In Newtonian mechanics, force is a manifestation of the interaction between two objects.

One distinguishes between different kinds of forces depending on the nature of the interaction. The mathematical expressions representing different kinds of forces are different.

In macrosopic physics, as opposed to elementary particle physics, one distinguishes the following kinds of forces:

1. Two fundamental forces: gravitational and electromagnetic. These are long-range forces, which means that objects can interact via these forces when the objects are macrosopic distances away from each other, maybe a few centimeters or many thousands of kilometers.
   
   
2. Contact forces. These act only when the two interacting objects are in contact with each other. Examples are friction forces, whose direction always lies in the surface of contact, and normal forces, whose direction is perpendicular to the surface of contact. (The "normal", in mathematics, is the line perpendicular to a surface.)
   
   Contact forces arise from the electromagnetic interaction between atoms and molecules in the contacting surface layers of the two objects. Although electromagnetic in origin, these forces are not long-range when the objects in contact carry zero net electric charge.

Terminology. An object has momentum and has kinetic energy, but it does not have force. Instead, one says that a force "is exerted on an object". This terminology suggests that there is another object involved that exerts the force. The complete expression is to say that a force "is exerted on object A by object B". This way of talking about force makes it clear what force one is talking about.

Quantitative Definition. Force is a vector quantity. Assuming there is only one force exerted on a given object A, i.e., object A is interacting with only one other object B, then this force is defined to be numerically equal to the product of mass m and acceleration of object A,

= m.

The symbol in this equation stands for a mathematical expression, the force expression, that charaterizes the particular force involved. If the force exerted on object A is a fundamental force, the force expression involves parameters relating to object A and the other object B that is interacting with object A. E.g., if we are dealing with the gravitational force, the force expression is known as Newton's law of gravitation and involves the masses of objects A and B and the displacement vector from one object to the other. In the case of a contact force, e.g., the force applied to object A by a rope attached to this object, the force expression may be simply a specific vector whose magnitude has been determined empirically and whose direction is known also.

Note that the product m is not considered to be a force. This product is what the force expression is equal to.

If more than one force is acting on object A, i.e., object A is interacting with more than one other object, the product m is equal to the vector sum of the corresponding force expressions. This sum is called the net force. The statement that the net force acting on an object is equal to the product of mass and acceleration of the object is known as Newton's second law of motion.

The SI-unit of force is the newton (N).