Basic Laws of Engineering Mechanics

The structure of engineering mechanics rests on relatively few basic laws. They are given below.

1) Newton’s Laws of Motion

2) Newton’s Law of Gravitation

3) Principle of Transmissibility of Forces

4) Parallelogram Law of Forces

5) Principles of Physical Independence of Forces

6) Principles of Superposition

1) Newton’s Laws of Motion

i) Newton’s First Law of Motion

Newton’s first law states that ‘everybody continues in its state of rest or of uniform motion in a straight line unless it is compelled by an external agency acting on it’. This leads to the definition of force as ‘force is an external agency which changes or tends to change the state of rest or uniform linear motion of the body’.

ii) Newton’s Second Law of Motion

Magnitude of force is defined by Newton’s second law. It states that ‘the rate of change of momentum of a body is directly proportional to the impressed force and it takes place in the direction of the force acting on it’. As the rate of change of velocity is acceleration and the product of mass and velocity is momentum we can derive expression for the force as given below.

From Newton’s second law of motion,

                                            Force ∝ rate of change of momentum

                                            Force ∝ rate of change of (mass × velocity)

Since mass do not change,

                                           Force ∝ mass × rate of change of velocity

                                                      ∝ mass × acceleration

                                                   F ∝ m × a

                                                   F = k × m × a

where 'F' is the force, 'm' is the mass, 'a' is the acceleration and 'k' is the constant of proportionality.

In all the systems, unit of force is so selected that the constant of the proportionality becomes unity. For example, in S.I. system, unit of force is Newton, which is defined as the force that is required to move one kilogram (kg) mass at an acceleration of 1 m/sec².

                                 ∴ One newton = 1 kg mass × 1 m/sec²

   Thus k = 1

                                                     F = m × a

ii) Newton’s Third Law of Motion

Newton’s first law gave definition of the force and second law gave basis for quantifying the force. Newton’s third law states that ‘for every action there is an equal and opposite reaction’.

Consider the two bodies in contact with each other. Let one body apply a force F on another. According to this law the second body develops a reactive force R which is equal in magnitude to force F and acts in the line same as F but in the opposite direction. Fig.1 shows the action of a ball on the floor and the reaction of floor to this action. In Fig. 2 the action of a ladder on the wall and the floor and the reactions from the wall and the floor are shown.

Fig. 1 Action of a Ball on the Floor and the Reaction of Floor

Fig. 2 Action of a Ladder on the Wall and the Floor and the Reactions from the Wall and the Floor

2) Newton’s Law of Gravitation

It states that everybody attracts the other body. ‘The force of attraction between any two bodies is directly proportional to their masses and inversely proportional to the square of the distance between them’. Thus the force of attraction between the bodies of mass m1 and mass m2 at distance ‘d’ between them as shown in Fig. 3 is

where G is the constant of proportionality and is known as constant of gravitation.

Fig. 3 Force of Attraction between Two Bodies

From above equation,

It has been proved by experiments that the value of G = 6.673 × 10^–11 Nm²/kg². Thus if two bodies one of mass 10 kg and the other of 5 kg are at a distance of 1 m, they exert a force 

on each other.

Similarly, 1 kg-mass on earth surface experiences a force of

Since, mass of earth = 5.96504 × 10²⁴ kg and radius of earth = 6371 × 10³ m. This force of attraction is always directed towards the centre of earth. In common usage the force exerted by earth on a body is known as weight of the body. Thus weight of 1 kg-mass on/near earth surface is 9.80665 N, which is approximated as 9.81 N for all practical problems. Compared to this force the force exerted by two bodies on each other is negligible. Thus in statics

• Weight of a body W = mg
• Its direction is towards the centre of the earth, in other words, vertically downward. 
• The force of attraction between the other two objects on the earth is negligible.

3) Principle of Transmissibility of Forces

According to this law ‘the state of rest or motion of the rigid body is unaltered, if a force acting on the body is replaced by another force of the same magnitude and direction but acting anywhere on the body along the line of action of the replaced force’.

Let F be the force acting on a rigid body at point A as shown in Fig. 4. According to this law, this force has the same effect on the state of body as the force F applied at point B, where AB is in the line of force F.

Fig. 4 Representation of Principle of Transmissibility of Forces

In using law of transmissibility it should be carefully noted that it is applicable only if the body can be treated as rigid. Hence if we are interested in the study of internal forces developed in a body, the deformation of body is to be considered and hence this law cannot be applied in such studies.

3) Parallelogram Law of Forces

The parallelogram law of forces enables us to determine the single force called resultant force which can replace the two forces acting at a point with the same effect as that of the two forces. This law was formulated based on experimental results on a body subjected to two forces. This law can be applied not only to the forces but to any two vectors like velocities, acceleration, momentum etc. 

This law states that ‘if two forces (vectors) acting simultaneously on a body at a point are represented in magnitude and directions by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes thorough the point of intersection of the two sides representing the forces (vectors)’.

Fig. 5 Representation of Parallelogram Law of Forces

In the Fig. 5, the force F1 = 4 units and the force F2 = 3 units are acting on a body at a point A. To get the resultant of these forces, according to this law, construct the parallelogram ABCD such that AB is equal to 4 units to the linear scale and AC is equal to 3 units. Then according to this law, the diagonal AD represents the resultant in magnitude and direction. Thus the resultant of the forces F1 and F2 is equal to the units corresponding to AD in the direction α to F1.

4) Principles of Physical Independence of Forces

It states that the action of a force on a body is not affected by the action of any other force on the body.

5) Principles of Superposition of Forces

It states that ‘the net effect of a system of forces on a body is same as the combined of individual forces acting on the body’. Since a system of forces in equilibrium do not have any effect on a rigid body this principle is stated in the following form also: ‘The effect of a given system of forces on a rigid body is not changed by adding or subtracting another system of forces in equilibrium.’

Fig. 6 Representation of Principle of Superposition of Forces