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**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.

**Newtonian Mechanics**

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)

**Remember:**

- Mass is a property of matter that does not change from one location to another.
- Weight refers to the gravitational attraction of the earth on a body or quantity of mass. Its magnitude depends upon the elevation at which the mass is located
- Weight of a body is the gravitational force acting on it.

**Newtonian Mechanics**

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:

**First Law**

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.

**Second Law**

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 Σ**F**_{i}=**F **is an unbalanced force (i.e. **F** is a non-zero resultant force), then

**F**=m**a **…Equation(1.1)

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

F+(-ma)=0 …Equation(1.2)

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.

**Third Law**

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:**

- The first law is a special case of the second law, where the acceleration a of the particle is zero.
- The three laws of motion can be illustrated by a rocket in its launching state, Figure 1.12. Initially, the rocket neither moves nor changes direction; it moves only after being acted upon by an external force F, which is reaction to the push T of its engine (First law). After it launches, the acceleration a experienced by the rocket is directly proportional to the reaction (Second law). The Third law is illustrated by the statement that every action (push of the engine force) produces a reaction which is of the same magnitude but of opposite sense (motion of the rocket).

**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

G=6.673×10^{-11} m^{3}/(kg.s^{2})

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:

m_{e} = the mass of the earth

r_{e} = the mean radius of the earth

G = (Gm_{e}/r_{e}^{2}) = 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/s^{2 }at the equator and rises towards the poles. Its value at the latitude 45^{o} is 9.81 m/s^{2 }and at the poles 9.93 m/s^{2}. The acceleration g is called the gravitational acceleration.

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