a seminar report on basic mechanics

7/24/2019 A Seminar Report On Basic Mechanics

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SEMINAR

ON

ENGINEERING

MECHANICS

SUBMITTED BY:

SIDDHARTH PANWAR

SECTION L

ROLL NO. 38


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INDEX

Introduction

Resolving vectors by AccurateDrawing

Resolving vectors by

trigonometry

Principles of equilibrium

Free body diagram

Equilibrium and its equations Lamis teorem

!onditions of equilibrium


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INTRODUCTION

Forces in equilibrium mean that they arebalanced. !oplanar forces act in the sameplane. Two balanced forces are equal inmagnitudebut opposite in directionto the other.

We can see easily from the free body diagram thatthe resultant force is "ero.

If we are considering tree coplanar forces in

equilibrium, use the triangle of forces rule:

If 3 forces acting at a point can berepresented in size or direction by the sidesof a closed triangle, then the forces are inequilibrium, provided their directions canform a closed triangle.

This means that the forces can follow each otherround a triangle


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Notice how:

The forces form into a closed triangle. The directions of the forces go round the

triangle.

In statics, remember that all forces add up to zero.That does not mean that there are no forces theforces balance each other out.

In statics problems, we need to !now how toresol"e forces, which can be done by:

accurate drawing use of trigonometry.

Resolving te #ectors by AccurateDrawing

#onsider three forces acting in equilibrium:


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We see that the forces $from the weights% form aclosed triangle and the directions form a closedloop. Therefore the forces are balanced. We canshow this as a triangle of forces.

&. #hoose a scale $e.g. ' cm ( & N%'. )se graph paper.*.)se a sharp pencil.+.)se a compass.. )se a protractor if angles are mentioned.

-. raw the arrows in the direction speci/ed.


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0ere is the equilibrium situation abo"e representedby accurate drawing:

The angles are measured with a protractor to gi"ethe "alues shown.

Resolving Vectors by Trigonometry

The problem with accurate drawing is that ofaccuracy. If you get the answer to the nearestdegree, you1re doing well. 2nd accurate drawing isnot easy. If you are challenged by measuring,


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resolution of "ectors using trigonometry is theanswer.2ny "ector in any direction can be resol"ed intovertical and ori"ontal components at 34degrees to each other.

For three forces in equilibrium we can draw a forcevector diagram:


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For this situation we !now that weight always acts"ertically downwards. We can resol"e the twoother "ectors into their horizontal and "erticalcomponents:

&.T1resol"es into T1cosq1$horizontal% and T1sin

q1$"ertical%'.T2resol"es into T2cosq2$horizontal% and T2sin

q2$"ertical%.

We !now that the three forces add up to zero, so

we can say:

T1cos q1+ - T2cos q2 ( 4. This means thatthe forces are equal and opposite.

5 T1cosq1= T2cosq2

T1sin q1+ T2sinq2= W


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6e careful that you don1t assume that W is splite"enly between T1sin q1and T2sin q2. This is onlytrue when the weight is half way between theends.

PRINCIPLES OF EQUILIRIU!

There are three main principles of equilibrium.

$%& 'wo force principle(

2ccording to this principle, if a body is in

equilibrium under the action of two forces, thenthey must be equal, opposite and collinear.

$)& 'ree force principle(


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2ccording to this principle, if a body is inequilibrium under the action of three forces thenthe resultant of any two forces must be equal,opposite and collinear with the third force.

$*& Four force principle(

2ccording to this principle, if a body is inequilibrium under the action of four forces then theresultant of any two forces must be equal, oppositeand collinear with the resultant of the other twoforces.

FREE OD" DI#$R#!

The new body diagram is a simple diagrammaticrepresentation of an isolated body or combinationof bodies $treated as a single body% to show all theforces imposed on the body from the surrounding.

The forces may be either internal or e7ternal to thebody under consideration. 2ll the forces $includingreactions% acting on it are drawn. To draw it thesupports are remo"ed and replaced by reactionswhich they e7ert on the body. The condition ofequilibrium of the body is attained when the acti"eforces and reacti"e forces together represent asystem of forces in equilibrium.


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Thus it can be concluded that

2 free body diagram represents a force system inequilibrium.

$'% If a system of bodies is in equilibrium then eachsubsystem must also be in equilibrium.

$*% If the body is isolated from earth thegra"itational force should be treated as an e7ternalforce on the body.


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EQUILIRIU! #ND ITS EQU#TIONS

The equilibrium state may be de/ned as thecondition in which the resultant of all the forces

acting on a body is zero, i.e., all the forces andmoments applied to the body are in balance. Thefree body diagram which represents a system offorces and couples acting on the body, can bereplaced by a single force and a single moment.Now, the equilibrium can e7ist only if both theresultant force and the resultant couple "anish.

That is:


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8 F ( 4

M = 0

E%&ilibri&m E%&'tion For conc&rrent (orce systems)

These forces may be collinear, coplanar or spatialand the "ector or algebraic method can beemployed. Te condition for equilibrium is

9( 8Fi( F&; F'; F+; ...4

This "ector equation for equilibrium can be used togenerated three scalar equations for a general

concurrent force system, i.e.,

97( 8 F7( F&7; F'7; F*7; F+7; ... ( 4

9y( 8 Fy( F&y; F'y; F*y; F+y; ... ( 4

9z( 8 Fz( F&z; F'z; F*z; F+z; ... ( 4

Note)


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If 9 < 4. Then the equi"alent force F&( =9 mustpass through the point of concurrency in order tobring the body in equilibrium

E%&ilibri&m E%&'tion For Co*l'n'r (orce system)

For a force system which comprises of forces with

their lines of action in the same plane and themoments due to couples which are directedperpendicular to the plane, the conditions ofequilibrium can be written as:

8 F7 ( 4 $sum of all horizontal forces%

8 Fy( 4 $sum of all "er/cal forces%

and 8 > ( 4 $sum of all moments%

the summation of moments may be ta!en about

any point on the ?@ plane but it should be aboutthe A=a7is through the chosen point. The body thuscan only be in equilibrium if the algebraic sum ofall the e7ternal forces and their moments aboutany point in their plane is zero.


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E%&ilibri&m E%&'tion For *'r'llel (orce system)

For such a system the resultant of the forces maybe a non=zero or a zero force. It may not beaccompanied by a couple moment. Thenequilibrium will occur only if

8 F& ( 4

8 >B ( 4

i.e., 8 >& ( 4, 8 >' ( 4, 8 >*( 4

where B represents the point about which themoment of all the forces in the plane is ta!en.

In fact there can be only two of these equations

mutually independent. Thus, for a plane parallelforce system, the conditions of equilibrium reduceto

8 >&( 4 and 8 >'( 4


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where & and ' represent any two suitably chosenpoints.

L'mi+s t,eorem

In statics, Lami+s teoremis an equation relatingthe magnitudes of three coplanar, concurrent andnon=collinear forces, which !eeps an obBect in static

equilibrium, with the angles directly opposite to thecorresponding forces.


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where 2, 6 and # are the magnitudes of threecoplanar, concurrent and non=collinear forces,which !eep the obBect in static equilibrium, and C,D and E are the angles directly opposite to theforces 2, 6 and # respecti"ely.

ami1s theorem is applied in static analysis of

mechanical and structural systems. The theorem isnamed after 6ernard amy.

Proof of Lami's Theorem

Guppose there are three coplanar, concurrent andnon=collinear forces, which !eeps the obBect instatic equilibrium. 6y the triangle law, we can re=

construct the diagram as follow:
http://en.wikipedia.org/wiki/File:Lami.png


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6y the law of sines,

CONDITIONS OF EQUILIRIU!

#onsider a body acted upon by a number ofcoplanar non=concurrent forces. 2 little

consideration will show the as a result of these
http://en.wikipedia.org/wiki/File:LamiProof.png


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forces, the body may ha"e any one of the followingstates:

&. The body may mo"e in any one direction.'. The body may rotate about itself without

mo"ing.*. The body may mo"e in any one direction and

at the same time it may also rotate aboutitself.

+. The body may be completely at rest.

Now, we shall study the abo"e mentioned fourstates one by one.

&. If a body mo"es in any direction, it means thatthere is a resultant force acting on it. 2 littleconsideration will show that if the body is to beat rest or in equilibrium, the resultant forcecausing mo"ement must be zero. Hr in otherwords, the horizontal component of all theforces and "ertical forces must be zero.>athematically,

0 ( 4 and J ( 4

'. If the body rotates about itself, withoutmo"ing, it means that there is a singleresultant couple acting on it with no resultantforce. 2 little consideration will show that if thebody is to be at rest or in equilibrium, themoment of the couple causing rotation must bezero. >athematically,

> ( 4

*. If the body mo"es in any direction and at thesame time it rotates about itself, it means that


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there is a resultant force and also a resultantcouple acting on it. 2 little consideration willshow that if the body is to be at rest or inequilibrium, th e resultant force must be zero. Hrin other words, horizontal component of all theforces $0%, "ertical component of all theforces$J% and resultant moment of all theforces$>% must be zero. >athematically,

0 ( 4 J ( 4 and > (4

+. If the body is completely at rest, it necessarilymeans that there is neither a resultant force nora couple acting on it. 2 little consideration willshow that in this case the following conditionsare satis/ed :

0 ( 4 J ( 4 and > (4

The abo"e mentioned three equations are !nownas the conditions of equilibrium.

a seminar report on basic mechanics

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