engineering mechanics lecture 2

8/17/2019 Engineering Mechanics Lecture 2

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Force is the action of one body on another. A force tends to move a body in

action. The action of a force is characterized by its magnitude, by the directio

by its point of application. Thus force is a vector quantity. The SI units of force

Principe of Transmissibiity

This aw states that a force may be appied at any point on its !iven ine of acti

the resutant effects of the force e"terna to the ri!id body on which it acts.

For e"ampe, the force P actin! on the ri!id pate in Fi!. may be appied at A

other point on its ine of action, and the net e"terna effects of P on the brac#et


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$oncurrent Forces

Two or more forces are said to be concurrent at a point if their ines of acti

point. The forces and shown in Fi!. have a common point of appication an

the point A . Thus, they can be added usin! the paraeo!ram aw in their comm

their sum or resutant R , as shown in Fi! a. The resutant ies in the same pane

Suppose the two concurrent forces ie in the same pane but are appied at two

in Fi! b. &y the principe of transmissibiity, we may move them aon! their

compete their vector sum R at the point of concurrency A, as shown in Fi! b. '

with the resutant R without aterin! the e"terna effects on the body upon w

can aso use the trian!e aw to obtain R , but we need to move the ine of aforces, as shown in Fi! c.

'e can e"press the sum of the two forces mathematicay by the vector equatio


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(ectan!uar $omponents

The most common two)dimensiona resoution of a force vector is into rectan!u

foows from the paraeo!ram rue that the vector F of Fi!. may be written as

where and are vector components of F in the x) and y)directions. *ach

components may be written as a scaar times the appropriate unit vector. In

vectors i and j of Fi! , and , and thus we may write

where the scaars and are the x and y scalar components of the vector F.The scaar components can be positive or ne!ative, dependin! on the quad

points. For the force vector of Fi! , the x and y scaar components are both posi

to the ma!nitude and direction of F by


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Probem+- etermine the ma!nitude of the resutant force actin! on the screw eye and its direction measured coc#wise from the x a"is.

Son-

Step+- raw the force trian!e dia!ram

Step+/-


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Step+0-

Step+1-

i.e. direction of resutant with ")a"is


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Probem+/- Two forces act on the hoo#. etermine the ma!nitude of the resutant force.

Son.

Step+ Fi!ure representin! the force trian!e

Step+/

Step+0


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Step+1

Step+2

Therefore the resutant force is actin! at an!e .

Probem+0- etermine the ma!nitude of the resutant force and its direction measured countercoc#wise from the positive x a"is.

Son.Step+ raw the force trian!e


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Step+/ &y cosine rue

Step+1

i.e. direction of resutant with ")a"is.

Probem+1- (esove the 03)b force into components aon! the u and v a"es, and determine the ma!nitude of each of these components.

Son.

Step+


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Step+/

Step+0

Step+1

Probem+2- The force acts on the frame. (esove this force into components actin! aon! members AB and AC , and determine the ma!nitude


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Son.

Step+ raw the force trian!e

Step+/

Step+0

Probem+4- If force F is to have a component aon! the u a"is of , determine the ma!nitude of F and the ma!nitude of its component aon! the v a"is.


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Son.

Step+ raw the force dia!ram

Step+/

Step+0


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Probem+5- etermine the ma!nitude of the component force F in Fi!. and the ma!nitude of the resutant force if is directed aon! the positive y

Son.


Step+/ The magnitudes of and F

are the twounknowns They can be determined by applying the law of sines.

Probem+6- It is required that the resutant force actin! on the eyebot in Fi!. be directed aon! the positive x a"is and that have a mini

ma!nitude, the an!e , and the correspondin! resutant force.


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Son.


Step+/ The trian!e rue for 7 8

Since the ma!nitudes 9en!ths: of and are not specified, then can actuay be any vector that has its head touchin! the ine of act

ma!nitude of is a minimum or the shortest en!th when its ine of action is perpendicular to the ine of action of , that is, when

Since the vector addition now forms the shaded ri!ht trian!e, the two un#nown ma!nitudes can be obtained by tri!onometry.

Probem+< If and F, determine the magnitude of the resultant force and its direction, measured coupositive x axis.


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Son.

Step+ S#etch the forces actin! on the an!e section as foows-

Step+/

$onstruct the paraeo!ram as foows-

. $onsider the "y coordinate system with ori!in as =.

/. (epresent the force F passin! throu!h = ma#in! an an!e of with positive ")a"is in counter coc#wise direction.

0. (epresent the force 533 N passin! throu!h = ma#in! an an!e of with the ne!ative ")a"is in counter coc#wise direction.

1. From the head of force F , draw a ine parae to the force 533 N.

2. From the head of force 533 N, draw a ine parae to force F.

4. >ar# the intersection point of the above two ines as A.

5. The resutant force is represented by the dia!ona =A of the paraeo!ram.Step+0 (epresent the resutant force usin! paraeo!ram aw of addition.


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Step+1 $onsider the vector trian!e as foows-

Step+2 $acuate the resutant force by appyin! the aw of cosines to the vector trian!e.

Substitute for OA, 123 N for AB, 533 N for OB, for in the above equation.


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Step+4 Appy Sine rue to the vector trian!e

Substitute for OA, 123 N for AB, , for in the above equation.

$acuate the incination of resutant force with positive ")a"is in counter coc#wise direction as foows-

Therefore the incination of resutant force with positive ")a"is in counter coc#wise direction is .

Probem+3 If the ma!nitude of the resutant force is to be 233 N, directed aon! the positive y a"is, determine the ma!nitude of force


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Son.

Step+ raw the free body dia!ram of the section to use paraeo!ram aw.

Step+/


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Probem+3 etermine the ma!nitude of the resutant forceand its direction, measured counter coc#wise from the positive x a"is.

Son.

Step+

Appy paraeo!ram aw and represent the direction and ma!nitude of the resutant force actin! on the system.

Step+/


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Step+0

Substitute/23b for for %

Step+1 $acuate the incination of the resutant force measured counter coc#wise from the positive ")a"is .


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SA>P?* P(=&?*>

F

A

&


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engineering mechanics lecture 2

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