Newton's first law of motion is a subset of second law in the sense that first law is just a specific description of motion, when net external force on the body is zero.

F = m a = 0 F = m a = 0

This means that if there is no net external force on the body, then acceleration of the body is zero. Equivalently, we can state that if there is no net external force on the body, then velocity of the body can not change. Further, it is also easy to infer that if there is no net external force on the body, it will maintain its state of motion. These are exactly the statements in which first law of motion are stated.

The body under investigation may be acted upon by a number of forces. We must use vector sum of all external forces, while applying second law of motion. A more general form of Second law of motion valid for a system of force is :

The vector addition of forces as required in the left hand side of the equation excludes the forces that the body applies on other bodies. We shall know from Third law of motion that force always exists in the pair of “action” and “reaction”. Hence, if there are “n” numbers of forces acting on the body, then there are “n” numbers of forces that the body exerts on other bodies. We must carefully exclude all such forces that body as a reaction applies on other bodies.

Consider two blocks “A” and “B” lying on horizontal surface as shown in the figure. There are many forces in action here :

Figure 1: There are many forces acting on different bodies.

Forces acting on two blocks and Earth

Let us now consider that we want to apply law of motion to block “B”. There are total of six forces related to “B” : 3, 4, 5, 6, 7 and 8. Of these, three forces 4, 5 and 7 act on “B”, while the remaining equal numbers of forces are applied by “B” on other bodies.

Figure 2: There are three external forces on B as shown with red arrow.

External forces acting on block B

From the point of view of the law of motion, there are, thus, only three forces as shown with red arrow in the figure, which are external force. These are the forces that the surrounding applies on the block "B".

Also as pointed out before in the course, we must not include internal forces like intermolecular forces in the consideration. In order to understand the role and implication of internal forces, we consider interaction of two blocks together with Earth’s surface. In this case, the forces 3 (B pushes up A) and 4 (A pushes down B) are pair of normal forces acting at the interface between blocks A and B. They are internal to the combined system of two bodies and as such are not taken into account for using law of motion. Here, external forces are 1 (Earth pulls down A), 5(Earth pulls down B) and 7(Earth pushes up B).

Figure 3: External and internal forces acting on blocks A and B

External and internal forces acting on blocks A and B

In nutshell, we must exclude (i) forces applied by the body and (ii) internal forces. Clearly, we must only consider external forces applied on the body while using equation of motion as given by Newton's second law.

As the force and acceleration are vector quantities, we can represent them with three components in mutually perpendicular directions. The consideration of dimension is decided by the force system as applied to the body.

If forces are collinear i.e. acting along a particular direction, then we use equation of motion in one direction. In such situation, it is possible to represent vector quantities with equivalent signed scalar quantities, in which sign indicates the direction. This is the simplest case.

If forces are coplanar i.e. acting in a plane, then we use component equations of motion in two directions. In such situation, we use component equations of motion in two directions.

Since each coordinate direction is bi-directional, we treat component vectors by equivalent scalar representation whose sign indicates direction. The important aspect of component equation of motion is that acceleration in a particular component direction is caused by net of force components in that direction and is independent of other net of component force in perpendicular direction.

In case forces are distributed in three dimensional volume, then we must consider component equation of motion in third perpendicular direction also :

We implicitly consider that forces are applied at a single point object. The forces that act on a point object are concurrent by virtue of the fact that a point is dimensionless entity. This may appear to be confusing as we have actually used the word “body” – not “point” in the definition of second law. Here, we need to appreciate the intended meaning clearly.

Actually second law is defined in the context of translational motion, in which a three dimensional real body behaves like a point. We shall subsequently learn that application of a force system (forces) on a body in translation is equivalent to a point, where all mass of the body can be considered to be concentrated. In that case, the acceleration of the body is associated with that point, which is termed as “center of mass (C)”. The Newton's second law is suitably modified as :

where a c a c is the acceleration of the center of mass (we shall elaborate about the concept of center of mass in separate module). In general, application of a force system on a real body can involve both translational and rotational motion. In such situation, the concurrency of the system of forces with respect to points of application on the body assumes significance. If the forces are concurrent (meeting at a common point), then the force system can be equivalently represented by a single force, applied at the common point. Further, if the common point coincides with “center of mass (C)”, then body undergoes pure translation. Otherwise, there is a turning effect (angular/rotational effect) also involved.

Figure 4

Concurrent force system
(a) Common point coincides with center of mass (b) Common point does not coincide with center of mass

What if the forces are not concurrent? In this case, there are both translational and rotational effects to be considered. The translational motion is measured in terms of center of mass as in pure translation, whereas the turning effect is studied in terms of “moment of force” or “torque”. This is defined as :

Figure 5: Forces as extended do not meet at a common point.

Non-concurrent force system

where r ⊥ r ⊥ is perpendicular distance from the point of rotation.

Most importantly, same force or force system is responsible for both translational effect (force acting as "force" as defined by the second law) and angular/rotational effect (force manifesting as "torque" as defined by the angular form of Newton's second law in the module titled Second law of motion in angular form ). We leave the details of these aspects of application of force as we will study it separately. But the point is made. Linear acceleration is not the only “effect” of the application of force (cause).

Also, force causes “effect” not necessarily as cause of acceleration – but can manifest in many ways : as torque to cause rotation; as pressure to change volume, as stress to deform a body etc. We should, therefore, always keep in mind that the study of translational effect of force is specific and not inclusive of other possible effects of force(s).

In the following listing, we intend to clarify the context of the study of the motional effect of force :

1: The body is negligibly small to approximate a point. We apply Newton’s second law for translation as defined without any consideration of turning effect.

2: The body is a real three dimensional entity. The force system is concurrent at a common point. This common point coincides with the center of mass. We apply Newton’s second law for translation as defined without any consideration of turning effect. Here, we implicitly refer the concurrent point as the center of mass.

3: The body is a real three dimensional entity. The force system is concurrent at a common point. But this common point does not coincide with the center of mass. The context of study in this case is also same as that for the case in which force system is not concurrent. We apply Newton’s second law for translation as defined for the center of mass and Newton’s second law for angular motion (we shall define this law at a later stage) for angular or rotational effect.

The discussion so far assumes that the body under consideration is free to translate and rotate. There are, however, real time situations in which rotational effect due to external force is counter-balanced by restoring torque. For example, consider the case of a sliding block on an incline. Application of an external force on the body along a line, not passing through center of mass, may not cause the body to overturn (rotate as it moves). The moment of external force i.e. applied torque may not be sufficient enough to overcome restoring torque due to gravity. As such, if it is stated that body is only translating under the given force system, then we assume that the body is a point mass and we apply Newton’s second law straight way as if the body were a point mass.

Unless otherwise stated or specified, we shall assume that the body is a point mass and forces are concurrent. We shall, therefore, apply Newton’s second law, considering forces to be concurrent, even if they are not. Similarly, we will consider that the body is a point mass, even if it is not. For example, we may consider a block, which is sliding on an incline. Here, “friction force” is along the interface, whereas the “normal force” and “weight” of the block act through center of mass (C). Obviously these forces are not concurrent. We, however, apply Newton’s second law for translation, as if forces were concurrent.