SlideShare a Scribd company logo
CHAPTER 1 
RESULTANT OF COPLANAR FORCES 
CONTENT OF THE TOPIC: 
 Introduction to Applied Mechanics 
 Mechanics or Engineering Mechanics 
 Branches of Mechanics 
 SI system of Units, Basic units, Derived units 
 Body, Rigid body, particle 
 Scalar quantity, vector quantity 
 Force and Graphical representation of force. 
 Moment of forces 
 Couple and moment of couple 
 Law of Parallelogram of forces 
 Law of transmissibility of forces 
 Varignon’s theorem, 
 Composition of forces 
 Coplanar force system 
 Coplanar Non-concurrent force system 
 Analytical method 
 Graphical Method: Triangle law of forces, polygon law of forces 
 Bow’s notation 
 Problems 
 Problems on calculation of resultant 
 Problems on Varignon’s Theorem 
Applied Mechanics: 
It is the branch of engineering which studies the effect of external forces applied in any 
manner on a particle or a body. 
Engineering Mechanics/ Mechanics: 
It is the branch of physical science which deals with the behavior of a body when the body 
is at rest or in motion. 
Depending upon the body to which the mechanics is applied, the Engineering Mechanics/ 
Mechanics is classified as 
a) Mechanics of solids 
b) Mechanics of fluids 
Mechanics of solids (rigid bodies) further classified in two groups: 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 1
Statics: 
It is a branch of Mechanics which deals with the studies of the bodies or rigid bodies in 
equilibrium under the action of external forces. 
Dynamics: 
It is a branch of Mechanics which deals with the studies of the bodies or rigid bodies in 
motion. 
Dynamics has two parts: 
a) Kinematics 
b) Kinetics 
Kinematics: 
The study of the body in motion, when the forces which cause the motion are not 
considered, is called as Kinematics. 
Kinetics: 
The study of the body in motion, when the forces which cause the motion are 
considered, is called as Kinetics. 
SI system of Units: 
It is an internal system of units. It is universally approved and accepted. It is adopted by 
large number of countries. 
System: 
Measuring systems are adopted for the measurement of physical quantities. 
Unit/Quantity: 
It is standard for the measurement of physical quantities. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 2
Basic Unit/ Fundamental units/ Basic quantities: 
Basic quantities/ Basic Unit: 
The quantities which do not depend upon other quantities for their measurement is known 
as basic quantities and their corresponding units are known as the basic units. 
Eg. Length, Mass, Time, Temperature, Electric current, plane angle etc. 
Derived quantities/ Derived Unit: 
The quantities which depend upon one or more basic quantities for their measurement is 
known as derived quantities and their corresponding units are known as the derived units. 
Eg. Velocity, Acceleration, Force, Work & Energy, Power etc. 
Body: 
A body is defined as an object, which cannot retain its shape and size under the action of a 
force system. 
Rigid body: 
A rigid body is defined as a body, which can retain its shape and size even if subjected to 
external forces. 
In practice, there is small deformation of body under the action of a force system. Such 
deformation is neglected and the body is treated as rigid body. 
Particle: 
A particle is defined as a very small amount of matter, which may be assumed to occupy 
a single point in space. 
Practically, any object having very small dimensions as compared to its range of motion 
can be called as a Particle. 
Eg. Stars, planets, Rockets, Bullets etc. 
Scalar quantity: 
It is the quantity having magnitude only. It has no direction. 
Eg. Mass, speed etc. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 3
Vector quantity: 
It is the quantity having magnitude and direction. It is shown by vector. 
Eg. Force, Velocity, acceleration etc. 
Force: 
The external agency, which tends to change the state of a body is known as force. 
A force is completely defined only when the following four characteristics are specified: 
- Magnitude 
- Point of application 
- Line of action 
- Direction 
A force (F) is a vector quantity which is represented graphically by a straight line say ‘ab’ 
whose length is proportional to the magnitude of force and the arrow shows the direction of 
force ‘ab’ as shown in Figure above. Unit of force is Newton (N). 
Force System: 
When several forces of different magnitude and direction act upon a body, they constitute a 
system of forces. 
Main types of force systems are as follows: 
1) Coplanar Force System: 
Lines of action of all the forces lie in the same plane in this system as shown in Fig. (A) 
below. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 4
2) Collinear Force System: 
Lines of action of all the forces lie in the same straight line in this system as shown in Fig. 
(B) above. 
3) Concurrent Force System: 
Lines of action of all the forces meet at a point in this system. The concurrent forces may 
not be collinear or coplanar as shown in Fig. (C) above. 
4) Parallel Force System: 
Lines of action of all the forces are in parallel as shown in Fig. (D) above. 
5) Non- Coplanar Force System: 
Lines of action of all the forces does not lie in the same plane as shown in Fig. (E) above. 
6) Non- Concurrent Force System: 
Lines of action of all the forces do not meet at a point in this system as shown in Fig. (E & 
F) above. 
7) Non-Parallel Force System: 
Lines of action of all the forces are not in parallel as shown in Fig. (H) above. 
8) Coplanar Concurrent Force System: 
Lines of action of all the forces lie in the same plane and meet at a point shown in Fig. (G) 
above. 
9) Coplanar Non-Concurrent Force System: 
Lines of action of all the forces lie in the same plane, but do not meet at a a point as shown 
in Fig. (A) above. They may be in parallel. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 5
10) Coplanar parallel Force System: 
Lines of action of all the forces are in parallel in the same plane shown in Fig. (D) above. 
11) Coplanar, non-concurrent, non-parallel Force System: 
The lines of action of all the forces are not in parallel, they do not meet at a point but they 
are in the same plane as shown in Fig. (A) above. 
12) Non- Coplanar, non-concurrent Force System: 
The lines of action of all the forces do not lie in the same plane and do not meet at a point 
as shown in Fig. (E) above. 
Fundamental Laws of Mechanics: 
 Newton’s First Law 
 Newton’s Second Law 
 Newton’s Third Law 
 Newton’s Law of gravitation 
 Law of transmissibility of Force 
 Parallelogram law of Forces 
1) Newton’s First Law: 
It states that every body continues in its state of rest or of uniform motion in a straight line 
unless it is compelled by external agency acting on it. 
 Newton’s First Law for rotation: 
Newton’s laws of motion of rotation which state that, “Every body continues in its state of 
rest or of uniform motion of rotation about an axis unless it is acted upon by some external 
torque” 
2) Newton’s Second Law: 
It states that the rate of change of momentum of a body is directly proportional to the 
impressed force and it takes place in the direction of the force acting on it. 
Force α rate of change of momentum 
But, 
Momentum = Mass x velocity 
As mass do not change, 
Force α Mass x rate of change velocity 
Force α Mass x acceleration 
F α ma 
F = ma 
3) Newton’s Third Law: 
It states that for every action there is an equal and opposite reaction. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 6
4) Newton’s Law of gravitation: 
Everybody attracts the other body. The force of attraction between any two bodies is 
directly proportional to their masses and inversely proportional to the square of the distance 
between them. 
Where, G is the constant of proportionality, it is known as constant of gravitation. 
Experimentally, it is proved that the value of G = 6.673 x 10-11 Nm2/kg2 
F= G 
푚1 푚2 
푑2 
5) Law of transmissibility of Force: 
Statement: 
“The point of application of force may be transmitted along its line of action without 
changing its effect on the rigid body to which the force is applied”. 
Explanation: 
A force is acting at point A along line of action AB on rigid body as shown in Fig. (a). 
Two equal and opposite forces of magnitude ‘P’ are added at point ‘B’ along line of action 
AB according to the law of superposition as shown in Fig (b). 
Figure (a) Figure (b) 
Two equal and opposite forces of the magnitude ‘P’ at point A and B can be subtracted 
without changing action of original force P according to the law of superposition as shown in 
Fig (c). 
Figure (c) 
Thus the point of application of force P is transmitted along its line of action from A to B. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 7
Varignon’s Theorem of Moments/ Principle of Moments: 
Statement: 
“The algebraic sum of the moments of all the forces about any point is equal to the 
moment of their resultant about the same point”. 
i.e. ΣM = Σ (Moments of forces) = Moment of R 
Proof: 
In above Figure AB and AC represents forces P and Q resp. and ‘O’ is the point about 
which moment is taken. ABCD represents a parallelogram. 
A diagonal AD represents resultant of forces P and Q. Now extend CD up to the point ‘O’ 
which is the line of CD. Join OA and OB. 
Now, we know that, 
Moment of force = 2(Area of triangle) 
Moment of force P = 2 x Area of Triangle AOB 
And Moment of force Q = 2 x Area of Triangle AOC 
Algebraic sum of Moments of forces P and Q = Σ M = 2 x (Area of ΔADB + Area of ΔAOC) 
Now, 
Area of Δ AOB = Area of ΔADB = Area of ΔACD 
Since, AB = CD (base is same) and height is same 
Σ M = 2 x (Area of Δ ACD + Area of Δ AOD) = 2 x (Area of Δ AOD) 
Σ M = Moment of Resultant Force ‘R’ 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 8
Application: 
1) It is generally used to locate the point of application of resultant. 
2) In case of coplanar non-concurrent system of forces this concept is used to locate the line 
of action of the resultant. 
Parallelogram law of Forces Statement: 
Statement: 
“If two forces acting simultaneously on a body at a point are presented in magnitude and direction 
by the two adjacent sides of parallelogram, their resultant is represented in magnitude and 
direction by the diagonal of the parallelogram which passes through the point of intersection of the 
two sides representing the forces”. 
Fig. (a) Fig. (b) 
The length of diagonal in Fig. (b) will indicate the magnitude of resultant of ‘R’. 
Derivation: 
From right angle triangle BCD 
BD = Q sinθ 
CD = Q cosθ 
Using Pythagorus theorem to the ΔOCD 
OC2 = CD2 + OD2 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 9
OC2 = CD2 + (OB + BD) 2 
R2 = (P + Q cosθ)2 + (Q sinθ)2 
R2 = P2 + Q2 cos2θ + 2PQ cosθ + Q2 sin2θ 
R2 = P2 + Q2 + 2PQ cosθ 
R = √P2 + Q2 + 2PQ cosθ ---------------------------------------(1) 
Angle α of resultant R with force P is given by, 
α = tan-1[ Q sinθ 
푃+ Q cosθ 
] ---------------------------------------(2) 
Particular cases: 
1) When θ = 900 R= √푃2 + 푄2 
2) When θ = 00 R= P + Q (acting along Same Direction) 
3) When θ = 1800 R= P – Q (acting in Opposite Direction) 
Moment: 
The turning effect caused by a force on the body is called as a moment of force. 
Definition: 
The moment of a force (M) is equal to the magnitude of the force (F) multiplied by the 
perpendicular distance (d) between the line of action of the force and the axis of rotation. 
Moment = Force x Perpendicular Distance 
M = F x d 
Sign convention: 
If the moment of the force producing clockwise rotation is the clockwise moment and it is taken as 
positive as shown in Fig. (a). 
If the moment of the force producing anticlockwise rotation is the anticlockwise moment and it is 
taken as negative as shown in Fig. (b). 
Figure (a) Figure (b) 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 10
Unit: 
If the force is measured in Newton and the distance in meter, the SI unit of the moment is Newton 
meter (Nm). 
Geometrical Representation of Moment: 
As shown in Fig. below, AB represents force F and O is the point about which the moment of 
force M is taken. Let OC be the perpendicular distance‘d’. Moment of Force F is given by, 
M = F x d 
M = AB x OC 
M = 2 x (½ AB x OC) 
M = 2 (Area of triangle OAB) 
Thus Moment of Force about any point is geometrically equal to twice the area of the triangle 
having base representing a point about which moment is taken. 
Couple: 
Two equal, opposite and parallel (non-collinear) forces are said to form a couple as shown in Fig. 
below. 
Figure (a) 
Arm of couple: 
The distance ‘a’ between the lines of action of the two forces of a couple is known as ‘arm of 
couple’. 
Properties: 
a) Couple cannot be replaced by a single resultant force. 
b) Couple cannot produce rotation or moment but it cannot produce straight line motion. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 11
Moment of Couple: 
From above Fig. moment of couple about any point ‘O’ (moment of centre) is given by 
F (a +d) – (Fd) = Fa + Fd – Fd = Fa Nm 
Moment of Couple = Force x Arm 
Thus the moment of the couple has a constant value irrespective of the point about which moment 
is taken. 
Bow’s notation: 
Any force F divides the space into two parts A and B as shown in Fig. below. This force is named 
as force AB according to this method. 
Space Diagram Force Diagram 
In this method line ‘ab’ is drawn parallel to force F such that the length of line ab represents the 
magnitude and the direction is from ‘a’ to ‘b’ which indicates the direction of the force ‘F’ as 
shown in Fig. below. 
Suitability: 
This notation is useful for solving the problems in statics by graphical method. 
Composition of Forces: 
(Resultant of coplanar concurrent forces) 
The process of determining the Resultant of number of forces acting simultaneously on a body is 
known as Composition of Forces. It is the method of reducing the given force system to its 
equivalent simplest system of single force (or couple). 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 12
Combining the forces of any given system is termed as composition of forces. 
There are two main methods of determining the resultant force: 
a) Analytical Methods and 
b) Graphical Methods 
Analytical Methods: 
There are two Analytical Methods: 
1) Parallelogram law of Forces (as explained above) 
2) Component Law or Resolution Methods 
1. Component Law of Forces: 
1. Forces such as F1, F2, F3 and F4 acting at point ‘O’ as shown in Fig. above. are resolved 
along x-axis (horizontally). The algebraic sum of horizontal components is ΣH or ΣFX. 
Figure (a) 
2. Similarly, the forces are resolved along y-axis (vertically). The algebraic sum of vertical 
components is ΣV or ΣFy. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 13
3. Resultant ‘R’ is given by, 
R = √ΣF푋2 
2 
+ ΣF푌 
4. Angle of inclination θ with x-axis is given by, 
θ = tan-1[ΣFy/ ΣFX] 
Particular cases: 
1) When θ = 900 R= √푃2 + 푄2 
2) When θ = 00 R= P + Q (acting along Same Direction) 
3) When θ = 1800 R= P – Q (acting in Opposite Direction) 
Sign convention: 
While taking ΣFX, forces acting from left to right are taken as positive and those are acting from 
right to left are considered negative. While taking ΣFy forces acting upwards are assumed 
positive and those acting downwards are assumed negative. 
Resolution of Forces: 
The process of splitting or subdividing a force into its components without changing its effect on 
the body is known as Resolution of Forces. It is the replacement of a single force by several 
components having the same effect as that of single force. 
Generally, a force ‘F’ is resolved into two components Fx and Fy which are mutually perpendicular 
to each other as shown in Figure below. 
Horizontal component Fx = F cos θ 
Vertical component FY = F sin θ 
Consider a rigid body as shown in Fig. above. Let F1, F2 and F3 be three forces acting on a 
rigid body. Let ‘R’ be there resultant. Then we can say that F1, F2 and F3 are resolved parts of 
R or components of ‘R’ in three different directions. 
Generally, a force is replaced in to two rectangular components. 
Graphical Methods: 
1) Law of triangle of Forces: 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 14
If two forces P and Q (acting at point ‘O’) as shown in Figure below in which they represents 
the magnitude and direction of the two sides of the triangle taken in order, then the third side 
taken in opposite sense represents the resultant ‘R’ of the two forces in magnitude and 
direction. 
2) Law of Polygon of Forces: 
If numbers of forces are acting on a body, are represented in magnitude and direction by the 
sides of the polygon taken in order, then the closing side taken in opposite sense represents 
the resultant of all the forces in magnitude and direction. 
Fig. a Fig. b Fig. c 
Above Figure (a) shows the system of four forces in magnitude and direction and Figure 
(b) shows the polygon of same forces. The closing side ‘R’ represents the resultant. 
We can use the triangle law of forces in this polygon, such that, the resultant of forces F1 and 
F2 is R as shown in Figure c. Similarly, the resultant of R1 and F3 is R2 and finally, the resultant 
of R2 and F4 is R by the triangle law of forces. 
Conclusion: 
The polygon law of forces is the application of triangle law of forces. 
Idealizations in mechanics: 
1) The body is rigid. 
2) The body is treated as continuum. 
Continuum: when the body is assumed to consist of a continuous distribution of matter is 
called as continuum. 
3) If the size of the body is small as compared to other distances involved in the problem, it 
may be treated as a particle. 
4) If the area over which force is acting on a body is small as compared to the size of the 
body, it may be treated as a point force. 
5) Support conditions are idealized as simple, hinged, fixed etc. 
CHAPTER NO. 1 Resolution of Coplanar Forces Page 15

More Related Content

What's hot

Moment of inertia
Moment of inertia Moment of inertia
Moment of inertia
Ankita Dungarwal
 
System of forces
System of forcesSystem of forces
System of forces
Rinkita Panchal
 
STRAIN ENERGY CONCEPT STRENGTH OF MATERIAL
STRAIN ENERGY CONCEPT STRENGTH OF MATERIALSTRAIN ENERGY CONCEPT STRENGTH OF MATERIAL
STRAIN ENERGY CONCEPT STRENGTH OF MATERIAL
dhavalprajapati100
 
Som ppt
Som pptSom ppt
Som ppt
27072707
 
Centroid & Centre of Gravity
Centroid & Centre of GravityCentroid & Centre of Gravity
Centroid & Centre of Gravity
Akash Patel
 
Structures and Materials- Section 6 Axially Loaded Structural Members
Structures and Materials- Section 6 Axially Loaded Structural MembersStructures and Materials- Section 6 Axially Loaded Structural Members
Structures and Materials- Section 6 Axially Loaded Structural Members
The Engineering Centre for Excellence in Teaching and Learning
 
moments couples and force couple systems by ahmad khan
moments couples and force couple systems by ahmad khanmoments couples and force couple systems by ahmad khan
moments couples and force couple systems by ahmad khan
Self-employed
 
Principle stresses and planes
Principle stresses and planesPrinciple stresses and planes
Principle stresses and planes
PRAJWAL SHRIRAO
 
Engineering Mechanics Fundamentals
Engineering Mechanics FundamentalsEngineering Mechanics Fundamentals
Engineering Mechanics Fundamentals
Yasir Hashmi
 
INTRODUCTION TO ENGINEERING MECHANICS - SPP.pptx
INTRODUCTION TO ENGINEERING MECHANICS - SPP.pptxINTRODUCTION TO ENGINEERING MECHANICS - SPP.pptx
INTRODUCTION TO ENGINEERING MECHANICS - SPP.pptx
Samirsinh Parmar
 
Strength of materials by A.Vinoth Jebaraj
Strength of materials by A.Vinoth JebarajStrength of materials by A.Vinoth Jebaraj
Strength of materials by A.Vinoth Jebaraj
Vinoth Jebaraj A
 
Some basics of Strength Of Materials..
Some basics of Strength Of Materials..Some basics of Strength Of Materials..
Some basics of Strength Of Materials..
Mohammed Mubeen
 
Lami's Theorem | Mechanical Engineering
Lami's Theorem | Mechanical EngineeringLami's Theorem | Mechanical Engineering
Lami's Theorem | Mechanical Engineering
Transweb Global Inc
 
Strength of Materials
Strength of MaterialsStrength of Materials
Strength of Materials
Tanzania Atomic Energy Commission
 
Strain energy
Strain energyStrain energy
Deflection
DeflectionDeflection
Deflection
Shivendra Nandan
 
Lecture 2 principal stress and strain
Lecture 2 principal stress and strainLecture 2 principal stress and strain
Lecture 2 principal stress and strain
Deepak Agarwal
 
Coplanar Non-concurrent Forces
Coplanar Non-concurrent ForcesCoplanar Non-concurrent Forces
Coplanar Non-concurrent Forces
Mahesh Bajariya
 
FORCE, TYPES, & SYSTEM OF FORCES
FORCE, TYPES, & SYSTEM OF FORCESFORCE, TYPES, & SYSTEM OF FORCES
FORCE, TYPES, & SYSTEM OF FORCES
Manish Jha
 
Chapter 3
Chapter 3Chapter 3
Chapter 3
krishn_desai
 

What's hot (20)

Moment of inertia
Moment of inertia Moment of inertia
Moment of inertia
 
System of forces
System of forcesSystem of forces
System of forces
 
STRAIN ENERGY CONCEPT STRENGTH OF MATERIAL
STRAIN ENERGY CONCEPT STRENGTH OF MATERIALSTRAIN ENERGY CONCEPT STRENGTH OF MATERIAL
STRAIN ENERGY CONCEPT STRENGTH OF MATERIAL
 
Som ppt
Som pptSom ppt
Som ppt
 
Centroid & Centre of Gravity
Centroid & Centre of GravityCentroid & Centre of Gravity
Centroid & Centre of Gravity
 
Structures and Materials- Section 6 Axially Loaded Structural Members
Structures and Materials- Section 6 Axially Loaded Structural MembersStructures and Materials- Section 6 Axially Loaded Structural Members
Structures and Materials- Section 6 Axially Loaded Structural Members
 
moments couples and force couple systems by ahmad khan
moments couples and force couple systems by ahmad khanmoments couples and force couple systems by ahmad khan
moments couples and force couple systems by ahmad khan
 
Principle stresses and planes
Principle stresses and planesPrinciple stresses and planes
Principle stresses and planes
 
Engineering Mechanics Fundamentals
Engineering Mechanics FundamentalsEngineering Mechanics Fundamentals
Engineering Mechanics Fundamentals
 
INTRODUCTION TO ENGINEERING MECHANICS - SPP.pptx
INTRODUCTION TO ENGINEERING MECHANICS - SPP.pptxINTRODUCTION TO ENGINEERING MECHANICS - SPP.pptx
INTRODUCTION TO ENGINEERING MECHANICS - SPP.pptx
 
Strength of materials by A.Vinoth Jebaraj
Strength of materials by A.Vinoth JebarajStrength of materials by A.Vinoth Jebaraj
Strength of materials by A.Vinoth Jebaraj
 
Some basics of Strength Of Materials..
Some basics of Strength Of Materials..Some basics of Strength Of Materials..
Some basics of Strength Of Materials..
 
Lami's Theorem | Mechanical Engineering
Lami's Theorem | Mechanical EngineeringLami's Theorem | Mechanical Engineering
Lami's Theorem | Mechanical Engineering
 
Strength of Materials
Strength of MaterialsStrength of Materials
Strength of Materials
 
Strain energy
Strain energyStrain energy
Strain energy
 
Deflection
DeflectionDeflection
Deflection
 
Lecture 2 principal stress and strain
Lecture 2 principal stress and strainLecture 2 principal stress and strain
Lecture 2 principal stress and strain
 
Coplanar Non-concurrent Forces
Coplanar Non-concurrent ForcesCoplanar Non-concurrent Forces
Coplanar Non-concurrent Forces
 
FORCE, TYPES, & SYSTEM OF FORCES
FORCE, TYPES, & SYSTEM OF FORCESFORCE, TYPES, & SYSTEM OF FORCES
FORCE, TYPES, & SYSTEM OF FORCES
 
Chapter 3
Chapter 3Chapter 3
Chapter 3
 

Similar to Applied mechanics

Applied mechanics
Applied mechanicsApplied mechanics
Applied mechanics
Pralhad Kore
 
Module 3.pptx
Module 3.pptxModule 3.pptx
Module 3.pptx
AkshitRajput23
 
Unit 1 ( introduction basic)
Unit 1 ( introduction basic)Unit 1 ( introduction basic)
Unit 1 ( introduction basic)
V.Mohan Kumar
 
Statics of particle
Statics of particle Statics of particle
Statics of particle
ArundevKozhunthuvel
 
Engineering Mechanics - Intro to Statics.pdf
Engineering Mechanics - Intro to Statics.pdfEngineering Mechanics - Intro to Statics.pdf
Engineering Mechanics - Intro to Statics.pdf
Yogesh Kulkarni
 
Ctm 154[1]
Ctm 154[1]Ctm 154[1]
Ctm 154[1]
0242694327
 
Engineering mechanics
Engineering mechanics Engineering mechanics
Engineering mechanics
Ashish Mishra
 
Engineering Mechanics.pptx
Engineering Mechanics.pptxEngineering Mechanics.pptx
Engineering Mechanics.pptx
Yogesh Kulkarni
 
Force System-Engineering Mechanics
Force System-Engineering MechanicsForce System-Engineering Mechanics
Force System-Engineering Mechanics
Vinod Shikhare
 
Basic Principles of Statics
Basic Principles of StaticsBasic Principles of Statics
Basic Principles of Statics
UTM International Campus
 
Lecture 4 - Resultant of Forces - Part 1.pptx
Lecture 4 - Resultant of Forces - Part 1.pptxLecture 4 - Resultant of Forces - Part 1.pptx
Lecture 4 - Resultant of Forces - Part 1.pptx
MarjorieJeanAnog
 
Structures and Materials- Section 1 Statics
Structures and Materials- Section 1 StaticsStructures and Materials- Section 1 Statics
Structures and Materials- Section 1 Statics
The Engineering Centre for Excellence in Teaching and Learning
 
engineering mechanics - statics and dynamics
engineering mechanics - statics and dynamicsengineering mechanics - statics and dynamics
engineering mechanics - statics and dynamics
VelmuruganV15
 
2. statics.pdf
2. statics.pdf2. statics.pdf
2. statics.pdf
EmmanuelKiamaMutinda
 
ME3351 ENGG. MECHANICS MLM.pdf
ME3351 ENGG. MECHANICS MLM.pdfME3351 ENGG. MECHANICS MLM.pdf
ME3351 ENGG. MECHANICS MLM.pdf
SathiyaGK
 
2 coplanar
2 coplanar2 coplanar
FORCE-1.pptx
FORCE-1.pptxFORCE-1.pptx
FORCE-1.pptx
PREETAMSAHU20
 
APPLIED MECHANICS (1).ppt
APPLIED MECHANICS (1).pptAPPLIED MECHANICS (1).ppt
APPLIED MECHANICS (1).ppt
SatishKotwal
 
APPLIED MECHANICS online lecture.pptx
APPLIED MECHANICS online lecture.pptxAPPLIED MECHANICS online lecture.pptx
APPLIED MECHANICS online lecture.pptx
SatishKotwal
 
Analyzing motion of system of particles
Analyzing motion of system of particlesAnalyzing motion of system of particles
Analyzing motion of system of particles
vikasaucea
 

Similar to Applied mechanics (20)

Applied mechanics
Applied mechanicsApplied mechanics
Applied mechanics
 
Module 3.pptx
Module 3.pptxModule 3.pptx
Module 3.pptx
 
Unit 1 ( introduction basic)
Unit 1 ( introduction basic)Unit 1 ( introduction basic)
Unit 1 ( introduction basic)
 
Statics of particle
Statics of particle Statics of particle
Statics of particle
 
Engineering Mechanics - Intro to Statics.pdf
Engineering Mechanics - Intro to Statics.pdfEngineering Mechanics - Intro to Statics.pdf
Engineering Mechanics - Intro to Statics.pdf
 
Ctm 154[1]
Ctm 154[1]Ctm 154[1]
Ctm 154[1]
 
Engineering mechanics
Engineering mechanics Engineering mechanics
Engineering mechanics
 
Engineering Mechanics.pptx
Engineering Mechanics.pptxEngineering Mechanics.pptx
Engineering Mechanics.pptx
 
Force System-Engineering Mechanics
Force System-Engineering MechanicsForce System-Engineering Mechanics
Force System-Engineering Mechanics
 
Basic Principles of Statics
Basic Principles of StaticsBasic Principles of Statics
Basic Principles of Statics
 
Lecture 4 - Resultant of Forces - Part 1.pptx
Lecture 4 - Resultant of Forces - Part 1.pptxLecture 4 - Resultant of Forces - Part 1.pptx
Lecture 4 - Resultant of Forces - Part 1.pptx
 
Structures and Materials- Section 1 Statics
Structures and Materials- Section 1 StaticsStructures and Materials- Section 1 Statics
Structures and Materials- Section 1 Statics
 
engineering mechanics - statics and dynamics
engineering mechanics - statics and dynamicsengineering mechanics - statics and dynamics
engineering mechanics - statics and dynamics
 
2. statics.pdf
2. statics.pdf2. statics.pdf
2. statics.pdf
 
ME3351 ENGG. MECHANICS MLM.pdf
ME3351 ENGG. MECHANICS MLM.pdfME3351 ENGG. MECHANICS MLM.pdf
ME3351 ENGG. MECHANICS MLM.pdf
 
2 coplanar
2 coplanar2 coplanar
2 coplanar
 
FORCE-1.pptx
FORCE-1.pptxFORCE-1.pptx
FORCE-1.pptx
 
APPLIED MECHANICS (1).ppt
APPLIED MECHANICS (1).pptAPPLIED MECHANICS (1).ppt
APPLIED MECHANICS (1).ppt
 
APPLIED MECHANICS online lecture.pptx
APPLIED MECHANICS online lecture.pptxAPPLIED MECHANICS online lecture.pptx
APPLIED MECHANICS online lecture.pptx
 
Analyzing motion of system of particles
Analyzing motion of system of particlesAnalyzing motion of system of particles
Analyzing motion of system of particles
 

More from Pralhad Kore

Transportation engineering
Transportation engineeringTransportation engineering
Transportation engineering
Pralhad Kore
 
Chapter wise question papers_bce
Chapter wise question papers_bceChapter wise question papers_bce
Chapter wise question papers_bce
Pralhad Kore
 
Design of staircase_practical_example
Design of staircase_practical_exampleDesign of staircase_practical_example
Design of staircase_practical_example
Pralhad Kore
 
Presentation "Use of coupler Splices for Reinforcement"
Presentation "Use of coupler Splices for Reinforcement"Presentation "Use of coupler Splices for Reinforcement"
Presentation "Use of coupler Splices for Reinforcement"
Pralhad Kore
 
Guidelines_for_building_design
Guidelines_for_building_designGuidelines_for_building_design
Guidelines_for_building_design
Pralhad Kore
 
Strength of materials_I
Strength of materials_IStrength of materials_I
Strength of materials_I
Pralhad Kore
 
Presentation_on_Cellwise_Braced_frames
Presentation_on_Cellwise_Braced_framesPresentation_on_Cellwise_Braced_frames
Presentation_on_Cellwise_Braced_frames
Pralhad Kore
 
Study of MORT_&_H
Study of MORT_&_HStudy of MORT_&_H
Study of MORT_&_H
Pralhad Kore
 
List of various_IRCs_&_sps
List of various_IRCs_&_spsList of various_IRCs_&_sps
List of various_IRCs_&_sps
Pralhad Kore
 
Analysis of multi storey building frames subjected to gravity and seismic loa...
Analysis of multi storey building frames subjected to gravity and seismic loa...Analysis of multi storey building frames subjected to gravity and seismic loa...
Analysis of multi storey building frames subjected to gravity and seismic loa...
Pralhad Kore
 
Seismic response of _reinforced_concrete_concentrically_a_braced_frames
Seismic  response  of _reinforced_concrete_concentrically_a_braced_framesSeismic  response  of _reinforced_concrete_concentrically_a_braced_frames
Seismic response of _reinforced_concrete_concentrically_a_braced_frames
Pralhad Kore
 
Use of mechanical_splices_for_reinforcing_steel
Use of mechanical_splices_for_reinforcing_steelUse of mechanical_splices_for_reinforcing_steel
Use of mechanical_splices_for_reinforcing_steel
Pralhad Kore
 
Guide lines bridge_design
Guide lines bridge_designGuide lines bridge_design
Guide lines bridge_design
Pralhad Kore
 
Dissertation report
Dissertation reportDissertation report
Dissertation report
Pralhad Kore
 
Seismic response of cellwise braced reinforced concrete frames
Seismic response of cellwise braced reinforced concrete framesSeismic response of cellwise braced reinforced concrete frames
Seismic response of cellwise braced reinforced concrete frames
Pralhad Kore
 
Water Management
Water ManagementWater Management
Water Management
Pralhad Kore
 
Chaper wise qpapers_bce
Chaper wise qpapers_bceChaper wise qpapers_bce
Chaper wise qpapers_bce
Pralhad Kore
 
Basic Loads Cases
Basic Loads CasesBasic Loads Cases
Basic Loads Cases
Pralhad Kore
 
Earthquake analysis by Response Spectrum Method
Earthquake analysis by Response Spectrum MethodEarthquake analysis by Response Spectrum Method
Earthquake analysis by Response Spectrum Method
Pralhad Kore
 
Earthquake analysis by psudeo static method
Earthquake analysis by psudeo static methodEarthquake analysis by psudeo static method
Earthquake analysis by psudeo static method
Pralhad Kore
 

More from Pralhad Kore (20)

Transportation engineering
Transportation engineeringTransportation engineering
Transportation engineering
 
Chapter wise question papers_bce
Chapter wise question papers_bceChapter wise question papers_bce
Chapter wise question papers_bce
 
Design of staircase_practical_example
Design of staircase_practical_exampleDesign of staircase_practical_example
Design of staircase_practical_example
 
Presentation "Use of coupler Splices for Reinforcement"
Presentation "Use of coupler Splices for Reinforcement"Presentation "Use of coupler Splices for Reinforcement"
Presentation "Use of coupler Splices for Reinforcement"
 
Guidelines_for_building_design
Guidelines_for_building_designGuidelines_for_building_design
Guidelines_for_building_design
 
Strength of materials_I
Strength of materials_IStrength of materials_I
Strength of materials_I
 
Presentation_on_Cellwise_Braced_frames
Presentation_on_Cellwise_Braced_framesPresentation_on_Cellwise_Braced_frames
Presentation_on_Cellwise_Braced_frames
 
Study of MORT_&_H
Study of MORT_&_HStudy of MORT_&_H
Study of MORT_&_H
 
List of various_IRCs_&_sps
List of various_IRCs_&_spsList of various_IRCs_&_sps
List of various_IRCs_&_sps
 
Analysis of multi storey building frames subjected to gravity and seismic loa...
Analysis of multi storey building frames subjected to gravity and seismic loa...Analysis of multi storey building frames subjected to gravity and seismic loa...
Analysis of multi storey building frames subjected to gravity and seismic loa...
 
Seismic response of _reinforced_concrete_concentrically_a_braced_frames
Seismic  response  of _reinforced_concrete_concentrically_a_braced_framesSeismic  response  of _reinforced_concrete_concentrically_a_braced_frames
Seismic response of _reinforced_concrete_concentrically_a_braced_frames
 
Use of mechanical_splices_for_reinforcing_steel
Use of mechanical_splices_for_reinforcing_steelUse of mechanical_splices_for_reinforcing_steel
Use of mechanical_splices_for_reinforcing_steel
 
Guide lines bridge_design
Guide lines bridge_designGuide lines bridge_design
Guide lines bridge_design
 
Dissertation report
Dissertation reportDissertation report
Dissertation report
 
Seismic response of cellwise braced reinforced concrete frames
Seismic response of cellwise braced reinforced concrete framesSeismic response of cellwise braced reinforced concrete frames
Seismic response of cellwise braced reinforced concrete frames
 
Water Management
Water ManagementWater Management
Water Management
 
Chaper wise qpapers_bce
Chaper wise qpapers_bceChaper wise qpapers_bce
Chaper wise qpapers_bce
 
Basic Loads Cases
Basic Loads CasesBasic Loads Cases
Basic Loads Cases
 
Earthquake analysis by Response Spectrum Method
Earthquake analysis by Response Spectrum MethodEarthquake analysis by Response Spectrum Method
Earthquake analysis by Response Spectrum Method
 
Earthquake analysis by psudeo static method
Earthquake analysis by psudeo static methodEarthquake analysis by psudeo static method
Earthquake analysis by psudeo static method
 

Recently uploaded

This study Examines the Effectiveness of Talent Procurement through the Imple...
This study Examines the Effectiveness of Talent Procurement through the Imple...This study Examines the Effectiveness of Talent Procurement through the Imple...
This study Examines the Effectiveness of Talent Procurement through the Imple...
DharmaBanothu
 
Blood finder application project report (1).pdf
Blood finder application project report (1).pdfBlood finder application project report (1).pdf
Blood finder application project report (1).pdf
Kamal Acharya
 
一比一原版(USF毕业证)旧金山大学毕业证如何办理
一比一原版(USF毕业证)旧金山大学毕业证如何办理一比一原版(USF毕业证)旧金山大学毕业证如何办理
一比一原版(USF毕业证)旧金山大学毕业证如何办理
uqyfuc
 
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
ijseajournal
 
DELTA V MES EMERSON EDUARDO RODRIGUES ENGINEER
DELTA V MES EMERSON EDUARDO RODRIGUES ENGINEERDELTA V MES EMERSON EDUARDO RODRIGUES ENGINEER
DELTA V MES EMERSON EDUARDO RODRIGUES ENGINEER
EMERSON EDUARDO RODRIGUES
 
UNIT-III- DATA CONVERTERS ANALOG TO DIGITAL CONVERTER
UNIT-III- DATA CONVERTERS ANALOG TO DIGITAL CONVERTERUNIT-III- DATA CONVERTERS ANALOG TO DIGITAL CONVERTER
UNIT-III- DATA CONVERTERS ANALOG TO DIGITAL CONVERTER
vmspraneeth
 
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...
IJCNCJournal
 
OOPS_Lab_Manual - programs using C++ programming language
OOPS_Lab_Manual - programs using C++ programming languageOOPS_Lab_Manual - programs using C++ programming language
OOPS_Lab_Manual - programs using C++ programming language
PreethaV16
 
Properties of Fluids, Fluid Statics, Pressure Measurement
Properties of Fluids, Fluid Statics, Pressure MeasurementProperties of Fluids, Fluid Statics, Pressure Measurement
Properties of Fluids, Fluid Statics, Pressure Measurement
Indrajeet sahu
 
ELS: 2.4.1 POWER ELECTRONICS Course objectives: This course will enable stude...
ELS: 2.4.1 POWER ELECTRONICS Course objectives: This course will enable stude...ELS: 2.4.1 POWER ELECTRONICS Course objectives: This course will enable stude...
ELS: 2.4.1 POWER ELECTRONICS Course objectives: This course will enable stude...
Kuvempu University
 
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
upoux
 
原版制作(Humboldt毕业证书)柏林大学毕业证学位证一模一样
原版制作(Humboldt毕业证书)柏林大学毕业证学位证一模一样原版制作(Humboldt毕业证书)柏林大学毕业证学位证一模一样
原版制作(Humboldt毕业证书)柏林大学毕业证学位证一模一样
ydzowc
 
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICSUNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
vmspraneeth
 
AN INTRODUCTION OF AI & SEARCHING TECHIQUES
AN INTRODUCTION OF AI & SEARCHING TECHIQUESAN INTRODUCTION OF AI & SEARCHING TECHIQUES
AN INTRODUCTION OF AI & SEARCHING TECHIQUES
drshikhapandey2022
 
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Transcat
 
一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理
一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理
一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理
nedcocy
 
Introduction to Artificial Intelligence.
Introduction to Artificial Intelligence.Introduction to Artificial Intelligence.
Introduction to Artificial Intelligence.
supriyaDicholkar1
 
Determination of Equivalent Circuit parameters and performance characteristic...
Determination of Equivalent Circuit parameters and performance characteristic...Determination of Equivalent Circuit parameters and performance characteristic...
Determination of Equivalent Circuit parameters and performance characteristic...
pvpriya2
 
Unit -II Spectroscopy - EC I B.Tech.pdf
Unit -II Spectroscopy - EC  I B.Tech.pdfUnit -II Spectroscopy - EC  I B.Tech.pdf
Unit -II Spectroscopy - EC I B.Tech.pdf
TeluguBadi
 
comptia-security-sy0-701-exam-objectives-(5-0).pdf
comptia-security-sy0-701-exam-objectives-(5-0).pdfcomptia-security-sy0-701-exam-objectives-(5-0).pdf
comptia-security-sy0-701-exam-objectives-(5-0).pdf
foxlyon
 

Recently uploaded (20)

This study Examines the Effectiveness of Talent Procurement through the Imple...
This study Examines the Effectiveness of Talent Procurement through the Imple...This study Examines the Effectiveness of Talent Procurement through the Imple...
This study Examines the Effectiveness of Talent Procurement through the Imple...
 
Blood finder application project report (1).pdf
Blood finder application project report (1).pdfBlood finder application project report (1).pdf
Blood finder application project report (1).pdf
 
一比一原版(USF毕业证)旧金山大学毕业证如何办理
一比一原版(USF毕业证)旧金山大学毕业证如何办理一比一原版(USF毕业证)旧金山大学毕业证如何办理
一比一原版(USF毕业证)旧金山大学毕业证如何办理
 
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
 
DELTA V MES EMERSON EDUARDO RODRIGUES ENGINEER
DELTA V MES EMERSON EDUARDO RODRIGUES ENGINEERDELTA V MES EMERSON EDUARDO RODRIGUES ENGINEER
DELTA V MES EMERSON EDUARDO RODRIGUES ENGINEER
 
UNIT-III- DATA CONVERTERS ANALOG TO DIGITAL CONVERTER
UNIT-III- DATA CONVERTERS ANALOG TO DIGITAL CONVERTERUNIT-III- DATA CONVERTERS ANALOG TO DIGITAL CONVERTER
UNIT-III- DATA CONVERTERS ANALOG TO DIGITAL CONVERTER
 
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...
 
OOPS_Lab_Manual - programs using C++ programming language
OOPS_Lab_Manual - programs using C++ programming languageOOPS_Lab_Manual - programs using C++ programming language
OOPS_Lab_Manual - programs using C++ programming language
 
Properties of Fluids, Fluid Statics, Pressure Measurement
Properties of Fluids, Fluid Statics, Pressure MeasurementProperties of Fluids, Fluid Statics, Pressure Measurement
Properties of Fluids, Fluid Statics, Pressure Measurement
 
ELS: 2.4.1 POWER ELECTRONICS Course objectives: This course will enable stude...
ELS: 2.4.1 POWER ELECTRONICS Course objectives: This course will enable stude...ELS: 2.4.1 POWER ELECTRONICS Course objectives: This course will enable stude...
ELS: 2.4.1 POWER ELECTRONICS Course objectives: This course will enable stude...
 
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
 
原版制作(Humboldt毕业证书)柏林大学毕业证学位证一模一样
原版制作(Humboldt毕业证书)柏林大学毕业证学位证一模一样原版制作(Humboldt毕业证书)柏林大学毕业证学位证一模一样
原版制作(Humboldt毕业证书)柏林大学毕业证学位证一模一样
 
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICSUNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
 
AN INTRODUCTION OF AI & SEARCHING TECHIQUES
AN INTRODUCTION OF AI & SEARCHING TECHIQUESAN INTRODUCTION OF AI & SEARCHING TECHIQUES
AN INTRODUCTION OF AI & SEARCHING TECHIQUES
 
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
 
一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理
一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理
一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理
 
Introduction to Artificial Intelligence.
Introduction to Artificial Intelligence.Introduction to Artificial Intelligence.
Introduction to Artificial Intelligence.
 
Determination of Equivalent Circuit parameters and performance characteristic...
Determination of Equivalent Circuit parameters and performance characteristic...Determination of Equivalent Circuit parameters and performance characteristic...
Determination of Equivalent Circuit parameters and performance characteristic...
 
Unit -II Spectroscopy - EC I B.Tech.pdf
Unit -II Spectroscopy - EC  I B.Tech.pdfUnit -II Spectroscopy - EC  I B.Tech.pdf
Unit -II Spectroscopy - EC I B.Tech.pdf
 
comptia-security-sy0-701-exam-objectives-(5-0).pdf
comptia-security-sy0-701-exam-objectives-(5-0).pdfcomptia-security-sy0-701-exam-objectives-(5-0).pdf
comptia-security-sy0-701-exam-objectives-(5-0).pdf
 

Applied mechanics

  • 1. CHAPTER 1 RESULTANT OF COPLANAR FORCES CONTENT OF THE TOPIC:  Introduction to Applied Mechanics  Mechanics or Engineering Mechanics  Branches of Mechanics  SI system of Units, Basic units, Derived units  Body, Rigid body, particle  Scalar quantity, vector quantity  Force and Graphical representation of force.  Moment of forces  Couple and moment of couple  Law of Parallelogram of forces  Law of transmissibility of forces  Varignon’s theorem,  Composition of forces  Coplanar force system  Coplanar Non-concurrent force system  Analytical method  Graphical Method: Triangle law of forces, polygon law of forces  Bow’s notation  Problems  Problems on calculation of resultant  Problems on Varignon’s Theorem Applied Mechanics: It is the branch of engineering which studies the effect of external forces applied in any manner on a particle or a body. Engineering Mechanics/ Mechanics: It is the branch of physical science which deals with the behavior of a body when the body is at rest or in motion. Depending upon the body to which the mechanics is applied, the Engineering Mechanics/ Mechanics is classified as a) Mechanics of solids b) Mechanics of fluids Mechanics of solids (rigid bodies) further classified in two groups: CHAPTER NO. 1 Resolution of Coplanar Forces Page 1
  • 2. Statics: It is a branch of Mechanics which deals with the studies of the bodies or rigid bodies in equilibrium under the action of external forces. Dynamics: It is a branch of Mechanics which deals with the studies of the bodies or rigid bodies in motion. Dynamics has two parts: a) Kinematics b) Kinetics Kinematics: The study of the body in motion, when the forces which cause the motion are not considered, is called as Kinematics. Kinetics: The study of the body in motion, when the forces which cause the motion are considered, is called as Kinetics. SI system of Units: It is an internal system of units. It is universally approved and accepted. It is adopted by large number of countries. System: Measuring systems are adopted for the measurement of physical quantities. Unit/Quantity: It is standard for the measurement of physical quantities. CHAPTER NO. 1 Resolution of Coplanar Forces Page 2
  • 3. Basic Unit/ Fundamental units/ Basic quantities: Basic quantities/ Basic Unit: The quantities which do not depend upon other quantities for their measurement is known as basic quantities and their corresponding units are known as the basic units. Eg. Length, Mass, Time, Temperature, Electric current, plane angle etc. Derived quantities/ Derived Unit: The quantities which depend upon one or more basic quantities for their measurement is known as derived quantities and their corresponding units are known as the derived units. Eg. Velocity, Acceleration, Force, Work & Energy, Power etc. Body: A body is defined as an object, which cannot retain its shape and size under the action of a force system. Rigid body: A rigid body is defined as a body, which can retain its shape and size even if subjected to external forces. In practice, there is small deformation of body under the action of a force system. Such deformation is neglected and the body is treated as rigid body. Particle: A particle is defined as a very small amount of matter, which may be assumed to occupy a single point in space. Practically, any object having very small dimensions as compared to its range of motion can be called as a Particle. Eg. Stars, planets, Rockets, Bullets etc. Scalar quantity: It is the quantity having magnitude only. It has no direction. Eg. Mass, speed etc. CHAPTER NO. 1 Resolution of Coplanar Forces Page 3
  • 4. Vector quantity: It is the quantity having magnitude and direction. It is shown by vector. Eg. Force, Velocity, acceleration etc. Force: The external agency, which tends to change the state of a body is known as force. A force is completely defined only when the following four characteristics are specified: - Magnitude - Point of application - Line of action - Direction A force (F) is a vector quantity which is represented graphically by a straight line say ‘ab’ whose length is proportional to the magnitude of force and the arrow shows the direction of force ‘ab’ as shown in Figure above. Unit of force is Newton (N). Force System: When several forces of different magnitude and direction act upon a body, they constitute a system of forces. Main types of force systems are as follows: 1) Coplanar Force System: Lines of action of all the forces lie in the same plane in this system as shown in Fig. (A) below. CHAPTER NO. 1 Resolution of Coplanar Forces Page 4
  • 5. 2) Collinear Force System: Lines of action of all the forces lie in the same straight line in this system as shown in Fig. (B) above. 3) Concurrent Force System: Lines of action of all the forces meet at a point in this system. The concurrent forces may not be collinear or coplanar as shown in Fig. (C) above. 4) Parallel Force System: Lines of action of all the forces are in parallel as shown in Fig. (D) above. 5) Non- Coplanar Force System: Lines of action of all the forces does not lie in the same plane as shown in Fig. (E) above. 6) Non- Concurrent Force System: Lines of action of all the forces do not meet at a point in this system as shown in Fig. (E & F) above. 7) Non-Parallel Force System: Lines of action of all the forces are not in parallel as shown in Fig. (H) above. 8) Coplanar Concurrent Force System: Lines of action of all the forces lie in the same plane and meet at a point shown in Fig. (G) above. 9) Coplanar Non-Concurrent Force System: Lines of action of all the forces lie in the same plane, but do not meet at a a point as shown in Fig. (A) above. They may be in parallel. CHAPTER NO. 1 Resolution of Coplanar Forces Page 5
  • 6. 10) Coplanar parallel Force System: Lines of action of all the forces are in parallel in the same plane shown in Fig. (D) above. 11) Coplanar, non-concurrent, non-parallel Force System: The lines of action of all the forces are not in parallel, they do not meet at a point but they are in the same plane as shown in Fig. (A) above. 12) Non- Coplanar, non-concurrent Force System: The lines of action of all the forces do not lie in the same plane and do not meet at a point as shown in Fig. (E) above. Fundamental Laws of Mechanics:  Newton’s First Law  Newton’s Second Law  Newton’s Third Law  Newton’s Law of gravitation  Law of transmissibility of Force  Parallelogram law of Forces 1) Newton’s First Law: It states that every body continues in its state of rest or of uniform motion in a straight line unless it is compelled by external agency acting on it.  Newton’s First Law for rotation: Newton’s laws of motion of rotation which state that, “Every body continues in its state of rest or of uniform motion of rotation about an axis unless it is acted upon by some external torque” 2) Newton’s Second Law: It states that the rate of change of momentum of a body is directly proportional to the impressed force and it takes place in the direction of the force acting on it. Force α rate of change of momentum But, Momentum = Mass x velocity As mass do not change, Force α Mass x rate of change velocity Force α Mass x acceleration F α ma F = ma 3) Newton’s Third Law: It states that for every action there is an equal and opposite reaction. CHAPTER NO. 1 Resolution of Coplanar Forces Page 6
  • 7. 4) Newton’s Law of gravitation: Everybody attracts the other body. The force of attraction between any two bodies is directly proportional to their masses and inversely proportional to the square of the distance between them. Where, G is the constant of proportionality, it is known as constant of gravitation. Experimentally, it is proved that the value of G = 6.673 x 10-11 Nm2/kg2 F= G 푚1 푚2 푑2 5) Law of transmissibility of Force: Statement: “The point of application of force may be transmitted along its line of action without changing its effect on the rigid body to which the force is applied”. Explanation: A force is acting at point A along line of action AB on rigid body as shown in Fig. (a). Two equal and opposite forces of magnitude ‘P’ are added at point ‘B’ along line of action AB according to the law of superposition as shown in Fig (b). Figure (a) Figure (b) Two equal and opposite forces of the magnitude ‘P’ at point A and B can be subtracted without changing action of original force P according to the law of superposition as shown in Fig (c). Figure (c) Thus the point of application of force P is transmitted along its line of action from A to B. CHAPTER NO. 1 Resolution of Coplanar Forces Page 7
  • 8. Varignon’s Theorem of Moments/ Principle of Moments: Statement: “The algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant about the same point”. i.e. ΣM = Σ (Moments of forces) = Moment of R Proof: In above Figure AB and AC represents forces P and Q resp. and ‘O’ is the point about which moment is taken. ABCD represents a parallelogram. A diagonal AD represents resultant of forces P and Q. Now extend CD up to the point ‘O’ which is the line of CD. Join OA and OB. Now, we know that, Moment of force = 2(Area of triangle) Moment of force P = 2 x Area of Triangle AOB And Moment of force Q = 2 x Area of Triangle AOC Algebraic sum of Moments of forces P and Q = Σ M = 2 x (Area of ΔADB + Area of ΔAOC) Now, Area of Δ AOB = Area of ΔADB = Area of ΔACD Since, AB = CD (base is same) and height is same Σ M = 2 x (Area of Δ ACD + Area of Δ AOD) = 2 x (Area of Δ AOD) Σ M = Moment of Resultant Force ‘R’ CHAPTER NO. 1 Resolution of Coplanar Forces Page 8
  • 9. Application: 1) It is generally used to locate the point of application of resultant. 2) In case of coplanar non-concurrent system of forces this concept is used to locate the line of action of the resultant. Parallelogram law of Forces Statement: Statement: “If two forces acting simultaneously on a body at a point are presented in magnitude and direction by the two adjacent sides of parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes through the point of intersection of the two sides representing the forces”. Fig. (a) Fig. (b) The length of diagonal in Fig. (b) will indicate the magnitude of resultant of ‘R’. Derivation: From right angle triangle BCD BD = Q sinθ CD = Q cosθ Using Pythagorus theorem to the ΔOCD OC2 = CD2 + OD2 CHAPTER NO. 1 Resolution of Coplanar Forces Page 9
  • 10. OC2 = CD2 + (OB + BD) 2 R2 = (P + Q cosθ)2 + (Q sinθ)2 R2 = P2 + Q2 cos2θ + 2PQ cosθ + Q2 sin2θ R2 = P2 + Q2 + 2PQ cosθ R = √P2 + Q2 + 2PQ cosθ ---------------------------------------(1) Angle α of resultant R with force P is given by, α = tan-1[ Q sinθ 푃+ Q cosθ ] ---------------------------------------(2) Particular cases: 1) When θ = 900 R= √푃2 + 푄2 2) When θ = 00 R= P + Q (acting along Same Direction) 3) When θ = 1800 R= P – Q (acting in Opposite Direction) Moment: The turning effect caused by a force on the body is called as a moment of force. Definition: The moment of a force (M) is equal to the magnitude of the force (F) multiplied by the perpendicular distance (d) between the line of action of the force and the axis of rotation. Moment = Force x Perpendicular Distance M = F x d Sign convention: If the moment of the force producing clockwise rotation is the clockwise moment and it is taken as positive as shown in Fig. (a). If the moment of the force producing anticlockwise rotation is the anticlockwise moment and it is taken as negative as shown in Fig. (b). Figure (a) Figure (b) CHAPTER NO. 1 Resolution of Coplanar Forces Page 10
  • 11. Unit: If the force is measured in Newton and the distance in meter, the SI unit of the moment is Newton meter (Nm). Geometrical Representation of Moment: As shown in Fig. below, AB represents force F and O is the point about which the moment of force M is taken. Let OC be the perpendicular distance‘d’. Moment of Force F is given by, M = F x d M = AB x OC M = 2 x (½ AB x OC) M = 2 (Area of triangle OAB) Thus Moment of Force about any point is geometrically equal to twice the area of the triangle having base representing a point about which moment is taken. Couple: Two equal, opposite and parallel (non-collinear) forces are said to form a couple as shown in Fig. below. Figure (a) Arm of couple: The distance ‘a’ between the lines of action of the two forces of a couple is known as ‘arm of couple’. Properties: a) Couple cannot be replaced by a single resultant force. b) Couple cannot produce rotation or moment but it cannot produce straight line motion. CHAPTER NO. 1 Resolution of Coplanar Forces Page 11
  • 12. Moment of Couple: From above Fig. moment of couple about any point ‘O’ (moment of centre) is given by F (a +d) – (Fd) = Fa + Fd – Fd = Fa Nm Moment of Couple = Force x Arm Thus the moment of the couple has a constant value irrespective of the point about which moment is taken. Bow’s notation: Any force F divides the space into two parts A and B as shown in Fig. below. This force is named as force AB according to this method. Space Diagram Force Diagram In this method line ‘ab’ is drawn parallel to force F such that the length of line ab represents the magnitude and the direction is from ‘a’ to ‘b’ which indicates the direction of the force ‘F’ as shown in Fig. below. Suitability: This notation is useful for solving the problems in statics by graphical method. Composition of Forces: (Resultant of coplanar concurrent forces) The process of determining the Resultant of number of forces acting simultaneously on a body is known as Composition of Forces. It is the method of reducing the given force system to its equivalent simplest system of single force (or couple). CHAPTER NO. 1 Resolution of Coplanar Forces Page 12
  • 13. Combining the forces of any given system is termed as composition of forces. There are two main methods of determining the resultant force: a) Analytical Methods and b) Graphical Methods Analytical Methods: There are two Analytical Methods: 1) Parallelogram law of Forces (as explained above) 2) Component Law or Resolution Methods 1. Component Law of Forces: 1. Forces such as F1, F2, F3 and F4 acting at point ‘O’ as shown in Fig. above. are resolved along x-axis (horizontally). The algebraic sum of horizontal components is ΣH or ΣFX. Figure (a) 2. Similarly, the forces are resolved along y-axis (vertically). The algebraic sum of vertical components is ΣV or ΣFy. CHAPTER NO. 1 Resolution of Coplanar Forces Page 13
  • 14. 3. Resultant ‘R’ is given by, R = √ΣF푋2 2 + ΣF푌 4. Angle of inclination θ with x-axis is given by, θ = tan-1[ΣFy/ ΣFX] Particular cases: 1) When θ = 900 R= √푃2 + 푄2 2) When θ = 00 R= P + Q (acting along Same Direction) 3) When θ = 1800 R= P – Q (acting in Opposite Direction) Sign convention: While taking ΣFX, forces acting from left to right are taken as positive and those are acting from right to left are considered negative. While taking ΣFy forces acting upwards are assumed positive and those acting downwards are assumed negative. Resolution of Forces: The process of splitting or subdividing a force into its components without changing its effect on the body is known as Resolution of Forces. It is the replacement of a single force by several components having the same effect as that of single force. Generally, a force ‘F’ is resolved into two components Fx and Fy which are mutually perpendicular to each other as shown in Figure below. Horizontal component Fx = F cos θ Vertical component FY = F sin θ Consider a rigid body as shown in Fig. above. Let F1, F2 and F3 be three forces acting on a rigid body. Let ‘R’ be there resultant. Then we can say that F1, F2 and F3 are resolved parts of R or components of ‘R’ in three different directions. Generally, a force is replaced in to two rectangular components. Graphical Methods: 1) Law of triangle of Forces: CHAPTER NO. 1 Resolution of Coplanar Forces Page 14
  • 15. If two forces P and Q (acting at point ‘O’) as shown in Figure below in which they represents the magnitude and direction of the two sides of the triangle taken in order, then the third side taken in opposite sense represents the resultant ‘R’ of the two forces in magnitude and direction. 2) Law of Polygon of Forces: If numbers of forces are acting on a body, are represented in magnitude and direction by the sides of the polygon taken in order, then the closing side taken in opposite sense represents the resultant of all the forces in magnitude and direction. Fig. a Fig. b Fig. c Above Figure (a) shows the system of four forces in magnitude and direction and Figure (b) shows the polygon of same forces. The closing side ‘R’ represents the resultant. We can use the triangle law of forces in this polygon, such that, the resultant of forces F1 and F2 is R as shown in Figure c. Similarly, the resultant of R1 and F3 is R2 and finally, the resultant of R2 and F4 is R by the triangle law of forces. Conclusion: The polygon law of forces is the application of triangle law of forces. Idealizations in mechanics: 1) The body is rigid. 2) The body is treated as continuum. Continuum: when the body is assumed to consist of a continuous distribution of matter is called as continuum. 3) If the size of the body is small as compared to other distances involved in the problem, it may be treated as a particle. 4) If the area over which force is acting on a body is small as compared to the size of the body, it may be treated as a point force. 5) Support conditions are idealized as simple, hinged, fixed etc. CHAPTER NO. 1 Resolution of Coplanar Forces Page 15