CHAPTER 1 – INTRODUCTION
DEFINITION OF MECHANICS
Mechanics may be defined as the physical science which describes and predicts
the conditions of rest or motion of bodies under the action of force systems. In
other words, where there is motion or force, there is mechanics.
In engineering, mechanics is generally based on Newton’s Laws and is often called
Newtonian (or Classical) Mechanics after the English scientist Sir Isaac Newton
(1642-1727).
Conditions involving speed of bodies close to the speed of light (about 300 x 106
m/s) and conditions requiring consideration of bodies with extremely small mass
and size (such as subatomic particles with sizes in the order of 10-12 m and
smaller) cannot be adequately described by Newton’s Laws. These extreme
conditions are treated in Relativistic Mechanics and Quantum Mechanics.
However, for a vast range of problems between these extremes, Newton’s Laws
give accurate results and are far simpler to apply. Although the fundamental
principles of Newtonian Mechanics are surprisingly few in numbers, they have
exceedingly wide range of applications. Modern research and development in the
fields of vibrations, stability and strength of machines and structures, rocket and
spacecraft design, robots, automatic control, engine performance, fluid flow,
electrical machines and apparatus are highly dependent upon the basic principles
of mechanics.
Mechanics is divided into three parts as shown below:
Mechs. of Rigid Bodies Mechs. of Deformable Bodies Mechs. of Fluids
Statics
Dynamics
Theory of Elasticity
Strength of Materials
Theory of Plasticity
Theory of Failure
Compressible Fluids
Incompressible Fluids
As seen, mechanics of rigid bodies is divided into two parts as “Statics” and
“Dynamics”.
Statics, is the branch of mechanics that deals with the bodies that are acted on
by balanced forces. A force system acting on a body is said to be balanced if it
has no tendency to change the state of rest or motion of the body in any way. If
a body is in equilibrium, the force system acting on it must be balanced.
Furthermore, a body in a state of equilibrium must be either at rest or moving
along a straight path with a constant velocity. Most problems in Statics concern
bodies at rest.
Statics constitutes a very important part of mechanics since it presents solution
methods for the determination of support forces at bodies in equilibrium and
establishment of relationships between external loads and internal force
distribution. Numerous practical engineering problems involving load carrying
members can be solved by using the Principles of Statics.
Dynamics is concerned with moving bodies. It is subdivided into two parts as
Kinematics and Kinetics. Kinematics deals with the geometry of motion without
taking into consideration the loading that causes this motion. Kinetics considers
the loads that cause the motion.
In rigid body mechanics, it is assumed that bodies are completely rigid –
nondeformable. In reality though, no structure or machine is completely rigid and
it will somewhat deform under the effect of forces it is subjected to. However
these deformations are generally extremely small and they neither affect the
body in concern nor the equilibrium conditions appreciably. On the other hand, if
the fact considered in the analysis is the amount of deformation in the member
or its resistance against failure, these deformations gain importance and in this
case, such bodies become the subject of mechanics of deformable bodies.
The third part of mechanics, mechanics of fluids is concerned with liquids and
gases at rest or in motion.
A comprehensive knowledge of mathematics is essential for the solution of almost
all problems in mechanics.
FUNDAMENTAL CONCEPTS
Certain concepts and definitions are basic to the study of mechanics and they
should be understood at the outset.
The basic concepts in mechanics are space, time, mass and force. These concepts
cannot be truly defined. They should be accepted on the basis of our intuition and
experience, and used as a mental frame of reference for our study of mechanics.
In Newtonian mechanics, space, time and mass are absolute quantities, which
mean that they are independent of each other (this is not true in Relativistic
Mechanics, where the time of an event depends upon its position and the mass of
a body varies with its velocity) and cannot be defined in terms of other quantities
or in simpler terms. Force is a derived quantity.
Space: is the geometric region occupied by bodies whose positions are described
by linear or angular measurements relative to a specific coordinate system. For
three dimensional problems, three independent coordinates are needed. For two
dimensional problems only two coordinates will be required.
Time: is a concept for measuring the succession and the duration of events. Time
is not directly involved in the analysis of problems in Statics.
Mass: is a measure of the translational inertia of the body, which is its resistance
to a change in 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.
The concept of mass is used to characterize and compare two bodies on the basis
of certain fundamental mechanical experiments depending on the definitions given
above. For example: 1) Two bodies of the same mass will be attracted by the
Earth in the same manner. 2) They will also offer the same resistance to a change
in translational motion.
Force: A force represents the action of one body on another. Force can be
generated either by the direct contact of bodies or by their effect at a distance.
Forces always occur in pairs. Forces of a pair are always equal in magnitude and
opposite in direction. Force is a vector quantity. The action of a force is
characterized by its magnitude, by the direction of its action and by its point of
application. A force tends to move a body in the direction of its action (the push –
pull effect of the force). In addition, a force tends to rotate the body about any
axis which does not intersect the line of action of the force and which is not
parallel to it (the moment effect of the force).
Idealization in Mechanics: The mathematical description of a real engineering
problem can become very complex. Thus, idealization (or models) and assumptions
are used in mathematics in order to simplify the application of the theory.
1 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 the 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. A
body considered as a particle is taken as a unique point, which is generally the
mass center of the body. A particle has mass but no shape and dimensions. In so
doing, the principles of mechanics are reduced to a rather simplified form, since
the geometry of the body will not be involved in the analysis of the problem. The
line of actions of all the forces applied to the body must pass through this point.
Forces can only exert push – pull effects on a particle.
Rigid Body: is an idealized body composed of a large number of particles all of
which always remain at fixed distances from each other. In addition to the
tendency to move a body in the direction of its application, a force may also tend
to rotate a body about an axis. A rigid body is assumed to undergo no deformation
under the action of applied forces. Its shape and dimensions remain fixed under
all loading conditions and at all times.
Point Force: is an idealized force assumed to act at a point on a body. A constant
force exerted on a body by another is actually distributed over the area of
contact between two bodies. If the area of contact is relatively small, the
contact force between the two bodies may be considered as a point
(concentrated) force.
LAWS OF MECHANICS
1) The Parallelogram Law: Two vectors
v
v
A and B , treated as free vectors, can
v
be replaced by their equivalent R ,
which is the diagonal of the
v
v
parallelogram formed by A and B as its
v
two sides, as shown. R is called the
v
v
resultant of A and B . Hence, the
v
combined effect of two forces A and
v
B (for example acting on a particle) is
equivalent to the effect of their
resultant.
v
B
v
R
v
B
=
O
2) The Principle of Transmissibility: The
effect of a force on a rigid body will remain
unchanged if the forced is moved to act on its
line of action. In other words, a force may be
applied at any point on its given line of action
without altering the resultant effects on the
rigid body on which it acts.
v
A
v
A
v v v
R = A+B
rigid
body
C
x
x
v
F
B
3) Newton’s First Law: If the resultant force acting on a particle is zero, then
v
v
the particle is in equilibrium. Stated mathematically as ( F = 0 ), where F is the
vector sum (the resultant) of all the forces acting on the particle.
4) Newton’s Second Law: The acceleration of a particle is proportional to the
resultant force acting on it and is in the direction of this force.
v
v
F = ma
5) Newton’s Third Law: The forces of action and reaction between interacting
bodies are equal in magnitude, opposite in direction and collinear.
6) Newton’s Law of Gravitation: This law
states that two particles of mass m1 and m2
are mutually attracted with equal and opposite
v
v
forces F and F of magnitude F, given by the
formula.
F =G
m1
v
F
v
F
m2
r
m1m2
r2
where,
“r” is the centroidal distance between the two particles
“G” is the universal constant of gravitation equal to 6.673 × 10
11
m3
kg s 2
A particular case of great importance is that of the attraction of the Earth on a
particle located at its surface. The force F exerted by the Earth on the particle
is then defined as the weight W of the particle.
W = mg
Where,
“m” s the mass of the particle
“g” is the gravitational acceleration of the Earth
INERTIAL REFERENCE FRAME
To describe accurately the state of rest or motion of particles or rigid bodies, a
proper reference frame must be used. An inertial reference frame is one in which
Newton’s Laws of Motion are valid or highly accurate. Experiments indicate that a
nonaccelerating reference frame, which is ether stationary or translating
uniformly relative to the Sun is an approximate inertial reference frame in most
situations.
DIMENSIONS, UNITS AND DIMENSIONAL HOMOGENEITY
A dimension is a qualitative description of a physical quantity, which can be
quantified by a certain standard of measure called unit. The seven base
dimensions are length, time, mass, electric current, thermodynamic temperature,
amount of matter and luminous intensity. We use only length, time and mass in
Statics. A unit of a dimension is a standard of measure for the quantitative
description of dimension. The primary unit of a base dimension is called a basic
unit. The SI uses meter (m), second (s), kilogram (kg), ampere (A), Kelvin (K), mole
(mol) and candela (cd), respectively for the above seven base dimensions.
A useful check of our computations may be obtained if we carry out the
computations with the units as well as the numerical values. The Law of
Dimensional Homogeneity states that because natural phenomena proceed with no
regard for man-made units, basic equations representing physical phenomena must
be valid for all systems of units. And all equations derived analytically from these
fundamental laws must also be dimensionally homogeneous.
According to this Law:
Every grouping in a physical equation must have the same dimensional
representation. In other words, “Apples cannot be equal to oranges”.
Numerical Accuracy: The accuracy of the solution of a problem depends upon
two items:
1) The accuracy of the given data
2) The accuracy the computations performed
The solution cannot be more accurate than the less accurate of these two items.
For example: If the loading of a bridge is known to be 375000 N with a possible
error of 500 N either way, the relative error which measures the degree of
accuracy of the data is
500
= 0.13%
375000
It would then, be meaningless in computing the reaction at one of the bridge
supports, to record it as 71610 N. The accuracy of the solution cannot be greater
than 0.13%, no matter how accurate the computations are, and the possible error
in the answer may be as large as
0.13
× 71600 = 93 N
100
The solution should be properly recorded as
71600 ± 93 N
71600 ± 100 N