As the title states, my question is: Is the normal force always equal to the force of weight on an inclined ramp?
As an extension of this, when we are solving problems involving "a block slides down a frictionless, inclined plane that makes a 30 degree angle with the horizontal", why don't we have to take into consideration $mg\cos{\theta}$ or $mg$? I am told to solve it like this: $$a=\frac{F}{m}=\frac{mg\sin{\theta}}{m}=g\sin\theta$$
Best Answer
The weight of the object, $mg$, is split into components down the ramp and normal to the ramp. These components are $mg\sin{\theta}$ and $mg\cos{\theta}$ respectively. So to directly answer your question, the normal force is never equal to the weight of the object on an inclined plane (unless you count the limiting case of level ground). It is equal to the weight of the object times the cosine of the angle the inclined plane makes with horizontal.
When computing the acceleration of an object down a frictionless inclined plane, we are only interested in the component of force (weight) down the plane, namely $mg\sin{\theta}$. Since the plane is frictionless, there is no contribution whatsoever from the normal force.
See here to visualize how the weight of the object is split into components: