Theoretical Mechanics | Forces and Other Vector Quantities | CHARACTERISTICS OF FORCES Lecture Outline 1-NEWTON S THIRD LAW 2-FORCE RESULTANTS AND FORCE COMPONENTS 3-RECTANGULAR COMPONENTS OF A VECTOR 4-POLYGON LAW OF FORCES 5-NUMERICAL PROBLEMS

1. Lecture # 03 Semester : 2nd
Lecture Main Title :
Forces and Other Vector Quantities

2. Lecture Outline
 CHARACTERISTICS OF FORCES
 NEWTON S THIRD LAW
 FORCE RESULTANTS AND FORCE COMPONENTS
 RECTANGULAR COMPONENTS OF A VECTOR
 NUMERICAL PROBLEMS

3. Force Resultants and Force Components
Forces add by the parallelogram law. The addition of forces is called
composition of forces.
 Determine the resultant of a system of two or more forces using either graphical or
trigonometric methods.
 Resolve a force into its components along a specified set of axes.
 The investigations of Stevin, Descartes, Newton, and other scientists of the 17th
century led to the conclusion that the effects of several forces that act at a common
point P of a body invariably may be produced by a single force that acts at point P.
 The single force that is equivalent to the several forces is called the resultant force,
or simply the resultant.

4. Characteristics of Forces
THE PARALLELOGRAM LAW AND COMPOSITION OF FORCES
 Two forces F and G that act at a point P of a body are represented by two arrows
drawn outward from point P ( see Fig. below ) .
 These arrows form two sides of a parallelogram.
 The diagonal R of the parallelogram, drawn from point P, represents the resultant
of the forces F and G.
 In all respects, the single force R is equivalent to the resultant of the two forces F
and G.
 This statement expresses the parallelogram law of force, a principle that has been
verified by direct experiments.
 The process of combining two or more concurrent forces into a single resultant
force is called composition of forces.
Parallelogram construction of the resultant
of two concurrent forces

5. Characteristics of Forces
POLYGON LAW OF FORCES
 The parallelogram construction of the resultant R of two concurrent forces F and G is
illustrated in Fig. 2. 6a.
 Since the sides AB and PC are equal and parallel, we may simplify the construction by
drawing only the triangle PBA ( Fig. 2.6b) .
 The side BA represents the force F, although it does not designate the actual line of action of
F.
 For the graphical construction of the resultant R, we displace the line segment representing
force F.
 You must remember, however, that all forces act at point P .
 Alternatively, the resultant R may be obtained by displacing the line segment PB that
represents force G to coincide with CA ( Fig. 2. 6c) .
 This method is called the triangle construction of the resultant force.

6. Characteristics of Forces
POLYGON LAW OF FORCES
 The triangle construction is easily generalized to the case where there are more
than two concurrent forces.
 For example, suppose there are three forces F, G, and H acting at a point P ( Fig. 2.
7a) .
 The triangle construction first yields the resultant R ( dashed in Fig. 2. 7b) of the
two forces F and G.
 Application of the triangle construction again with the forces R and H gives the
resultant R of the three forces F , G , and H .

7. Characteristics of Forces
POLYGON LAW OF FORCES
 The intermediate resultant R is shown only for illustrative purposes. To obtain the
resultant R, you can simply arrange the directed arrows representing the forces F,
G, and H in a chain, maintaining their proper directions.
 Then the arrow from the initial point P to the terminal point of the chain
represents the resultant force.
 This construction is not limited to three forces; it applies for any number of
concurrent forces.
 Figure 2. 8 illustrates the construction of the resultant of five forces that act at a
point P.
(a) system of five forces
(b) Force polygon

8. Characteristics of Forces
PROBLEM-SOLVING TECHNIQUE
 Resultant of Two Concurrent Forces
To find the resultant of a pair of concurrent forces:
 Draw the two concurrent forces, to a convenient scale, from a common point of
application, with direction angles measured from a common reference line.
 Construct a parallelogram with the given forces as sides.
 Draw the resultant as the diagonal of the parallelogram, with the tail of the arrow
at the point of application of the two given forces.
 Measure the magnitude of the resultant with a scaled ruler; measure the direction
angle of the resultant, relative to the common reference line, with a protractor.
 Alternatively, find the magnitude and direction of the resultant by trigonometry.

9. Problem
Polygon Construction of the Resultant of Four Concurrent Forces
 Problem Statement Four coplanar, concurrent forces act at point O of a body ( see
Fig. E2. 2a) . Use the polygon construction and trigonometry to determine the
magnitude ( in newtons) and direction of their resultant R .

10. RECTANGULAR Components OF A VECTOR
 Determine the components of a vector with respect to an arbitrary rectangular
Cartesian coordinate system.
 Manipulate vectors using their rectangular Cartesian
projections.
 Represent and manipulate vectors using their direction cosines
ORTHOGONAL PROJECTION OF A VECTOR ON AN AXIS
 Some of the most important properties of vectors are concerned with orthogonal projections
of vectors on fixed axes ( Fig. 2. 9) .
 An orthogonal projection of a vector A on an axis L is constructed by drawing projection
lines (shown as dashed lines in Fig. 2.9) from the axis to the tail and tip of the vector;
 these projection lines are perpendicular ( orthogonal) to the axis. The projection has a
magnitude defined by the distance between the two projection lines.
 The projection has a sign determined by the orientation of the vector A with respect to the
axis L.

11. RECTANGULAR Components OF A VECTOR
ORTHOGONAL PROJECTION OF A VECTOR ON AN AXIS
 The sign of the projection is positive if the component of A parallel to L has the same sense as
the axis; otherwise, the sign of the projection is negative ( see Fig. 2. 9) .
 If the vector is perpendicular to the axis, its projection on the axis is zero.
 If a vector of magnitude A forms an angle with an axis, the projection of the vector on the
axis is represented in magnitude and in sign by A cos .
 This is true whether is interpreted to be the smaller angle or the reflex angle that the vector
forms with the axis.
 Therefore, the projection of a vector on an axis is regarded as a scalar since it is
characterized as a signed numerical value.
 Consequently, the projection of a vector has no point of application.
 In this book, a reference to the projection of a vector on an axis should be understood to
imply an orthogonal projection. When nonorthogonal projections are considered, they will
be explicitly identified as such.

12. RECTANGULAR Components OF A VECTOR
RECTANGULAR CARTESIAN PROJECTIONS OF A VECTOR
 The projections of a vector on rectangular Cartesian xyz axes in space are denoted by
appending the subscripts x, y, and z to the symbol that denotes the magnitude of the vector.
 For example, the projections of a vector F on the xyz axes are denoted by (Fx, Fy, Fz); see
Fig. 2.10.
 The terms Fx, Fy, and Fz are called the rectangular Cartesian projections of the vector F.

13. RECTANGULAR Components OF A VECTOR
RECTANGULAR CARTESIAN PROJECTIONS OF A VECTOR
 The angle between a vector and a Cartesian coordinate axis is called the direction angle of
the vector with respect to that axis.
 The three direction angles x , y , and z of a vector with reference to the three rectangular
Cartesian axes determine the direction of the vector ( see Fig. 2. 11 ) .
 The direction angles of a vector are specified to lie in the range from 0 to 180 , inclusive.
Consequently, a direction angle can be determined by its cosine.
 If the cosine is negative, the angle is greater than 90 .
 The cosines of the direction angles of a vector are called the direction cosines of the vector.
 If x , y , and z are the direction angles of vector F and if F denotes the magnitude of F , then
𝐅𝐱 = Fcos 𝛉𝐱 , 𝐅𝐲 = Fsin 𝛉𝐲 , 𝐅𝐳 = Ftan 𝛉𝐳 (2.1)
 In terms of the rectangular Cartesian projections, the magnitude of any vector F is
determined by the equation
𝐅𝟐
= 𝐅𝐱
𝟐
+ 𝐅𝐲
𝟐
+ 𝐅𝐳
𝟐

14. RECTANGULAR Components OF A VECTOR
ADDITION OF VECTORS USING RECTANGULAR CARTESIAN
PROJECTIONS
 The polygon construction of the resultant of several vectors is shown in Fig. 2.12.
 You can see from Fig. 2. 12 that the projection AE of the resultant R on the x axis is equal to
the algebraic sum of the projections AB, BC, CD, and DE of the vectors F, G, H, and J on
that axis.
 In other words, Rx = Fx +Gx +Hx +Jx, where Fx, Gx , Hx, and Jx are the projections of
vectors F, G, H, and J on the x axis, respectively.
 In the case of rectangular Cartesian xyz axes, the expression for Rx may be supplemented by
similar relations Ry and Rz for projections on the y and z axes.
 The complete set of relations is
Rx =Fx +Gx +Hx +Jx

15. RECTANGULAR Components OF A VECTOR
ADDITION OF VECTORS USING RECTANGULAR CARTESIAN
PROJECTIONS
 or, in more concise vector notation, Thus, the projections of a resultant vector on xyz
coordinate axes are equal to the sums of the
R =F+G+H+J
projections of the individual vectors on the same xyz axes. This leads to the following theorem.
The resultant of several vectors is a vector whose projection on any axis is the
algebraic sum of the projections of the original vectors on the axis.
 This theorem tells you that you can obtain the resultant of several vectors by projecting each
of the vectors onto rectangular xyz axes and, for each axis, adding the corresponding
projections to obtain the ( x, y, z) projections ( Rx , Ry , Rz) of the resultant R.
 By the Pythagorean theorem [ Eq. ( 2. 2) ] with ( Rx , Ry , Rz) you can obtain the magnitude
R of the resultant R. Then, by Eq. ( 2. 1) , you can compute the direction cosines ( cos x, cos y
, cos z ) and, hence, the angles ( x, y, z ) that R forms with the xyz axes.
 In view of Eq. (2.4), R is called the vector sum of F, G, H, and J.
 The expressions vector sum and resultant are used interchangeably. Equation ( 2. 4) suggests
use of the polygon construction to find the resultant vector R.
 However, the projection equations [ Eqs. ( 2 . 3) ] and the vector equation [ Eq. ( 2. 4 ) ] are
equivalent.
 The polygon construction is useful in visual ( graphical) work, but the Cartesian projections
are often more convenient for analytical ( algebraic) work.

16. Problem Solving Techniques

17. Problem

18. Problem

19. Assignment Work
Read the Chapter 2 Highlights ???
Solve the Problems which are given at the end of
chapter

20. Reference
Book Title : Engineering Mechanics
Author : (Boresi, R. P.), (Schmidt, R. J.)

21. End
Thank you very much for your attention!