{Net}=F_{Normal}-mg$, and this makes sense. But what is confusing me is why $F_{Net}=ma={mv^2\over r}$. I realise that ${mv^2\over r}$ is the centripetal force acting inwards (towards the centre of the loop), and $a$ is the centripetal acceleration, but why does this equal the net force?! Shouldn't we take into account the acceleration due to gravity?

eg. If we are trying to find the Net Force, why don't we find the Net acceleration, which would be something like "Centripetal acceleration - acceleration due to gravity" (at the bottom of the loop)?

So if I was to find the Net Force on 60kg person at the bottom of a roller coaster of radius 9m, travelling at 18.8 $m/s^2$, I could just use $F_{Net}={mv^2\over r}$? This seems to me like we just ignored gravity :/

Any help much appreciated, I feel as though I know enough to harm myself but not enough to properly understand it

Smeato

Answer

The question you bring up is a very common one, and the source of some difficulty to novices. I know it's been addressed here, but I can't find a good presentation. What follows is not a good one, but it might be enough to answer the question.

Centripetal force describes a force (often the net of several forces). It does not name a force. Gravity is a force. Friction is a force. Centripetal is not a force. "Centripetal" describes a net force that points toward the center of a circle. In the event that the speed of the object is not changing, or if observing only over a very short period of time, then kinematics demands that $a_c=v^2/r$ and Newton's second law says that there must be a net force that produces that acceleration. We know that there is a centripetal force by observing the motion of the object, not by studying the interactions between objects as we would for gravity or friction. In a sense, "centripetal force" is more a statement of kinematics than dynamics.

The motion tells us that there must be a centripetal force. What the nature of that force is presents a different question, one that is answered by analyzing the actual real forces, those due to interactions between objects.

Kinematics tells us that the following must be true: $$F_\mathrm{net} = \frac{mv^2}{r}$$

Analysis of forces at the bottom of the loop tells us that $$F_\mathrm{net}=N-mg$$