ENGINEERING MECHANICS UNIT I STATICS OF PARTICLES 5/16/2019 DR.R.GANESAMOORTHY Prof.MECHANICAL 1 Dr.R.Ganesamoorthy Professor Mechanical Engg. Saveetha Engineering College

Introduction 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL2 Engineering mechanics is the branch of science which deals with the behavior of a body with the state of rest or motion under the action of forces. The Classification of Engineering Mechanics

Introduction  Statics is the study which deals with the condition of bodies in equilibrium subjected to external forces.  Dynamics is also a branch of mechanics in which the forces and their effects on the bodies in motion are studied. Dynamics is sub-divided into two parts: (1) Kinematics and (2) Kinetics  Kinematics deals with the geometry of motion of bodies without and application of external forces.  Kinetics deals with the motion of bodies with the application of external forces. 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL3

UNITS AND DIMENSIONS OF QUANTITIES Units: Measurements are always made in comparison with certain standards. Example, ‘metre’ is the unit of length. There are four systems of units used for the measurement of physical quantities. viz. 1.FPS (Foot – Pound – Second) system, 2.CGS (Centimetre – Gram – Second) system, 3.MKS (Meter-Kilogram–Second) system 4.SI (System international units– the French name) The SI system of units is said to be an absolute system. 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL4

Units S.I Units (International System of Units) The fundamental units of the system are Metre (m) for length, Kilogram (kg) for mass and Second (s) for time. The unit for force is Newton (N). One Newton is the amount of force required to induce an acceleration of 1 m/sec2 on one kg mass. Weight of a body (in N) = Mass of the body (in kg) × Acceleration due to gravity (in /sec2). 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL5

Dimensions The branch of mathematics dealing with dimensions of quantities is called dimensional analysis. There are two systems of dimensional analysis viz. absolute system and gravitational system. Absolute system (MLT system) A system of units defined on the basis of length, time and mass is referred to as an absolute system. Gravitational system (FLT system) A system of units defined on the basis of length, time and force is referred to as a gravitational system. FLT system refers to the Force-Length-Time system. 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL6

Dimensions of quantities in MLT and FLT systems 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL7 QuantityMLT-SystemFLT-system LengthLL MassMFL -1 T 2 AreaL2L2 L2L2 VolumeL3L3 L3L3 VelocityLT -1 AccelerationLT -2 MomentumMLT –1 FT StressML -1 T -2 FL -2 WeightMLT -2 F ForceMLT -2 F PowerML 2 T -3 FLT -1 DensityML -3 FL -4 T 3

Scalar and Vector Quantity Various quantities used in engineering mechanics may be grouped into scalars and vectors.  Scalar Quantity: A quantity is said to be scalar if it is completely defined by its magnitude alone. Examples of scalar quantities are: Area, length, Mass, Moment of inertia, Energy, Power, Volume And Work etc.  Vector Quantity: A quantity is said to be vector if it is completely defined only when its magnitude as well as direction are specified. Examples of vector quantities include: Force, Moment, Momentum, Displacement, Velocity and Acceleration. 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL8

Laws of Mechanics The following are the fundamental laws of mechanics: (i) Newton’s first law (ii) Newton’s second law (iii) Newton’s third law (iv) Newton’s law of gravitation (v) Law of transmissibility of forces (vi) Parallelogram law of forces (vii) Lami’s theorem (viii) Triangular law of forces 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL9

Laws of Mechanics (i) Newton’s first law: It states that everybody continues in its state of rest or of uniform motion in a straight line unless it is compelled by external agency acting on it. (ii) 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. According to this law, Force = rate of change of momentum. But momentum = mass × velocity As mass do not change, Force = mass × rate of change of velocity i.e. Force = mass × acceleration F = m × a 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL10

Laws of Mechanics (iii) Newton’s third law: It states that for every action there is an equal and opposite reaction. (iv) Newton’s law of gravitation: it’s states that every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. particleforcedirectly proportionalinversely proportional The equation for universal gravitation thus takes the form: Where: F - The force between the masses; G - The gravitational constant gravitational constant m 1 - The first mass; m 2 - The second mass; r - The distance between the centres of the masses. 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL11

Law of Transmissibility of Forces The Law of the Transmissibility of the Forces is states that When change the point of application of the force acting on a body to any other point which lies on the line of action of the force without any change in the direction of the force then there is no change in the equilibrium state of the body. 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL12

Parallelogram law of forces Its states that “if two forces acting simultaneously at a point be represented in magnitude and direction by the two adjacent sides of a parallelogram,then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point ” 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL13 The direction of the resultant is

Lami’s Theorem  Its states that “ if three coplanar forces acting at a point be in equilibrium, then each force is proportional to the sine of the angle between the other two forces” 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL14

Triangular law of forces  Its states that “ if two forces acting simultaneously at a point be represented in magnitude and direction by two sides of triangle, taken in order, then their resultant is represented in magnitude and direction by the third side of the triangle, taken in opposite order ”. 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL15

VECTORS 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL16 Vectors are defined as mathematical expressions possessing magnitude and direction Representation of a vector A vector is represented by a direction line as shown.it may be noted that the length OA represents the magnitude of the vector The direction of the vector is is from O (i.e., starting point) to A (i.e., end point). It is also known as vector P.

VECTORS 2. Unit vector. A vector, whose magnitude is unity,is known as unit vector. 3. Equal vectors. The vectors, which are parallel to each other and have same direction (i.e., same sense) and equal magnitude are known as equal vectors. 4. Like vectors. The vectors, which are parallel to each other and have same sense but unequal magnitude, are known as like vectors. 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL17

Vector Addition 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL18

Vector subtraction 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL19

Unit vector and their representation in three coordinate axes 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL20

Position vector of a given point 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL21

Vectorial representation of forces 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL22

Vectorial representation of forces 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL23

Vectorial representation of moments 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL24 A force can translate and rotate a body about an axis. This rotational tendency of the force about an axis is called moment and is denoted by M. the moment is the cross product of the position vector and the force vector M = r x F

Vectorial representation of moments 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL25

Vectorial representation of moments 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL26

Vectorial product 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL27

Vectorial product 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL28

SYSTEM OF FORCES 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL29 When two or more forces act on a body, they are called to form a system of forces. Following systems of forces are important from the subject point of view :

SYSTEM OF FORCES 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL30 1.Coplanar parallel forces. The forces, whose lines of action lie on the same plane, are known as coplanar parallel forces. Ex. The system of forces acting on a beam subjected to vertical loads. 2. Collinear forces. The forces, whose lines of action lie on the same line, are known as collinear forces. Ex. Forces on a rope in a tug of war. 3. Concurrent forces. The forces, which meet at one point, are known as concurrent forces. The concurrent forces may or may not be collinear.

SYSTEM OF FORCES 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL31 4. Coplanar concurrent forces. The forces, which meet at one point and their lines of action also lie on the same plane, are known as coplanar concurrent forces. Ex. Forces on a rod resting against a wall. 5. Coplanar non-concurrent forces. The forces, which do not meet at one point, but their lines of action lie on the same plane, are known as coplanar non-concurrent forces. Ex. Forces on a ladder resting against a wall when a person stands on a rung which is not at its centre of gravity. 6. Non-coplanar concurrent forces. The forces, which meet at one point, but their lines of action do not lie on the same plane, are known as non- coplanar concurrent forces. Ex. A tripod is carrying a camera.

SYSTEM OF FORCES 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL32 7. Non-coplanar non-concurrent forces. The forces, which do not meet at one point and their lines of action do not lie on the same plane, are called non-coplanar non-concurrent forces. Ex. forces are acting on a moving bus 8. Non-coplanar parallel forces: All the forces are parallel to each other, but not in the same plane. Ex. The weight of benches in a classroom

Resultant force 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL33

Resultant force 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL34

Resultant force 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL35

Resultant force 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL36

Resultant force more than two concurrent force 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL37

Resultant force more than two concurrent force 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL38

Resultant force more than two concurrent force 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL39

Resultant force more than two concurrent force 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL40

Equilibrium of a particles 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL41 Conditions of equilibrium

Principles of Equilibrium 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL42

Proof Principles of Equilibrium – lami’s theorem 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL43

Proof Principles of Equilibrium – lami’s theorem 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL44

Proof Principles of Equilibrium – lami’s theorem 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL45

Forces in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL46 Effect of system of forces in space i.e., three dimensional plane The position of particle in space is defined by X,Y,Z direction measured from the origin

Forces in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL47

Forces in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL48

Forces in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL49

Forces in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL50

Forces in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL51

Forces in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL52

Forces in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL53

Equilibrium of Particle in Space 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL54

Principle of Transmissibility 5/16/2019DR.R.GANESAMOORTHY Prof.MECHANICAL55