Engineering Mechanics

Nilesh G.

Presented by Nilesh Gaddapawar

Classification of Mechanics

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Mechanics of Deformable Bodies:

The mechanics of deformable bodies deals with how forces are distributed inside bodies, and with the deformations caused by these internal force distributions.  These internal force produce "stresses" in the body, which could ultimately result in the failure of the material itself.

Principles of rigid body mechanics often provide the beginning steps in analyzing these internal stresses, and resulting deformations.  These will be studied in courses called Strength of Materials or Mechanics of Materials.

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Mechanics of Fluids: The mechanics of fluids is the branch of mechanics

that deals with liquids or gases.   Fluids are commonly used in engineering

applications.  They can be classified as incompressible, or compressible.  While all real fluids are compressible to some degree, most liquids can be analyzed as incompressible in many engineering applications.  

Applications of fluid mechanics abound, from hydraulics and general flow in pipes to air flow in ducts to advanced applications in turbines and aerospace.  

The study of the mechanics of fluids will be studied in courses called Fluid Mechanics, Compressible Flow, Hydraulics, and others.

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Mechanics of Rigid Bodies:A rigid body is a body which does

not deform under the influence of forces.  

In all real applications, there is always deformation, however, many stuctures exhibit very small deformations under normal loading conditions, and rigid body mechanics can be used with sufficient accuracy in those cases.  

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Fundamentals Length:  Length is the quantity used to describe the position of a point in space

relative to another point. The universally accepted standard unit for length is the meter.

Time:  Time is the interval between two events. The generally accepted standard unit for time is the second.

Space: The geometric region in which the study of body is involved is called Space.

Displacement: It is defined as the distance moved by a body/particle in the specified direction.  

Velocity: It is the rate of change of displacement with respect to time. Acceleration : It is the rate of change of velocity with respect to time. Momentum: The product of mass and velocity is called momentum. Continuum: A body consists of several matters. Particle: A particle may be defined as an object which has only mass and no

size. Such a body cannot exist theoretically. However in dealing with problems involving distances considerably larger compared to the size of the body, the body may be treated as particle. Examples of such situations are

A bomber aeroplane is a particle for a gunner operating from the ground. A ship in mid sea is a particle in the study of its relative motion from a control

tower.

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Vectors and Scalars quantitiesAll physical quantities (e.g. speed and force) are described by a magnitude and a unit.

VECTORS – also need to have their direction specified examples: displacement, velocity, acceleration, force.

SCALARS – do not have a directionexamples: distance, speed, mass, work, energy.

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Addition of vectorsWith two vectors acting at an angle to each other:Draw the first vector. Draw the second vector with its tail end on the arrow of the first vector. The resultant vector is the line drawn from the tail of the first vector to the arrow end of the second vector.This method also works with three or more vectors.

4N

3N

Resultant vector

= 5N

4N

3N

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Comparison of Mass and WeightSr. no.

Mass Weight

01 Mass is a property of matter. The mass of an object is the same everywhere.

Weight depends on the effect of gravity. Weight varies according to location.

02 Mass can never be zero. Weight can be zero if no gravity acts upon an object, as in space.

03 Mass does not change according to location.

Weight increases or decreases with higher or lower gravity.

04 Mass is a scalar quantity. It has magnitude.

Weight is a vector quantity. It has magnitude and is directed toward the center of the Earth or other gravity well.

05 Mass may be measured using an ordinary balance.

Weight is measured using a spring balance.

06 Mass usually is measured in grams and kilograms.

Weight often is measured in Newton, a unit of force.

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Comparison of Distance and Displacement:

Sr. no.

Distance Displacement

01 Distance is the length of the path travelled by a body while moving from an initial position to a final position.

Displacement is the shortest distance between the initial position and the final position of the body.

02 Distance is a scalar quantity. Displacement is a vector quantity.

03 Distance measured is always positive.

Displacement can be positive or negative depending on the reference point.

04 The total distance covered is equal to the algebraic sum of all the distances travelled in different directions.

The net displacement is the vector sum of the individual displacements in different directions.

05 There is always a distance covered whenever there is a motion.

Displacement will be zero if the body comes back to its initial position.

06 Unit: metre (m) Unit: metre (m)Nilesh G.

For example: Q. Suppose you are observing an ant on the table, as shown in the diagram below. The ant moves from one corner of the table to the other corner. The blue irregular line shows the path of the ant.

For figure (B) & (C), Find Distance ?Displacement?

Answer :•For Length of this blue line is the distance covered by the ant.•The straight green line, which is the minimum distance between the two corners of the table is the displacement of the ant. Called the displacement.

Figure (B)

Figure (A)Figure (C)

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Comparison of Speed and Velocity

Sr. no Speed velocity01 Speed is refers to "how fast an

object is moving."Velocity refers to "the rate at which an object changes its position."

02 Speed is a scalar quantity. Velocity is a vector quantity.

03 Speed is the rate of motion, or the rate of change of position.

velocity is the rate of change of displacement.

04 Speed is thus the magnitude component of velocity.

Velocity contains both the magnitude and direction components.

05 Explanation :How fast my hand is moving to slapped on your face, this is speed

Explanation :When you get the slap and changes your face from right to left.. i.e the rate at which your face changes its position, this is Velocity....

06 speed= total distance/time taken

velocity= displacement(shortest root from initial to the final position) /time taken including direction.

07 Unit: km/hr like 60km/hr, Unit: 60km/hr in east direction.Nilesh G.

Units of Measure: The force unit is called a newton (N), and is defined

as the force required to accelerate a mass of 1 kg at a rate of 1 meter/sec.  So, we can write:

1 N = (1 kg)(1 m/s2)  or 1 N = 1kg.m/s2         The weight of an object is the gravitational force

which is exerted on that object which causes it to accelerate downward at the acceleration due to gravity So, we can write for the weight of a 1 kg mass:

W = mgW = (1 kg)(9.807 m/s2)W = 9.807 N

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SI prefixes: Multiplier

Prefix

Symbol

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 pico p

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FUNDAMENTAL LAWS OF MECHANICS

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Sir Isaac Newton

Sir Isaac Newton 25 December 1642 – 20 March 1727 was an English physicist and mathematician (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and as a key figure in the scientific revolution.

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1. Newton's first law “An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This law is often called the law of inertia.”

Objects tend to "keep on doing what they're doing." In fact, it is the natural tendency of objects to resist changes in their state of motion. This tendency to resist changes in their state of motion is described as inertia.

Example :The motion of a kite when the wind changes can also be described by the first law.

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Explanation to first law :

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“ The rate of change of momentum is directly proportional to impressed force and it takes place in the direction of force acting on it”.

Force ∝ rate of change of momentum. But momentum = mass × velocityAs mass do not change,

Force ∝ mass × rate of change of velocity

i.e., Force ∝ mass × acceleration

F ∝ m × a

2. Newton's second law

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“For every action there is an equal and opposite reaction”.

3. Newton's third law

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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”.

According to this law the force of attraction between the bodies of mass m1 and mass m2 at a distance d as shown in Fig.

where G is the constant of proportionality and is known as constant of gravitation.

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5. Principle or Law of transmissibility of forces:

“The state of rest or of Uniform motion of a rigid body is unaltered if the point of application of the force is transmitted to any other point along the line of action of the force”.

Let F be the force acting on a rigid body at point A. According to the law of transmissibility of force, this force has the same effect on the state of body as the force F appliedat point B.

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6. Law of superposition. The law of transmissibility of forces can be proved using

the law of superposition, which can be stated as “the action of a given system of forces on a rigid body is not changed by adding or subtracting another system of forces in equilibrium.”

Consider the rigid body shown in Fig. (a). It is subjected to a force F at A. B is another point on the line of action of the force. From the law of superposition it is obvious that if two equal and opposite forces of magnitude F are applied at B along the line of action of given force F, [Ref. Fig. (b)] the effect of given force on the body is not altered. Force F at A and opposite force F at B form a system of forces in equilibrium. If these two forces are subtracted from the system, the resulting system is as shown in Fig. 1.7 (c).Nilesh G.

7.Parallelogram Law of Forces This law states that “ if two forces acting simultaneously on a body at a

point are presented in magnitude and direction by the two adjacent sides of a 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”.

In Fig. the force F1 = 4 units and force F2 = 3 units are acting on a body at point A. Then to get resultant of these forces parallelogram ABCD is constructed such that AB is equal to 4 units to linear scale and AC is equal to 3 units. Then according to this law, the diagonal AD represents the resultant in the direction and magnitude. Thus the resultant of the forces F1and F2 on the body is equal to units corresponding to AD in the direction α to F1.Nilesh G.

DERIVED LAWS The Triangle Law of Forces may be stated as “If

two forces acting on a body are represented one after another by the sides of a triangle, their resultant is represented by the closing side of the triangle taken from first point to the last point”.

The polygon of law of forces and it may be stated as “If a number of concurrent forces acting simultaneously on a body are represented in magnitude and direction by the sides of a polygon, taken in a order, then the resultant is represented in magnitude and direction by the closing side of the polygon, taken from first point to last point”.

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Classifiaction of Mechanics of rigid bodies

Mechanics of rigid bodies

Statics Dynamics

Kinematics

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Static : It is the branch of mechanics which deals with the particles and bodies at rest.

Dynamics : It is the branch of mechanics which deals with the particles and bodies are in motion.

Kinematics : It is the study of motion of particles and bodies without reference to masses and effects of forces.

Kinetics: It is the study of motion of particles and bodies with reference to masses and effects of forces.

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What is a force According to Newton‟s I

law, Force is defined as an action or agent, which changes or tends to change the state of rest or of uniform motion of a body in a straight line.

Units of force: The gravitational (MKS) unit of force is the kilogram force and is denoted as „kgf‟.

The absolute (SI) unit of force is the Newton and is denoted as “N”. Note: 1 kgf = “g”N (But g = 9.81m/s2) Therefore 1 kgf = 9.81 N or 10 N.

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Characteristics of a force

It has four important characteristics, which can be listed as follows.

1) Magnitude: It can be denoted as 10 kgf or 100 N.

2) Point of application: It indicates the point on the body on which the force acts.

3) Line of action: The arrowhead placed on the line representing the direction represents it.

4) Direction: It is represented by a co-ordinate or cardinal system.

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Explanation to a force In figure, AB is a ladder kept against a wall.

At point C, a person weighing 600 N is standing. The force applied by the person on the ladder has the following characteristics.

Magnitude os 600 N Direction is vertically downwards. The line of action is vertical CD or C to D The point of application is at C.

The magnitude of the force is written near the arrow.

The line of arrow shows the line of action of the force and the arrow head shows the direction of the force.

The point of intersection of the force with the body shows the point of application of the force on the body.

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CLASSIFICATION OF FORCE

Force system

Coplanar force system

Concurrent/ non

concurrent

Co-linear / non co-

linear

parallel./ non

parallel

Non coplanar force system

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Figures to explain:

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Resolution of FORCEIt is often convenient to split a single FORCE into two perpendicular components.

Consider force F being split into vertical and horizontal components, FV and FH.

In rectangle ABCD opposite:

sin θ = BC / DB = DA / DB = FV / F

Therefore: FV = F sin θ

cos θ = DC / DB = FH / F

Therefore: FH = F cos θ

FFV

FHθ

C

BA

D

FV = F sin θ

FH = F cos θ

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QuestionCalculate the vertical and horizontal components if F = 4N and θ = 35o.

FV = F sin θ= 4 x sin 35o

= 4 x 0.5736FV = 2.29 N

FH = F cos θ= 4 x cos 35o

= 4 x 0.8192FH = 3.28 N

FFV

FHθ

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Composition of forcesThe reduction of a given system of forces to the simplest system that will be its equivalent is called composition of forces.

RESULTANT FORCE: It is possible to find a single force which will have the same effect as that of a number of forces acting on a body. Such a single force is called resultant force.

The process of finding out the resultant force is called composition offorces.

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Composition of Co-planar concurrent force system.

COMPOSITION OF TWO FORCES: It is possible to reduce a given system of forces i.e., two forces to the simplest system as its equivalent (resultant force) with the help of parallelogram law of forces.

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COMPOSITON OF FORCES BYRESOLUTION(Principle of resolved parts)

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Procedure to find magnitude and direction of resultant force : - Algebraic sum of the components of forces in X direction

Algebraic sum of the components of forces in Y direction

Now the system of forces is equal to two mutually perpendicular forces , the magnitude and direction of resultant force.

Magnitude: Direction:

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The moment of a forceAlso known as the turning effect of a force.

The moment of a force about any point is defined as: force x perpendicular distance from the turning point to the line of action of the force

moment = F x d

Unit: Newton-metre (Nm)

Moments can be either CLOCKWISE or ANTICLOCKWISE

Force F exerting an ANTICLOCKWISE moment through the spanner on the nut

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QuestionCalculate the moments of the 25N and 40N forces on the door in the diagram opposite.

moment = F x d

For the 25N force:

moment = 25N x 1.2m

= 30 Nm CLOCKWISE

For the 40N force:

moment = 40N x 0.70m

= 28 Nm ANTICLOCKWISE

hinge

door

40N

25N

0.70 m

1.2 m

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Couples and Torque“Two equal unlike parallel forces separated by a definite distance form a couple”.

In the diagram above:couple = F d

= One of the forces x the distance between the forces

Torque is another name for the total moment of a couple.

d

F

F

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Chracteristics of a Couple

1. The algebraic sum of magnitude of the member forces forming a couple is always zero.

2. The couple can be balanced by a couple only, which has same moment and opposite sense.

3. Moment of a couple is fixed and it does not depend on moment center

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Comparison of Torque and MomentSr.No.

Torque Moment

01 Torque is a movement force.

Moment is a static force.

02 Torque is often presented as Nm/revolution 

moment is typically presented as Nm.

03 tends to be used when there is an axle or pivot to be turned around

tends to be used in essentially non-rotational situations, such as analysis of forces on a beam.

04 Twisting effect produced by force

Turning effect produced by force

05

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Varignon’s theorem (Principle of moments)

Statement : “The sum of moment of all forces about a point is equal to the moment of their resultant about the same point”.

Proof: consider a force F acting at a point A and having components F1 and F2 in any two directions. Let us choose any point O, lying in the plane of the forces, as a moment center. Draw the perpendiculars d , d1 and d2 from moment center o to the forces F, F1 and F2 respectively. Let α,α1 and α3 be the angle made by the forces F, F1 and F2 respectively

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Moment of force F about O,

F × d = F × OA× cosα =OA × F × cosα = OA × Fx -----------------(1)

Moment of force F1 about O,

F1 × d1 = F1 × OA × cosα1 =OA × F1 × cosα1 = OA × Fx1 --------(2)

Moment of force F1 about O,

F2 × d2 = F2 × OA × cosα2 =OA × F2 × cosα2 = OA × Fx2 --------(3)

Adding (2) & (3),

F1 × d1 + F2 × d2 = OA ( Fx1 + Fx2 ) ----------------------------------(4)

Let ,

x-components of the resultant force F = sum of the x-components of the forces F1 and F2

Fx = Fx1 + Fx2 -----------------------------------------------------------(5)

From (4) & (5),

F1 × d1 + F2 × d2 = OA × Fx -------------------------------------------(6)

From (1) & (6),

F1 × d1 + F2 × d2 = F × d

i.e. ∑ MF@o = MR@o

Application : It is used for finding the point of application of the resultant in case of non concurrent and parallel force system.Nilesh G.