The forces between surface, or contact forces, are essential for the analysis of mechanical systems. They are divided into two components: the normal forces and frictional forces.
The figure above illustrates the superficial forces acting on the system foot and ground. The normal force \(\vec{N}\), which acts on the foot, is the orthogonal component of the reaction for the force that the foot does on the soil, \(\vec{N} = -\vec{F}_{f,g}^{\,\perp \, g} \) . And the frictional force, \(\vec{F}_{fric}\), is a force parallel to the surface, and is the reaction to the force that the foot does in the ground, \(\vec{F}_{fric} = -\vec{F}_{f,g}^{\,\parallel \, s}\) .
A component of the force that appears in the contact between two surfaces is Normal, which is always perpendicular, or orthogonal, to the contact surface. This force can be understood using the law of action and reaction. When an object pushes the surface of another, this object will also be pushed with a force of the same intensity in the opposite direction.
The algebraic value of the normal force can be found using Newton's second law. In the case of a system with mass \(m\), moving with acceleration \(a^{\, \perp \, s}\) perpendicular to the contact surface \(s\), the equation for the normal force is: $$N + \sum_i^n F_i^{\, \perp \, s} = m \, a^{\, \perp \, s},$$ where \( F_i^{\, \perp \, s}\) are the \(n\) algebraic values of the forces' modules perpendicular to the surface \(s\), besides the Normal, acting on the system. Important: in this equation is necessary to consider the module with the algebraic sign, indicating its direction, i.e., the algebraic value. The superscript \((\perp \, s )\) is to remind you that the calculation of the Normal force only uses the forces and accelerations perpendicular to the surface.
The frictional force is the force component parallel to the surface. It manifests itself between two contact surfaces when there is an external force, having a component parallel to the surface, is acting. This force is divided into two cases: