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Centripetal force is the invisible pull that keeps an object moving in a circular path; it's always directed towards the center of the circle. For objects in circular motion, like a car speeding around a track, centripetal force doesn't exist by itself; it's the result of other forces, such as tension, gravity, or in this case, the normal force exerted by the walls of a cylinder.

Think of it when you're swinging a ball attached to a string; your hand provides the force necessary to keep the ball in its circular path. Mathematically, this force can be described by the equation: \( F_c = m\frac{v^2}{r} \), where \( m \) is the mass of the object, \( v \) is its velocity, and \( r \) is the radius of the circle. Without sufficient centripetal force, an object will fly out of its circular path, following a straight line due to inertia.

Gravitational force is a universal force of attraction acting between all matter. According to Newton's law of universal gravitation, every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. On Earth, this force causes objects to fall towards the ground and is described by the simple equation \( F_g = m \times g \), where \( F_g \) is the gravitational force, \( m \) is the mass, and \( g \) is the acceleration due to gravity, which is approximately \(9.81 m/s^2\) near the Earth's surface.

In circular motion exercises, gravity can play a significant role, especially in vertical paths, as it constantly affects the centripetal force required to maintain motion.

Newton's second law of motion is the foundation of classical mechanics and states that the acceleration of an object is directly proportional to the net force applied and inversely proportional to its mass. Formally, the law is expressed by the equation \( F = m \times a \), where \( F \) represents the force applied, \( m \) is the mass, and \( a \) is the acceleration.

This law is pivotal in explaining how forces affect the motion of objects. When the forces are unbalanced, the object will accelerate. In the case of circular motion, this acceleration is directed towards the center of the path and is responsible for changing the direction of velocity, not its magnitude, because of the constant speed.

Circular motion physics involves studying movements where objects travel in a circular path. This can be either uniform motion, where the speed is constant, or non-uniform, where it changes. The critical thing to remember is that even if the speed is constant, velocity is not because velocity also depends on direction. Therefore, an object in circular motion is always accelerating, even at constant speed, because its direction is continually changing.

In the context of the remote-controlled car problem, we see how circular motion principles allow us to calculate the normal force at different points along the track. Furthermore, understanding the balance of forces involved in circular motion helps us predict the car's movement and the forces it experiences.