[Physics] Looking for an intuitive understanding of normal force

Question

I understand normal force to be the perpendicular force to a surface of contact. However, I have come across a problem which has caused me to rethink this.

My initial understanding of force is demonstrated by the following diagram of an object sliding down an inclined plane.

Here you can see that $F_N = F_g * cos(\theta)$

Now look at this banked curve problem.

This is taken directly from Wikipedia here. It says that $F_N = F_g/cos(\theta)$ which is bigger than in the first scenario.

Why is the normal force different here?

The normal force comes from Newton's third law, as a reactive force. Before there is any centripetal accel, $F_{N}$ was $F_{g}∗cosθ$, but after there is centripetal accel, $F_{N}$ changes to $F_{g}/cosθ$. This makes $F_{N}$ larger in the second scenario even though there is no more force acting in the direction of the ground. What causes the increase in $F_{N}$? Something must push against the ground in order for $F_{N}$ to push back.

My thoughts and attempts at solving this:

There has to be a force acting in the direction of the road in order to increase the value of the normal force. I think this force comes from the fact that the car's velocity is causing a force on the road, pushing in to the road because the car's velocity wants to go straight, but the road is shaped in a curved bank so the car cannot go straight; the normal force of the road pushes back, which allows for the centripetal acceleration.

I've tried accounting for the difference in $F_{N}$ and I've found that $F_{N}$ in the second scenario is $Fg*cos\theta + Fg*tan\theta*sin\theta$ the difference being the second term. I haven't been able to describe the difference in $F_{N}$ in any other way. I believe the difference in $F_{N}$ should should somehow be related to the centripetal acceleration.

I think the normal force is a result of the car's (the ball in this case) velocity against the circular road causing the road to push back. I lack the mathematical ability to describe it.

Answer 1

You have asked what causes the increase in force from the earlier case, try seeing it from the body's frame as it will be in rest there. The body exerts a centrifugal force on the plane along with gravitational force, the resultant of these two forces is matched up by the normal provided by the banked road, now since the resultant of centrifugal force and gravitational force is more than gravitational force alone, the normal force increases in magnitude.

In terms of equation :

$F_g \cos(\theta) + m a_c \sin(\theta) = F_n$
$ F_g \sin(\theta) = m a_c \cos(\theta)$

But if you see from ground frame the normal partly provides the centripetal force and also cancel outs the gravitational force i.e.

$F_n \cos(\theta) = F_g$
$F_n \sin(\theta) = m a_c$

In both frames as normal is involved in dealing with both forces it is greater than just the component of gravitational force.