6.3: What is Force?
Just like mass, the concept of force can be defined \({ }^{1}\) via Newton’s second law:
\[\sum_{j} \overrightarrow{\boldsymbol{F}}_{i j} \equiv m_{i} \overrightarrow{\boldsymbol{a}}_{i} \tag{6.4} \label{6.4}\]
So if an object with mass \(m_{i}\) is observed to accelerate with an acceleration vector \(\overrightarrow{\boldsymbol{a}}_{i}\), Newton’s second law tells us that the sum of the force vectors acting on the object is \(\sum_{i} \overrightarrow{\boldsymbol{F}}_{i j}\). To determine individual force vectors \(\overrightarrow{\boldsymbol{F}}_{i j}\) (instead of their sum) one should perform experiments where only one force is acting on \(m_{i}\).
6.3.2 Force and Newton’s third law
Newton’s third law gives us information on the way forces act, since it holds for all known forces. It states that all (fundamental) forces act between two point masses. If there are two point masses \(m_{i}\) and \(m_{j}\), the force vector \(\overrightarrow{\boldsymbol{F}}_{i j}\) generated on mass \(m_{i}\) by mass \(m_{j}\) is always equal in magnitude and opposite in direction to another force vector \(\overrightarrow{\boldsymbol{F}}_{j i}\) that is generated on \(m_{j}\) by mass \(m_{i}\). In vector notation, as shown in Figure 6.2:
\[\overrightarrow{\boldsymbol{F}}_{i j}=-\overrightarrow{\boldsymbol{F}}_{j i} \tag{6.5} \label{6.5}\]
Moreover, these two force vectors are collinear (i.e. have the same line of action) to the relative position vector \(\overrightarrow{\boldsymbol{r}}_{j / i}=\overrightarrow{\boldsymbol{r}}_{j}-\overrightarrow{\boldsymbol{r}}_{i}\) connecting the two point masses. Newton’s third law, Equation 6.5, also shows us that a force \(\overrightarrow{\boldsymbol{F}}_{i j}\) can never exist alone, and is always accompanied by a reaction force \(\overrightarrow{\boldsymbol{F}}_{j i}\) acting on another point mass, which sometimes can be left out of the analysis for simplicity. In words Newton’s third law is often stated as ’action force is equal and opposite to reaction force’.
6.3.3 Properties of forces
Newton’s third law tells us something about the properties of forces between point masses. There are only four known fundamental forces, which are the gravitational force, the Coulomb force, the weak nuclear force and the strong nuclear force. Furthermore, there are many mechanisms and phenomena, which can result in a variety of forces and force characteristics like spring forces and contact forces. Some of these forces will be discussed in Sec. 6.13.