STATICS OF RIGID BODIES
(EMech 1)
LESSON 1 – Resultants of Coplanar Concurrent
System of Forces
LESSON 1: RESULTANTS OF COPLANAR CONCURRENT FORCES
I. Topics/ Subject Matter
1. Introductory Concepts
2. Resultants of Coplanar Concurrent Forces
II. Specific Objectives
At the end of the Lesson, the student should be able to:
1. define technical terms under Lesson 1;
2. state the basic principles and concepts under Lesson 1
3. solve problems involving resolutions of forces
III. LESSON PRESENTATION
Mechanics- (in its broadest sense) is the science which
describes and predicts the conditions of rest or
motion of bodies under the action of forces.
- (in a narrower sense) it maybe divided into
areas which, with some overlap,
are of
separate interest to physicists and to
engineers.
 The topics of mechanics which interest physicists
and astronomers (and, to some extent, electrical
engineers) are particle mechanics, celestial
mechanics, quantum mechanics, and relativistic
mechanics.
 To engineers, the important areas of mechanics
are mechanics of solids and of fluids and their
concomitant, continuum mechanics
STATICS OF RIGID BODIES - It considers the effects and distribution of
forces on rigid bodies which are and
remain at rest.
DYNAMICS OF RIGID BODIES- It considers the various motions of
rigid bodies and the correlation of
these motions with the forces causing
them.
Rigid Body- It is a definite amount of matter the parts of which are
fixed in position relative to one another.
- Actually, solid bodies are never rigid; they deform under
the action of applied forces. In many cases, this
deformation is negligible compared to the size of the
body and the body may be assumed rigid.
Mass- It is that invariant property of a body which measures its
resistance to a change of motion.
Force- It is the action exerted by one body upon another.
External effect of a force upon a body- It is manifested by a
change in, or a tendency to change, the state of motion of a body upon
which it acts.
Internal effect of a force upon a body- To produce stress and
deformation in the body
Three (3) Characteristics of a Force
1. Its magnitude
2. The position of its line of action
3. The direction (or sense) in which the force acts along its line of action
The Principle of Transmissibility of a Force
“ The external effect of a force on a rigid body is the same for all
points of application along its line of action; i.e., it is independent of
the point of application”
Axioms of Mechanics
1. The Parallelogram Law: “ The resultant of two (2) forces is the diagonal
of the parallelogram whose initial sides are
the vectors of these forces.”
2. Two (2) forces are in equilibrium only when equal in magnitude,
opposite in direction, and collinear in action.
3. A set of forces in equilibrium may be added to any system of forces
without changing the effect of the original system.
4. Action and reaction forces are equal but oppositely directed.
Three (3) Types of Quantities Encountered in Mechanics
1. Scalars. These are quantities which posses magnitude only and can be
combined arithmetically.
2. Vectors. These are quantities having both magnitude and direction which
combine geometrically according to the Parallelogram Law.
Examples: force, moment, displacement, velocity, acceleration,
and momentum.
3. Tensors. These are quantities which posses magnitude but require two or
more directional aspects to describe them completely. Examples:
Inertia tensor, Stress tensor, and Strain tensor.
 Scalars and Vectors are special cases of Tensors.
INTRODUCTION TO FREE-BODY DIAGRAM
Free Body Diagram- is a sketch of the isolated body which shows only
the forces acting upon it by the removed
elements.
Applied Forces- the forces acting on the isolated or free body
- these are the action forces
Reaction Forces- are those exerted by the free-body upon other bodies
 One exception to this concept concerns ground support whose actions
on to the free body are commonly called reactions.
 The free-body may consist of an entire assembled structure or an
isolated part of it, depending on which quantities are to be
determined.
RESULTANTS OF FORCE SYSTEMS
 The effect of a system of forces on a body is usually expressed in
terms of its resultant, since the value of this resultant determines
the motion of the body.
• If the resultant is zero, the body will be in equilibrium and
will not change its original state of motion (statics).
• If the resultant of a force system is not zero, the body will
have a varying state of motion, thereby creating a problem in
dynamics.
PARALLELOGRAM LAW
The method of vector addition is based on the Parallelogram Law
which is one of the fundamental axioms of Mechanics. The
Parallelogram Law cannot be proved: it can only be demonstrated
by experiment
Parallelogram Law: “The resultant of two (2) forces is the
diagonal of the parallelogram whose
initial sides are the vectors of these
forces.”
 The diagonal to be used is that which emanates from the
intersection of the initial sides.
Free Vector- is one which may be freely moved in space
Localized Vector- a vector that is fixed or bound to a specific point
of application
Triangle Law – ( a convenient corollary of the Parallelogram Law)
“If two (2) forces are represented by their free vectors placed
tip-to-tail, their resultant is the vector directed from the tail
of the first vector to the tip of the second vector.”
FORCES AND COMPONENTS
 The Parallelogram Law shows how to combine two (2) forces
into a resultant force. Of equal importance is the inverse
operation, called resolution, in which a given force is
replaced by two (2) components which are equal or
which are equivalent to the given force.
RESULTANT OF COPLANAR CONCURRENT FORCES
The determination of the resultant of three (3) or more concurrent
forces that are not collinear requires determining the sum of three
(3) or more vectors. There are two (2) ways of accomplishing the
addition of three (3) or more vectors: graphically and analytically.
Graphically:
- Two vectors can be added to give a resultant; this resultant in
turn can be added to a third vector, etc., until all the vectors have
been added together to give an overall resultant. These vectors
can be added in any order.
Analytically:
- The vectors can be resolved into components that coincide
with arbitrarily chosen axes. The components of
each vector with respect to these axes can be added
algebraically, and the resulting additions will be the
components of the overall resultant vector.