The document discusses key concepts in mechanics including: - Mechanics deals with the motion and equilibrium of bodies under forces. - Statics analyzes equilibrium of rigid bodies at rest, while dynamics analyzes bodies in motion. - Basic terms like length, area, volume, force, mass, and weight are introduced. - Concepts like space, time, mass, particles, and forces are defined. - Coordinate systems including Cartesian, cylindrical and spherical are described to specify positions. - Newton's laws of motion and gravitational attraction form the basis of mechanics analyses.

1. Kosygin Leishangthem
Department of Civil Engineering

2.  What is mechanics?
Physical science deals with the
state of rest or motion of bodies
under the action of force
 Why we study mechanics?
This science form the
groundwork for further study in
the design and analysis of
structures
Mechanics
Statics Dynamics
Kinematics
Kinatics

3. BasicTerms
• Essential basic terms to be understood
• Statics: dealing with the equilibrium of a rigid-body at rest
• Rigid body: the relative movement between its parts are negligible
• Dynamics: dealing with a rigid-body in motion
• Length: applied to the linear dimension of a strait line or curved line
• Area: the two dimensional size of shape or surface
• Volume: the three dimensional size of the space occupied by substance
• Force: the action of one body on another whether it’s a push or a pull
force
• Mass: the amount of matter in a body
• Weight: the force with which a body is attracted toward the centre of
the Earth
• Particle: a body of negligible dimension

4. Concepts
• The following concepts and definitions are basic to the study of mechanics, and they should be
understood at the outset.
• Space A space is a geometrical region where events happen. It can be a one-dimensional line, a
two-dimensional plane, or a three-dimensional space. The term ‘space’ is normally used to refer
to a three-dimensional region. A one-dimensional region is normally referred to as a ‘line’ whilst
a two-dimensional region is simply called a ‘plane’. Note that the concept of a plane with n
dimensions does exist but is an abstract concept to describe the relationship between n
quantities
• 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. This force appears in many applications
in statics..

5. • 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.
• 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

6. Definition of a rigid body:
A rigid body is defined as a body on which the distance between two
points never changes whatever be the force applied on it.
Or you may say the body which does not deform under the influence of
forces is known as a rigid body.
But, in real life, there would be some force under which the body starts
to deform. For example, a bridge does not deform under the weight of
a single man but it may deform under the load of a truck or ten trucks.
However, the deformation is small.
Since, no object is rigid body in real life; we have to introduce another
concept that is concept of resistant body so that we would be able to
use it in engineering problems.

7. UNITS OF MEASUREMENT
Unit Systems
Various systems of units have been used the field of engineering. The
systems used differ between countries. For example, the United
Kingdoms previously used the gravitational system of feet, pounds, and
seconds (FPS) whilst most other European countries used the metric
absolute system.(1) However, from 1960, a modern version of the metric
system called the International System of Unit (SI) has been adopted by
most countries

8. Basic Units
• The SI measurement system involves seven basic units that has been
fixed arbitrarily. From these basic units, a number of derived units
are formed. These are mere combinations of the basic units.
Table 1.1. Basic Units
Quantity Unit Symbol
1. Length meter m
2. Mass kilogram kg
3. Time second s
4. Electrical Current Ampere A
5. Amount Of Substance mol mol
6. Temperature kelvin K
7. Light intensity candella cd
Note that the basic unit for
mass is kilogram, not gram.
The kilogram is the only unit
defined using a prefix. Prefixes
are used for other units only
when the magnitudes of the
quantities being measured are
too large or too small
compared to the quantity used
in practical situations.

9. Table 1.2. Types Of Prefixes
Power Prefix Symbol
1012 tera T
109 giga G
106 mega M
103 kilo k
10-2 centi c
10-3 milli m
10-6 micro μ
10-9 nano n
10-12 piko p
only three basic units are used, namely meter, kilogram, and second.
Other units used are units derived from those basic units.

10. the SI measurement system has only seven basic units of measurement. Hence,
the number of basic units is limited. It is the same in other systems of
measurement. Their numbers of basic units are limited. From the basic units,
other units are derived. These units are known as derived units. There are derives
units that have been given specific names.
For example, the unit kg.m.s-2 kg.m /s2 is named newton (N)
and
the unit N.m-2 is called Pascal (Pa).

11. Velocity & Acceleration
A derived unit is formed based on the definition of the quantity it measures. For
example, the acceleration a is defined as the rate of change in velocity v with respect
to time t, whilst the velocity is defined as the rate of change of position s with
respect to time t. Hence
v = ds/dt; a = dv/dt
The units of v are the unit of displacement (meter) divided by the unit of time
(seconds), i.e. m/s or m.s-1. Furthermore, the unit for a is obtained as unit of v
divided by the unit of time, i.e.
a = (m/s)/s = m.s-2

12. Besides, the derived units are also obtained from basic equations that
defined the quantities measured by the respective units. For example,
the unit for force is obtained from the definition of Newton’s second law
F = ma
F is the force acting on a body of mass m and causes it to move with
acceleration a
F = kg(m/s2) = kg.m/s2 = newton (N)

13. Table 1.3. Derived Units
Quantity Derived Unit Name Symbol
Force kg.m.s2 newton N
Work/Energy N.m joule J
Power J.s-1 watt W

14. SCALARS AND VECTORS
The quantities used in this book are of two types, namely scalars and
vectors. These quantities have their own unique
properties. Mathematically, the operations of these two quantities are
based on laws formed specially for them.
Scalars
Scalars are quantities that have magnitudes only. They do not have
directions. The effects of a scalar quantity on an analysis depend on its
magnitude alone. Scalar quantities found in this book include time,
mass, and density. For scalar quantities, their mathematical operations
obey rules of operations of standard algebra.

15. Vectors
Vectors are quantities that have magnitudes and
directions, and obey the parallelogram law of
addition. This law states that:
Two vectors can be replaced by one other vector
obtained by drawing the diagonal of a
parallelogram whose sides are formed by the two
original vectors. two vectors V1 and V2 are
equivalent to the single vector V, where
V=V1+V2. Among the vector quantities used in
this book are positions, displacements, forces, and
moments.

16. Displaying Vectors In Written Form.
In analyses, the vector quantities must be differentiated from
the scalar quantities. There are number of ways that vectors are
represented in writings.
A vector quantity is printed boldface, for example the vector V.
On the other hand, a scalar quantity is normally printed in italic
(or under-lined), for example the magnitude of the vector V is
the scalar V or V . In normal hand writing, a vector quantity is
written by putting a curvy line below it, whilst a scalar quantity
is not given any specific mark.

17. • Graphical Display Of A Vector. A vector quantity is displayed
graphically by using an arrow that defines its magnitude, direction,
and sense. The magnitude is represented by the length of the arrow
whilst the direction is represented by the line of action of the arrow;
the sense of the vector is shown by the direction of the arrow’s head.
Vector V in Figure 1.3, for example, has a magnitude V=3 unit, its
direction is 60o from the horizontal reference line, and the vector
points upwards sense-wise.

18. • Vector Mathematics. The mathematical operations of vector
quantities are based on the algebraic laws specific for them. Further
details are given in the Appendix at the end of this book. Students
requiring a study or revision on them are recommended to read the
Appendix before proceeding with the study.

19. COORDINATE SYSTEMS
• In the study of mechanics, we are always involved, directly or
indirectly, with the position of a particle. The position of a particle is
determined by referring it to a reference frame, or system of axes,
which is arbitrarily determined. The system of axes used is
dependent on the number of dimensions involved and the nature of
the problem to be analysed. Systems of axes most frequently used to
describe concepts of statics are given below.

20. One-dimensional System
• The position of a particle P in a one-dimensional region can change
along a specific curve (path). The position of the particle is specified
by comparing it to the position of a reference point O on the
path. The position P of the particle is the distance s from O to P,
Figure 1.4. For mathematical analysis, a sign must be given to s to
differentiate the position of P on the right side of O to its position on
the left side of O. The difference is obtained by specifying the
positive sense of the displacement of O from P. For example, if the
displacement to the right is taken as positive, then a displacement to
the left is taken as negative. Hence the position of P in Figure 1.4 is
+s. However, if P is on the left side of O, the magnitude s is given the
negative sign (-s).

21. Two-Dimensional Systems
• The position of particle P which is free to be move in two different
independent directions is determined completely by two
parameters. The particle changes its position if one, or both,
parameters are changed. There are two coordinate systems that can
be used to describe the position of the particle on its plane of
displacement. The coordinates systems are the Cartesian coordinate
system and the polar coordinate system.

22. Cartesian coordinates (x,y).
• Cartesian coordinates are the
ones that give the position of
a point by their x and y, or
points on x-axis and y-axis.

23. Polar coordinate (R,a).
• Polar coordinates give
the position of a point by
their distance from the
origin O and the angle
with respect to x-axis. (R
, a) a is an angle.
𝑦 = R. sin 𝑎
𝑥 = R. cos 𝑎
𝑦
𝑥
= tan 𝑎
𝑟2 = 𝑥2 + 𝑦2

24. Three-dimensional Systems
• There are three different coordinate systems that are normally used
to specify the position of a particle in space. Each of the system uses
three variables corresponding to the three independent directions for
the change in position of the particle. The systems are the Cartesian
coordinate system (3-D), the cylindrical coordinate system, and the
spherical coordinate system.
• Cartesian coordinates (x,y,z)
• Cylindrical coordinates ((r,θ,z)
• Spherical coordinates (ρ,φ,θ)

25. Relations between Cartesian, Cylindrical, and Spherical Coordinates
Consider a Cartesian, a cylindrical, and a spherical coordinate system, related as shown in Figure 1.
Standard relations between Cartesian, cylindrical, and
spherical coordinate systems. The origin is the same for all
three. The positive z-axes of the Cartesian and cylindrical
systems coincide with the positive polar axis of the spherical
system. The initial rays of the cylindrical and spherical
systems coincide with the positive x-axis of the Cartesian
system, and the rays =90° coincide with the positive y-axis.

27. Then the cartesian coordinates (x,y,z), the cylindrical coordinates (r,θ,z)(r, theta, z), and the spherical
coordinates (ρ,φ,θ) (rho, phi, theta) of a point are related as follows:
, ,

28. NEWTONIAN MECHANICS
• The whole structure of the study of mechanics is formed based on the
three Newton’s laws of motion.
First Law
Newton’s Laws : A particle remains at rest or continues to move
in a straight line with a uniform velocity if there is no
unbalanced force acting on it.
∑F=0 ⇔ a=0
If F=0, v is constant.

29. The forces of action and reaction between interacting bodies are equal
in magnitude, opposite in direction, and collinear.
action force = -(reaction force)

30. Points to note:
1. The first law is a special case of the second law, where the acceleration a
of the particle is zero.
2. The three laws of motion can be illustrated by a rocket in its launching
state. 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).

31. 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 m3/(kg.s2)

32. 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
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 /Re
2) = the gravitational constant of the earth

33. In centimeter-gram-second (CGS), G can be written as:
G can also be given as:

34. • 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.

35. The Concept of Force
1. Force is associated with the body till it is in motion.
2. When a body is at rest the force acting on it is zero.
3. Force is always in the same direction as the velocity of the body.
4. If the velocity is changing then the force is also changing.
5. Centripetal force and centrifugal force both act on the body moving
uniformly in a circle.
6. The action-reaction forces act on the same body.
7. The product of mass and acceleration is a force.
8. Only animate things like people and animals exert forces; passive
ones like tables, floors do not exert forces.
9. A force applied by, say a hand, still acts on the object even after the
object leaves the hand

36. A force is defined as the action of a body on another body. A
force is applied either through a direct contact or through a
remote action. Forces applied through a remote action are
gravitational, electrical, and magnetic forces. All other forces are
applied through direct contacts.
In simple terms, a force is a push, a pull, or a drag on an object.
There are three main types of force:

37. An applied force is an interaction of one object on another that causes the
second object to change its velocity.
A resistive force passively resists motion and works in a direction opposite
to that motion.
An inertial force resists a change in velocity. It is equal to and in an
opposite direction of the other two forces.
There is no such thing as a unidirectional force or a force that acts on only
one object. There must always be two objects involved, acting on each
other. One object acts on the other, while the second resists the action of the
first.

38. Types of applied force
There are several types of applied force:
The most common form of force is a push through physical contact. For example, you can push on a door to
open it. An object can also collide with another object, exerting a force and causing the second object to
accelerate. This is another type of push and can be called an impulse force, since the time interval is very
short.
You can pull on an object to change its velocity. Gravitation, magnetism, and static electricity are some of the
pulling forces that act at a distance with no physical contact required to move objects.
Finally, if two objects or materials are in contact, one can drag the other along by friction or other means.
Applied force
An applied force is an interaction that causes the second object to change its velocity.
Force equation : The force required to overcome the inertia of an object is according to the equation:
F = ma
where:
• F is the force
• m is the mass of the object
• a is the acceleration caused by the force

39. Resistive force
A resistive force passively inhibits or resists the motion of an object. It is a form
of push-back. It is considered passive, since it only responds to actions on the
object. Friction and fluid resistance are the major resistive forces.
Friction
When an object is being pushed along the surface of another object or material,
the resistive force of friction pushes back on the first object to resist its motion.
Fluid resistance
Fluid resistance pushes back on the moving object, which is basically trying to
plow through the fluid. It also included friction on the surface of the object.
Air resistance and water resistance are common forms of fluid resistance.
Inertial force
An inertial force works against a change in velocity, caused by an applied force,
as well as a resistive force.

40. Against applied force
According to Newton's Third Law of Motion or the Action-Reaction
Law:
Whenever one body exerts force upon a second body, the second
body exerts an equal and opposite force upon the first body.
This is often stated as: "For every action there is an equal and
opposite reaction."
Against resistive force
When a resistive force like friction, slows down the motion of an
object, the inertial force will push in the opposite direction and tend
to keep the object moving.

41. Internal Forces & External forces
Internal force is whatever causes an
object with mass to accelerate or move.
This force is equal to the product of the
mass of the object plus its rate of
acceleration.
• Internal Forces are forces between
objects found inside the system.
• cannot do work on system
• cannot change total energy of the
system
External forces include the force applied
to the system, air resistance of an object,
force of friction, tension and normal
force. Internal forces include the force of
gravity, spring force, and magnetic and
electrical field forces.
• forces that act on the system and their
agents are part of the environment.
• can do work on system
• can transfer energy into or out of
system, thus changing the total energy
of the system