The state of rest and the state of motion of the bodies under the action of different forces has engaged the attention of mathematicians and scientists for many centuries. The branch of physical science that deal with the state of rest or the state of motion of bodies is termed as mechanics. Starting from the analysis of rigid bodies under gravitational force and application of simple forces the mechanics has grown into the analysis of complex structures like multi-storey buildings, aircrafts, space crafts and robotics under complex system of forces like dynamic forces, atmospheric forces and temperature forces.
Archemedes (287–212 BC), Galileo (1564–1642), Sir Issac Newton (1642–1727) and Einstein (1878–1955) have contributed a lot to the development of mechanics. Contributions by Varignon, Euler, and D. Alemberts are also substantial. The mechanics developed by these researchers may be grouped as
(i) Classical mechanics/Newtonian mechanics
(ii) Relativistic mechanics
(iii) Quantum mechanics/Wave mechanics.
Sir Issac Newton, the principal architect of mechanics, consolidated the philosophy and experimental findings developed around the state of rest and state of motion of the bodies and putforth them in the form of three laws of motion as well as the law of gravitation. The mechanics based on these laws is called Classical mechanics or Newtonian mechanics.
Albert Einstein proved that Newtonian mechanics fails to explain the behaviour of high speed (speed of light) bodies. He putfourth the theory of Relativistic mechanics. Schrödinger (1887–1961) and Broglie (1892–1965) showed that Newtonian mechanics fails to explain the behaviour of particles when atomic distances are concerned. They putforth the theory of Quantum mechanics.
Engineers are keen to use the laws of mechanics to actual field problems. Application of laws of mechanics to field problems is termed as Engineering mechanics. For all the problems between atomic distances to high speed distances there are various engineering problems for which Newtonian mechanics has stood the test of time and hence is the mechanics used by engineers.
The various bodies on which engineers are interested to apply laws of mechanics may be classified as
(i) Solids and
(ii) Fluids.
The bodies which do not change their shape or size appreciably when the forces are applied are termed as Solids while the bodies which change their shape or size appreciably even when small forces are applied are termed as Fluids. Stone, steel, concrete etc. are the example of solids while water, gases are the examples of fluids.
In this book application of Newtonian mechanics to solids is dealt with.
Basic Terminologies In Mechanics
The following are the terms basic to the study of mechanics, which should be understood clearly.
Mass
The quantity of the matter possessed by a body is called mass. The mass of a body will not change unless the body is damaged and part of it is physically separated. If the body is taken out in a space craft, the mass will not change but its weight may change due to the change in gravitational force. The body may even become weightless when gravitational force vanishes but the mass remain the same.
Time
The time is the measure of succession of events. The successive event selected is the rotation of earth about its own axis and this is called a day. To have convenient units for various activities, a day is divided into 24 hours, an hour into 60 minutes and a minute into 60 seconds. Clocks are the instruments developed to measure time. To overcome difficulties due to irregularities in the earths rotation, the unit of time is taken as second which is defined as the duration of 9192631770 period of radiation of the cesium-133 atom.
Space
The geometric region in which study of body is involved is called space. A point in the space may be referred with respect to a predetermined point by a set of linear and angular measurements. The reference point is called the origin and the set of measurements as coordinates. If the coordinates involved are only in mutually perpendicular directions, they are known as cartesian coordination. If the coordinates involve angles as well as the distances, it is termed as Polar Coordinate System.
Length
It is a concept to measure linear distances. The diameter of a cylinder may be 300 mm, the height of a building may be 15 m, the distance between two cities may be 400 km. Actually metre is the unit of length. However depending upon the sizes involved micro, milli or kilo metre units are used for measurements. A metre is defined as length of the standard bar of platinum-iradium kept at the International Bureau of weights and measures. To overcome the difficulties of accessibility and reproduction now metre is defined as 1690763.73 wavelength of krypton-86 atom.
Continuum
A body consists of several matters. It is a well-known fact that each particle can be subdivided into molecules, atoms and electrons. It is not possible to solve any engineering problem by treating a body as conglomeration of such discrete particles. The body is assumed to be a continuous distribution of matter. In other words the body is treated as continuum.
Rigid Body
A body is said to be rigid, if the relative positions of any two particles do not change under the action of the forces acting on it.
In Fig. 1.1 (a), point A and B are the original positions in a body. After the application of forces F1, F2, F3, the body takes the position as shown in Fig. 1.1(b). A′ and B′ are the new positions of A and B. If the body is treated as rigid, the relative position of A′B′ and AB are the same i.e.
Particle
A particle may be defined as an object which has only mass and no size. Theoretically speaking such a body cannot exist. However in dealing with problems involving distances considerably larger compared to the size of the body, the body may be treated as a particle, without sacrificing accuracy.
For example:
· A bomber aeroplane is a particle for a gunner operating from the ground.
· A ship in mid sea is a particle in the study of its relative motion from a control tower.
· In the study of movement of the earth in celestial sphere, earth is treated as a particle.
Force
Force is an important term used in solid mechanics. 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’.
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. Noting that 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
∝ rate of change of (mass × velocity)
Characteristics of a Force
It may be noted that a force is completely specified only when the following four characteristics are specified
· Magnitude
· Point of application
· Line of action
· Direction.
In Fig. 1.2, AB is a ladder kept against a wall. At point C, a person weighing 600 N is standing. The force applied by the person on the ladder has the following characters:
· magnitude is 600 N
· the point of application is C which is at 2 m from A along the ladder
· the line of action is vertical
· the direction is downward.
It may be noted that in the figure
· magnitude is written near the arrow
· the line of arrow shows the line of application
· the arrow head shows the point of application
· the direction of arrow represents the direction of the force.
