chapter 2.c(equilibrium in plane 2d)
TRANSCRIPT
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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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!OME"OR' TUTORIAL (continued)
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$e
rods A and B!
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!OME"OR' TUTORIAL (continued)
(a) (b) (c ) (d ) (e)
T
T
T
T
T
L? (2!G) :
A 2E9g 'rate is supported by several ropeandpulley
arrangements as s$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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!OME"OR' TUTORIAL (continued)
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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