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Laws of Mechanics

The following basic laws and principles are considered to be the foundation of mechanics

1. Newton’s first and second laws of motion

2. Newton’s third law

3. The gravitational law of attraction

4. The parallelogram law

5. The principle of transmissibility of forces.

 

1.3.1. Newton’s First and Second Laws of Motion. Newton’s first law states, “Every body continues in a state of rest or uniform motion in a straight line unless it is compelled to change that state by some external force acting on it.”

Newton’s second law states, “The net external force acting on a body in a direction is directly proportional to the rate of change of momentum in that direction.”

 

1.3.2. Newton’s Third Law. Newton’s third law states, “To every action there is always equal and opposite reaction.”

Fig. 1.13 shows two bodies A and B which are placed one above the other on a horizontal surface.

Here                F 1 = Force exerted by horizontal surface on body A (action)

- F 1 =Force exerted by body A on horizontal surface (reaction)

F 2 =Force exerted by body A on body B (action)

- F2 = Force exerted by body Bon body A (reaction)

 1.3.3. The Gravitational Law of Attraction. It states that two bodies will be attracted towards each other along their connecting line with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

Refer to Fig. 1.13 (a).

Let       m 1 = Mass of first body

m2 = Mass of second body

r = Distance between the centre of bodies

F = Force of attraction between the bodies.

Then according to the law of gravitational attraction.

F= G m2 m 2 / r2

 

where G = Universal gravitational constant of proportionality.

In equation (1.6), F is expressed in N, m1 and m2 in kg and r in m. Hence dimensionally, equation ( 1.6) becomes as

 

N = G kg x kg / m2      or   G = Nm2 / kg2

1 N = 1 kg x 1m / s2   or   N = kg x m / s2

 

Substituting the value of N is equation (i),

G = (kg x m / s2 ) x m2 / kg2 = m2 / kg s2

Hence from equations (i) and (ii), dimension of Universal Gravitational Constant G is N m2/kg2 or m3/kg s2.

The value of G is 6.67 x l0- 11 N m2/kg2 or m3fkg s2.

In equation (1.6), if m1 = 1 kg, m2 = 1 kg and r = 1m, then F =G. This means that the force of attraction between two bodies of mass 1 kg each when they are at a distance of 1 m apart, will be 6.67 x 10-11 N i.e., 0.0000000000667 N. This force is very very small.

 

Weight

The weight of a body is defined with the help of law of gravitation. Weight is defined as the force with which a body is attracted towards the centre of earth.

Let                               M = Mass of the body

ME= Mass of the earth = 5.9761 x 1024 kg

r = Distance between the centres of the earth and the body

= 6.371 x 106 m         (i.e., radius of earth)

G =Universal gravitational constant= 6.67 x l0-11 N m2fkg2

F =Force of attraction which is equal to weight (W)

Substituting these values in equation (1.6), we get

W = G Mg x M / r2

= 6.67 x 10 -11 x 5.9761 x 10 24 M Nm2

/ 6.371 x 1012 x m2 Nm2 / kg x kg x kg = (9.81 x m) N

where 9.81 is acceleration due to gravity and is denoted by ‘g’.

. . W = g x M or M x g

Actually the term GME /r2  is equal to 9.81 m / s2,which is represented by ‘g’.

 

1.3.4. The Parallelogram Law. This law has been already defined. It states that if two forces acting at a point be represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

 

1.3.5. The Principle of Transmissibility of Forces. It states that if a force, acting at a point on a rigid* body, is shifted to any other point which is on the line of action of the force, the external effect of the force on the body remains unchanged.

 

*A body which does not deforms under the action of loads or external forces _is known as rigid body. Hence in case of a rigid body, the relative movement between the various points of the body are negligible or the distance between any two points remains the same for all the times.

For example, consider a force F acting at point O on a rigid body as shown in Fig. 1.14 (a). On this rigid body, “there is another point 0′ in the line of action of the force F. Suppose at this point 0′, two equal and opposite forces F 1 and F2 (each equal to F and collinear with F) are applied as shown in Fig. 1.14 (b). The force F and F2, being equal and opposite, will cancel each other, leaving a force F 1 at point 0′ as shown in Fig. 1.14 (c). But force F 1 is equal to force F.

 

The original force F acting at point 0, has been transferred to point 0 ‘ which is along the line of action of F without changing the effect of the force on the rigid body. Hence any force acting at a point on a rigid body can be transmitted to act at any other point along its line of action without changing its effect on the rigid body. This proves the principle of transmissibility of a force.