When more than one force acts on a particle, the resultant force is determined by a vector summation of all the forces; i.e., $\mathbf{F}_R = \Sigma \mathbf{F}$. For this more general case, the equation of motion may be written as
$$ \Sigma \mathbf{F} = m\mathbf{a} $$
To illustrate application of this equation, consider the particle, which has a mass $m$ and is subjected to the action of two forces, $\mathbf{F}_1$ and $\mathbf{F}_2$. We can graphically account for the magnitude and direction of each force acting on the particle by drawing the particle's free-body diagram.
Since the resultant of these forces produces the vector $m\mathbf{a}$, its magnitude and direction can be represented graphically on the kinetic diagram.
The equal sign written between the diagrams symbolizes the graphical equivalence between the free-body diagram and the kinetic diagram; i.e., $\Sigma \mathbf{F} = m\mathbf{a}$. In particular, note that if $F_R = \Sigma \mathbf{F} = 0$, then the acceleration is also zero, so that the particle will either remain at rest or move along a straight-line path with constant velocity. Such are the conditions of static equilibrium, Newton's first law of motion.
Inertial Reference Frame. When applying the equation of motion, it is important that the acceleration of the particle be measured with respect to a reference frame that is either fixed or translates with a constant velocity. In this way, the observer will not accelerate and measurements of the particle's acceleration will be the same from any reference of this type. Such a frame of reference is commonly known as a Newtonian or inertial reference frame.
When studying the motions of rockets and satellites, it is justifiable to consider the inertial reference frame as fixed to the stars, whereas dynamics problems concerned with motions on or near the surface of the earth may be solved by using an inertial reference frame which is assumed fixed to the earth.
Even though the earth both rotates about its own axis and revolves about the sun, the accelerations created by these rotations are relatively small and so they can be neglected for most applications.
The equation of motion will now be extended to include a system of particles isolated within an enclosed region in space. In particular, there is no restriction in the way the particles are connected, so the following analysis applies equally well to the motion of a solid, liquid, or gas system.
At the instant considered, the arbitrary $i$-th particle, having a mass $m_i$, is subjected to a system of internal forces and a resultant external force. The internal force, represented symbolically as $\mathbf{f}_i$, is the resultant of all the forces the other particles exert on the ith particle. The resultant external force $\mathbf{F}_i$ represents, for example, the effect of gravitational, electrical, magnetic, or contact forces between the ith particle and adjacent bodies or particles not included within the system.
The free-body and kinetic diagrams for the $i$th particle are shown in Fig. 13–4b. Applying the equation of motion,
$$ \Sigma \mathbf{F} = m\mathbf{a};\\ \mathbf{F}_i + \mathbf{f}_i = m_i \mathbf{a}_i $$
When the equation of motion is applied to each of the other particles of the system, similar equations will result. And, if all these equations are added together vectorially, we obtain
$$ \Sigma \mathbf{F}_i + \Sigma \mathbf{f}_i = \Sigma m_i \mathbf{a}_i $$
The summation of the internal forces, if carried out, will equal zero, since internal forces between any two particles occur in equal but opposite collinear pairs. Consequently, only the sum of the external forces will remain, and therefore the equation of motion, written for the system of particles, becomes
$$ \Sigma \mathbf{F}_i = \Sigma m_i \mathbf{a}_i $$
If $\mathbf{r}_G$ is a position vector which locates the center of mass $G$ of the particles, then by definition of the center of mass, $m\mathbf{r}_G = \Sigma m_i \mathbf{r}_i$, where $m = \Sigma m_i$ is the total mass of all the particles. Differentiating this equation twice with respect to time, assuming that no mass is entering or leaving the system, yields
$$ m\mathbf{a}_G = \Sigma m_i \mathbf{a}_i $$