Normal Force from Newton's Laws

Question

I have a simple question, if we put a body of mass m on a fixed incline, can we deduce from Newton's laws that the motion of the body has to be along the incline? Consider that we want to predict what happens and we have not made any observations whatsoever.

I know that we take the normal force to be equal to the component of gravity perpendicular to the incline by saying that there is no observed motion along the perpendicular, but in this case we have tacitly assumed that the motion has to be along the incline(by observation/experience). Can we justify this assumption using Newton's laws? So why should the normal force equal the component of gravity perpendicular to the incline?

PS:I know my question is stupid..but it would be helpful if someone can clarify my doubt.

Answer 1

Your question is absolutely not stupid. The answer is no, we cannot make such a deduction based purely on Newton's laws.

The fact that the body in question moves along the incline and not into it (or away from it, for that matter) is a constraint which we are imposing by hand. Only after we do so can we use Newton's laws to calculate the normal force which must exist between the incline and the block.

As a general rule of thumb, there are two broad categories of things in Newtonian mechanics. First, there are dynamic objects with masses and accelerations, to which we can apply Newton's laws to determine how they will move. Second, there are constraints, which we impose when we already know (either from experience or by physically motivated assumption) how they will behave to good approximation. This can dramatically simplify the analysis, but we should be careful to remember that we're doing it.

The incline in your problem is an artificial constraint, in the sense that it constrains the motion of the body to be along its surface and does not otherwise obey Newton's laws. In principle, it is possible to treat the incline as a dynamic object which can be compressed and deformed, but (a) this is far beyond what could reasonably be done in an introductory course and (b) the assumption that the incline serves only to constrain the body's motion is a pretty accurate assessment of what actually happens under standard laboratory conditions - though if the incline were unusually squishy or the mass unusually heavy, this assumption might break down.

Answer 2

Newton’s second law is

$$F=ma$$

Where $F$ is the net force acting on $m$ and $a$ is the resulting acceleration of $m$ in the direction of the net force. In the direction perpendicular to the incline there are two forces acting on $m$, the component of the force of gravity acting on $m$ perpendicular to the incline and the equal and opposite reaction force of the incline acting on $m$, for a net force of zero. Therefore $a$ perpendicular to the incline is zero.

Hope this helps.