Fundamental Concepts of Mechanics
The following concepts and deﬁnitions are basic to the study of mechanics, and they should be understood at the outset.
Space is the geometric region occupied by bodies whose positions are described by linear and angular measurements relative to a coordinate system. For three-dimensional problems, three independent coordinates are needed. For two-dimensional problems, only two coordinates are required.
Time is the measure of the succession of events and is a basic quantity in dynamics. Time is not directly involved in the analysis of statics problems.
Mass is a measure of the inertia of a body, which is its resistance to a change of velocity. Mass can also be thought of as the quantity of matter in a body. The mass of a body affects the gravitational attraction force between it and other bodies.
Force is the action of one body on another. A force tends to move a body in the direction of its action. The action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus force is a vector quantity.
A particle is a body of negligible dimensions. In the mathematical sense, a particle is a body whose dimensions are considered to be near zero so that we may analyze it as a mass concentrated at a point. We often choose a particle as a differential element of a body. We may treat a body as a particle when its dimensions are irrelevant to the description of its position or the action of forces applied to it.
Rigid body. A body is considered rigid when the change in distance between any two of its points is negligible for the purpose at hand. For instance, the calculation of the tension in the cable which supports the boom of a mobile crane under load is essentially unaffected by the small internal deformations in the structural members of the boom. For the purpose, then, of determining the external forces which act on the boom, we may treat it as a rigid body. Statics deals primarily with the calculation of external forces which act on rigid bodies in equilibrium.
Length, Time, and Mass are absolute concepts independent of each other
Force is a derived concept not independent of the other fundamental concepts. Force acting on a body is related to the mass of the body and the variation of its velocity with time.
Force can also occur between bodies that are physically separated (Ex: gravitational, electrical, and magnetic forces)
The whole structure of the study of mechanics is formed based on the three Newton’s laws of motion. The laws were found to satisfy most problems involving motions with velocities less than the velocity of light. The three laws of Newton, originally written in Latin, can be stated as follows:
A particle which is originally stationery, or moving with a constant velocity, will continue to be in that state except acted upon by an unbalanced force. Mathematically, if F is the total force acting on a particle and the velocity of the particle is v, the law is written as
If F=0, v is constant.
A particle which is acted upon by an unbalanced force will move with a velocity which is directly proportional to the magnitude of the force and in the direction of the force. Mathematically, the law is expressed as follows:
If ΣFi=F is an unbalanced force (i.e. F is a non-zero resultant force), then
where m is the mass of the particle and a is its acceleration.
This law can be written as F=d(mv)/dt. The product of the mass m and velocity v is known as the linear momentum of the particle.
The equation of motion F=ma can also be re-writtten as
In this form, the equation is similar to the Equation Of Equilibrium of two forces, F and –ma. In this form, the equation is called the d’Alembert’s principle. The term –ma is called the inertia force. The Alembert’s principle states that: “The inertia force balances the external forces”.
Note that the inertia force is an imaginary force formed to create, in the analysis, a state of equilibrium of a particle which in actual fact, is not in equilibrium. The imaginary equilibrium is termed dynamic equilibrium.
For every force acting on a particle, the particle exerts a reactive force (or reaction) of similar magnitude, opposite in direction, and collinear to the original force.
Points to note:
Laws of Gravitational Attraction
The gravitational law describes a reciprocating attraction between two particles. This law is expressed by the equation
where F is the magnitude of the force of attraction between a particle of mass and a second particle of mass and r is the distance between the centres of the particles, The constant of proportionality G is called the gravitational universal constant. The value of G has been determined by using experiments and found to be
Equation 1.3 states that all bodies, or particles, are attracting each other with a force which is proportional to the product of the masses of the bodies divided by the square of the distance between them. For bodies within the influence of the earth, the force of attraction which is most influential is the earth’s gravitational force of attraction. The force is known as the weight W and is the sole gravitational force which is taken into account in the study of mechanics. Hence the weight W of a body is the gravitational force applied to the body by the earth, as follows:
me = the mass of the earth
re = the mean radius of the earth
G = (Gme/re2) = the gravitational constant of the earth
The force of attraction of the earth W on a body has a magnitude that depends on the position of the body. This force of attraction, when acting alone, causes the body to undergo an acceleration g. The magnitude of g has been determined experimentally and found to be g=9.78 m/s2 at the equator and rises towards the poles. Its value at the latitude 45o is 9.81 m/s2 and at the poles 9.93 m/s2. The acceleration g is called the gravitational acceleration.