chapter 2.c(equilibrium in plane 2d)

8/19/2019 Chapter 2.c(Equilibrium in Plane 2D)

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Engineering Mechanics :Engineering Mechanics :STATICSSTATICS

Lecture #03By,

Noraniah Binti KassimUniversity Tun Hussein Onn Malaysia

(UTHM),


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EQUILIBRIUM OF A PARTICLE IN 2-D

Today’s Objectives:

Students will be able to :a) Draw a free body diagram (FBD), and,

b) Apply equations of equilibrium to solve

a 2D problem!

Learning Topics:

" #$at, w$y and $ow of a

FBD

" %quations of equilibrium

" Analysis of spring and

pulleys


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READING QUIZ

&) #$en a parti'le is in equilibrium, t$e sum of for'es a'ting

on it equals ! ($oose t$e most appropriate answer)

A) a 'onstant B) a positive number ) *ero

D) a negative number %) an integer!

2) For a fri'tionless pulley and 'able, tensions in t$e 'able(+& and +2) are related as !

A) +& +2

B) +& - +2

) +& . +2

D) +& - +2 sin θ


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APPLICATIONS

For a spool of given

weig$t, w$at are t$e

for'es in 'ables AB

and A /


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APPLICATIONS (continued)

For a given 'able

strengt$, w$at is t$ema0imum weig$t

t$at 'an be lifted /


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EQUILIBRIUM OF PARTICLE IN 2-D (Section 3.3

+$is is an e0ample of a 2D or

'oplanar for'e system! 1f t$e

w$ole assembly is in

equilibrium, t$en parti'le A is

also in equilibrium!

+o determine t$e tensions in

t$e 'ables for a given weig$t

of t$e engine, we need to

learn $ow to draw a free bodydiagram and apply equations

of equilibrium!


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T!E "!AT# "!$ AND !O" OF A FREE BOD$

DIAGRAM (FBD

Free Body Diagrams are one of t$e most important t$ings for

you to now $ow to draw and use!

#$at / 1t is a drawing t$at s$ows

all e0ternal for'es a'ting on t$e

parti'le!

#$y / 1t $elps you write t$eequations of equilibrium used to

solve for t$e unnowns (usually

for'es or angles)!


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3ow /

&! 1magine t$e parti'le to be isolated or 'ut free from its

surroundings!

2! S$ow all t$e for'es t$at a't on t$e parti'le!

A'tive for'es: +$ey want to move t$e parti'le!

4ea'tive for'es: +$ey tend to resist t$e motion!

5! 1dentify ea'$ for'e and s$ow all nown magnitudes

and dire'tions! S$ow all unnown magnitudes and 6

or dire'tions as variables !

FBD at A 7ote : %ngine mass - 289 g


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EQUATIONS OF 2-D EQUILIBRIUM

;r, written in a s'alar form,Σ F x = 9 and Σ F y - 9

+$ese are two s'alar equations of equilibrium (%of%)! +$ey

'an be used to solve for up to two unnowns!

Sin'e parti'le A is in equilibrium, t$e netfor'e at A is *ero!

So FAB < FAD


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E%AMPLE

#rite t$e s'alar %of%:

< → Σ F x - T

B 'os 59= > T

D - 9

< ↑ Σ F y - T B sin 59= > 2.?82 7 - 9

Solving t$e se'ond equation gives: +B - ?.@9 7

From t$e first equation, we get: +D - ?.28 7

7ote : %ngine mass - 289 g FBD at A


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SPRINGS# CABLES# AND PULLE$S

Spring For'e - spring 'onstant

deformation, or

F - S

#it$ a

fri'tionless

pulley, +& - +2!


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E%AMPLE

iven! Sa' A weig$s 29 7!

and geometry is ass$own!

Find! For'es in t$e 'ables and

weig$t of sa' B!

"#an!

&! Draw a FBD for oint %!

2! Apply %of% at oint % to solvefor t$e unnowns (+%C +%)!

5! 4epeat t$is pro'ess at !


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E%AMPLE (continued)

+$e s'alar %of% are:< → Σ F x - T EG sin 59= > T EC 'os ?8= - 9

< ↑ Σ F y - T EG 'os 59= > T EC sin ?8= > 29 7 - 9Solving t$ese two simultaneous equations for t$e

two unnowns yields:

T EC - 5E. 7

T EG - 8?. 7

A FBD at % s$ould loo lie t$e oneto t$e left! 7ote t$e assumed

dire'tions for t$e two 'able tensions!


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E%AMPLE (continued)

+ → Σ F x - 5E.? 'os ?8° > (?68) T CD - 9

+ ↑ Σ F y - (568) T CD < 5E!? sin ?8° > W B - 9

Solving t$e first equation and t$en t$e se'ond yields

T CD - 5?.2 7 and W B - ?G.E 7 !

+$e s'alar %of% are:

7ow move on to ring !

A FBD for s$ould loo

lie t$e one to t$e left!


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CONCEPT QUESTIONS

&999 7&999 7

&999 7

( A ) ( B ) ( )

&) Assuming you now t$e geometry of t$e ropes, you 'annotdetermine t$e for'es in t$e 'ables in w$i'$ system above/

A) +$e weig$t is too $eavy!

B) +$e 'ables are too t$in! ) +$ere are more unnowns t$an equations!

D) +$ere are too few 'ables for a &999 7

weig$t!

2) #$y/


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IN CLASS TUTORIAL (GROUP PROBLEM

SOL&ING

iven! +$e 'ar is towed at 'onstant

speed by t$e 99 7 for'eand t$e angle θ is 28H!

Find! +$e for'es in t$e ropes AB

and A!

"#an!

&! Draw a FBD for point A!

2! Apply t$e %of% to solve for t$e for'es in ropes AB and A!


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GROUP PROBLEM SOL&ING (continued)

59H28H

99 7

FAB

FA

AFBD at point A

Applying t$e s'alar %of% at A, we getI

< → ∑ F x - F AC 'os 59H > F AB 'os 28H - 9

< → ∑ F y - -F AC sin 59H > F AB sin 28H < 99 - 9

Solving t$e above equations, we getI

F AB - 5? 7

F AC - ? 7


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ATTENTION QUIZ

A

59°

?9°

&99 7

&! Sele't t$e 'orre't FBD of parti'le A!

A)A

&99 7

B)

59°

?9H

A

F& F2

) 59HA

F

&99 7

A

59H ?9HF& F2

&99 7

D)


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ATTENTION QUIZ

29 7

F&

89H

2! Jsing t$is FBD of oint , t$e sum of

for'es in t$e 0dire'tion (Σ FK) is !

Jse a sign 'onvention of < → !

A) F2 sin 89H > 29 - 9

B) F2 'os 89H > 29 - 9

) F2 sin 89H > F& - 9

D) F2 'os 89H < 29 - 9


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!OME"OR' TUTORIAL

L& (2!??) :

nowing t$at M - 28H, determine t$e tension

(a) in 'able AC ,

(b) in rope BC !


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!OME"OR' TUTORIAL (continued)

L2 (2!?) :

+wo 'ables are tied toget$er at C and are loaded as s$own!

nowing t$at M - 59H, determine t$e tension (a) in 'able AC , (b) in

'able BC !


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L5 (2!8&) :

+wo for'es " and $ are applied as s$own to an air'raft 'onne'tion!

nowing t$at t$e 'onne'tion is in equilibrium and t$e P - &!E7 and

Q = 2!5 7, determine t$e magnitudes of t$e for'es e0erted on t$e

rods A and B!


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(a) (b) (c ) (d ) (e)

T

T

T

T

T

L? (2!G) :

A 2E9g 'rate is supported by several ropeandpulley

arrangements as s$own! Determine for ea'$ arrangement t$e

tension in t$e rope!

( Hint : +$e tension in t$e rope is t$e same on ea'$ side of a simple

pulley!)


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L (5&G) :Determine t$e for'e in ea'$ 'able and t$e for'e F needed to $old

t$e lamp of mass M in t$e position s$own!

Hint : First analy*e t$e equilibrium at BI t$en, using t$e result for

t$e for'e in BC , analy*e t$e equilibrium at C !iven!

% :- ?'g

N :- 59

N& :- 9

N( :- 59


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chapter 2.c(equilibrium in plane 2d)

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