Car Moving Around a Banked Track

Basic Concepts

A car moving around a banked track follows a circular path, with its motion affected by several forces and principles of mechanics.
The car moves in a path that’s angled with respect to the vertical, hereafter defined as the angle θ.

The weight of the car acts vertically downwards, given by mg where m is the mass and g is the acceleration due to gravity.
There’s a normal reaction force (R) from the surface of the banked road which acts perpendicular to the inclined plane.
The frictional force may exist between the tyres and the road surface.
Because the car is moving in a circular path, an important force acting on the car is the resultant centripetal force, directed towards the centre of the circle.

The weight of the car can be decomposed into two components, one parallel to the inclined road (mg sin θ) and the other perpendicular to the road (mg cos θ).
At zero friction or when the car is just about to skid, the parallel component of the weight provides the necessary centripetal force for circular motion.

When no friction is involved or the car is about to skid, the component of the weight parallel to the incline, i.e., mg sin θ equals the required centripetal force mv²/r where v is the velocity and r is the radius of the circular path.
The normal reaction force balances the component of weight perpendicular to the incline, i.e., R = mg cos θ.

In reality, both friction and the normal reaction force contribute to providing the necessary centripetal force.
If a car moves at a speed higher than the one corresponding to zero friction, the frictional force also helps provide the required centripetal force, preventing the car from skidding.