Physics of Banked Curves at an angle [closed]

Question

When calculating the normal force of an object on a ramp inclined at an angle, the force of gravity is broken into components: $f_{gx}$ and $f_{gy}$, where $f_{gx} = f_g \sin (\theta)$ and $f_{gy} = f_g \cos (\theta)$ and the normal force is equal to $f_{gy}$ (since there's no acceleration in the $y$ direction).

However, when dealing with banked curves $F_n$ is not equal to $f_{gy}$, rather $F_{n}$ is equal to $f_g$. Why is this? Normal force is the opposite force of any force exerted on the surface. Since $f_{gy}$ is exerted on the surface, shouldn't $F_n$ be equal to $f_{gy}$ ? I am aware of centripetal force, but shouldn't that be taken care of by the horizontal component of $f_{gx}$?

Answer 1

If a vehicle is traveling around a frictionless banked curve then Newton's second law gives

$$ \sum F_x = N\sin\theta = ma \\ \sum F_y = N\cos\theta - mg = 0 $$

where $a$ is the horizontal acceleration (see figure). If the vehicle is to stay on the ground, then it cannot be that the normal force is equal to the gravitational force unless $\theta = 0$.