[Physics] Why don’t we resolve $mg$ i.e. weight while solving problems related to banking of road in Circular motion

Question

It requires concept from Newton's laws of motion and circular motion. While solving for net force on a body on an inclined plane , we always resolve mg (weight) parallel and perpendicular to the plane but in the banking of road concept, we resolve normal reaction. Why don't we simply resolve weight and then equate it with normal reaction?

Answer 1

What you solve for depends on the coordinate system you choose.

• Choose a tilted coordinate system along the incline, and you must resolve the weight into components (because it is not parallel to any axis) but not the normal force (which is parallel to the y-axis).
• Choose a horizontal/vertical coordinate system, and you don't resolve the weight (it is already parallel to the y-axis) but you do resolve the normal force (since it is not parallel to any axis).

So, it is all about the choice of coordinate system.

People often choose it to fit along the acceleration. Not because it has to, but because it can be easier to handle.

• For an object sliding down an incline, the acceleration is parallel to the incline. Thus the choice of tilted coordinate system that follows the incline.

• For an object in horizontal circular motion, the acceleration is horizontal towards the centre of the circle (not along the incline). We can then pick the horizontal/vertical coordinate system here.