GATE Engineering Mechanics Book

8/10/2019 GATE Engineering Mechanics Book

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ENGINEERING MECHANICS

for

Mechanical Engineering

By

www.thegateacademy.com
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Syllabus Engineering Mechanics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th

Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d

Syllabus for Engineering Mechanics

Equivalent force systems, free-body concepts, equations of equilibrium, trusses and frames,

virtual work and minimum potential energy. Kinematics and dynamics of particles and rigid

bodies, impulse and momentum (linear and angular), energy methods, central force motion.

Analysis of GATE Papers

(Engineering Mechanics)

Year Percentage of marks Overall Percentage

2013 3.00

3.18

2012 1.00

2011 5.00

2010 2.00

2009 3.00

2008 3.33

2007 2.00

2006 2.67

2005 6.67


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Contents Engineering Mechanics


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d

C O N T E N T S

Chapter Page No.

1. Introduction 1

2. Statics 2 - 65

Equivalent Force System 2 3

Equilibrium and Free Body Diagram 3 9

Friction 9

Virtual Work 9

10Trusses and Frames 10 12

Solved Examples 13 47

Assignment 1 48 52

Assignment 2 52 55

Answer Keys 56

Explanations 56 65

3. Dynamics 66 133

Kinematics of Rectilinear Motion 66 73

Kinematics of Curvilinear Motion 73

76Acceleration Analysis in Cylindrical Coordinates 76 83

Kinetics of Rectilinear Motion 83 90

Moment of Inertia 90 98

Friction 98 100

Impulse and Momentum 101 104

Collision of Elastic Bodies 104 107

Conversion of Angular Momentum 108 111

Conservative / Non conservative Force 111 117

Assignment 1 118 121

Assignment 2 121 124

Answer Keys 125

Explanations 125 133

Module Test 134 145

Test Questions 134 139

Answer Keys 140

Explanations 140 145

Reference Books 146


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Chapter 1 Engineering Mechanics


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d Page 1

CHAPTER

IntroductionMechanics is the science which deals with the action of forces on different types of bodies either

in motion or at rest.

Engineering mechanics is the application of mechanics to solve problems involving common

engineering elements.

Engineering Mechanics can be broadly classified as:

In this course material we will study about the mechanics of particles and rigid bodies.

Particle: It is a portion of matter which is indefinitely small in size.

Rigid body: A rigid body may be defined as a body in which the relative positions of any two

particles do not change under the action of forces. Statics deals primarily with the calculation of

external forces which act on rigid body in equilibrium.

Engineering Mechanics

Mechanics of fluidsMechanics of solids

Rigid bodies Deformable bodies

Statics Dynamics

Kinematics Kinetics

Strength of materials,

Theory of elasticity,

Theory of plasticity


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Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P 2

CHAPTER 2

STATICS

Introduction

Statics deals with system of forces that keeps a body in equilibrium. In other words the resultantof force systems on the body are zero.

Force:A force is completely defined only when the following three characters are specified.

Magnitude Point of application

Line of action/Direction

Scalar and Vector:

A quantity is said to be scalar if it is completely defined by its magnitudealone. e.g. length, energy, work etc. A quantity is said to be vector if it is completely defined only

when its magnitude and direction is specified. e.g. force, acceleration.

Equivalent Force System

Coplanar force system: If all the forces in the system lie in a single plane, it is called coplanarforce system.

Concurrent force system:

If line of action of all the forces in a system passes through a singlepoint it is called concurrent force system.

Collinear force system:

In a system, all the forces parallel to each other, if line of action of all

forces lie along a single line then it is called a collinear force system.

Force system Example

Coplanar like parallel force is straight. Weight of stationary train on rail off the track

Coplanar concurrent Forces on a rod resting against wall.

Coplanar non- concurrent force Forces on a ladder resting against a wall when

a person stands on a rung which is not at its

center of gravity.

Non- coplanar parallel The weight of benches in class room

Non- coplanar concurrent force A tripod carrying camera

Non- coplanar non-concurrent force Forces acting on moving bus


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Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


Newtons law of motion

First Law: Everybody continues in its state of rest or of uniform motion in a straight line unless itis compelled to change that state by force acting on it.

Second law:

The rate of change of momentum of a body is directly proportional to the applied

force & it takes place in the direction in which the force acts.

F(m dvdt)

Third law:

For every action, there is an equal and opposite reaction.

Principle of transmissibility of force: The state of rest or motion of rigid body is unaltered if a

force action on a body is replaced by another force of the same magnitude and direction butacting anywhere on the body along the line of action of applied force.

Parallelogram law of forces:

If two forces acting simultaneously on a body at a point are

represented 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.

Equilibrium and Free Body Diagrams

Coplanar Concurrent Forces

Triangle law of forces:If two forces acting simultaneously on a body are represented by the sides

of triangle taken in order, their resultant is represented by the closing side of the triangle takenin the opposite order.

Polygon law of forces

P

1

P2P3

P4P

2

P

4

C

P

1

P

3

E

R

B

A

R2

R1

D

A

B

P

P


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Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


If a number of forces acting at a point be represented in magnitude and direction by the sides of

a polygon in order, then the resultant of all these forces may be represented in magnitude and

direction by the closing side of the polygon taken in opposite order

Resultant (R) =

t an( )

= angle between two forces, = inclination of resultant with force P1When forces acting on a body are collinear, their resultant is equal to the algebraic sum of the

forces.

amis theorem: (only three coplanar concurrent forces) If a body is in equilibrium under the

action of three forces, then each force is proportional to the sine of the angle between the othertwo forces.

Psin

Psin

Psin

Free body diagram:A free body diagram is a pictorial representation used to analyze the forcesacting on afree body.Once we decide which body or combination of bodies to analyze, we thentreat this body or combination as a single body isolated from all our surrounding bodies

A free body diagram shows allcontact andnon-contactforces acting on the body.

P1P2

P3

P2

P1

P3

a

b

c

P2

BC

DE

P1

A
http://en.wikipedia.org/wiki/Free_bodyhttp://en.wikipedia.org/wiki/Contact_forcehttp://en.wikipedia.org/wiki/Non-contact_forcehttp://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Non-contact_forcehttp://en.wikipedia.org/wiki/Contact_forcehttp://en.wikipedia.org/wiki/Free_body


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Block, Bangalore-11


Sample Free body diagrams

A ladder resting on smooth wall

A cantilever beam

A block on a ramp

In a free body diagram all the contacts/supports are replaced by reaction forces which will exert

on the structure. A mechanical system comprises of different types of contacts/supports.

Types of contacts/supports

Following types of mechanical contacts can be found in various structures:

Flexible cable belt chain or rope

mgm

Free body diagram

of just the block

F3F2 F1

VV

V V

F

M=m

y

x

SMOOTH

S

P

600N

G

W600N

R1

P

R2


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Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


Force exerted by the cable is always a tension away from the body in the direction of the cable.

Smooth surfaces

Contact force is compressive and is normal to the surface.

Rough surfaces

Rough surfaces are capable of supporting a tangential component F (frictional force as well as anormal component N of the resultant R.

Roller support

Roller, rocker or ball support transmits a compressive force normal to supporting surface.

Freely sliding guide

N N

N

N

N

N

Weight of cable negligible

Weight of cable not negligible T

T


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Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11


Collar or slider support force normal to guide only. There is no tangential force as surfaces are

considered to be smooth.

Pin connection

A freely hinged pin supports a force in any direction in the plane normal to the axis; usually

shown as two components Rxand Ry. A pin not free to turn also supports a couple M.

Built in or fixed end

A built-in or fixed end supports an axial force F, a transverse force V, and a bending moment M.

Coplanar Non-Concurrent Forces

Varignons theorem:

The algebraic sum of the moments of a system of coplanar forces about a

momentum center in their plane is equal to the moment of their resultant forces about the same

moment center.

R.d = P1.d1+P2.d2

Effect of couple is unchanged if

Couple is rotated through any angle.

Couple is shifted to any position.

The couple is replaced by another pair of forces whose rotated effect is the same. Couple is free vector

A

B

R

A

FWeld

O

A

R

Rx Rx

R


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GATE Engineering Mechanics Book

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