L05 - Equilibrium

8/18/2019 L05 - Equilibrium

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January 20/21, 2014

ECOR 1101 Mechanics I

Sections C and F

Jack Vandener!

Lecture 04 – Equilibrium and FBDs

(Chapter 3 – Sections 3.!3.4"


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O"ecti#es

Learn the concept o# #ree!bod$ dia%rams

Learn to sol&e problems in&ol&in% particles in equilibrium

Learn to use equilibrium equations usin% cartesian &ector

coordinates

2

'D and 3D Equilibrium o# a particle

ECOR1101 –Mechanics I


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Equilibrium o# a article

) particle is in equilibrium i#*

+t remains at rest under action o# a s$stem o# #orces, or, +t continues in its state o# motion -ith constant &elocit$

under action o# a s$stem o# #orces.

For a particle to be in equilibrium the resultant o# all

#orces actin% on it must be ero.

Satis#ies /e-tons st la- o# motion

1he abo&e equation represents the necessar$ and

su##icient condition #or equilibrium o# a particle in

space.

3

F R= F x ∑ + F y + F z = 0∑∑


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)ccordin% to /e-tons 'nd la- o# motion, i# ΣF = ma = 0,

the particle is in equilibrium since a = 0 and ΣF = 0 i.e. the particle is under constant &elocit$ or is at rest

1he equilibrium equation can be used to sol&e problems

dealin% -ith equilibrium o# a particle in&ol&in% no more

than three un2no-ns

4

Equilibrium o# a article



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1o appl$ the equations o# equilibrium to a particle all

#orces (2no-n and un2no-n" must be accounted #or. 1he best -a$ to do this is to isolate the particle #rom its

surroundin%s to #orm a #ree!bod$ dia%ram (FBD".

1hen appl$ all the #orces (2no-n and un2no-n" actin%

on the particle

5

Free!bod$ Dia%rams (FBD"



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+solate particle #rom its surroundin%

S2etch outline shape o# particle -ith all #orces (acti&e

and reacti&e" indicated

Label all #orces (2no-n and un2no-n" -ith both their

ma%nitudes and directions

+# $ou 2no- that an un2no-n #orce is in tension, do $ou

dra- it a-a$ or to-ards the particle5

6

rocedure #or dra-in% FBD



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Free Bod$ Dia%rams (FBDs"

7


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FBDs

8

6i%id Bodies

Sprin%s 7oo2es la-

F 8 2 s

F 8 sprin% constant 9 displacement

Cables (assumptions"

:ust be in tension /e%li%ible -ei%ht

Do not stretch

Frictionless pulle$


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FBDs

9

Draw a FBD of the cable AB and of the joint C.


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Sample roblem

) '00!2% c$linder is hun% b$ means o# t-ocables AB and AC, -hich are attached to the

top o# a &ertical -all. ) horiontal #orce $

perpendicular to the -all holds the c$linder in

the position sho-n. Determine the ma%nitude

o# P and tension in each cable

10ECOR1101 –Mechanics I

!

10 !

1" !1." !

" !

A

B

C

A

B

C

P

P

w

TAC

TABk

i j

+ntroduce unit &ectors i, ", k alon%

ortho%onal a9es and resol&e #orces

P # P i $ 0 j $ 0k

% = 0i $ 0 j % mg k 8

8 0i $ 0 j ! '00×;.


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11ECOR1101 –Mechanics I

r AB # &%1."!'i % &!' j $ &10!' k

r AC # &%1."!'i $ &10!' j$ &10!'k

( ) ( )( ) ( )

mk, jir

u

mk, jir

u

m

m

ACAC

ABAB

7046.07046.00846.0

778.0622.0093.0

193.1410102.1

862.121082.1222

222

++−==

+−−==

=++−=

=+−+−=

AC

AB

AC

AB

r

r

r

r

0

0

=+++∴

=++=∑ ∑ ∑ ∑

ACAB

zyx

uuWP

FFFF

AC ABT T

(ince the c)linder is *nder e+*ilibri*!,

X AB AC AB AC

y AB AC AB AC

z Ab AC AB AC

AB

AC

F 0 P 0 0.093T 0.0846T 0 P 0.093T 0.0846T

F 0 0 0 0.622T 0.7046T 0 T 1.133T

F 0 0 1962 0.778T 0.7046T 0 T 0.9056T 2521.851

T 1401.6 N 1.40 kN

T 1237.1 N 1.24 kN

P 235 N

= ⇒ + − − = ⇒ = +

= ⇒ + − + = ⇒ =

= ⇒ − + + = ⇒ = − +

= =

= =

=

∑

∑∑


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Cables, Sprin%s and ulle$s

Cables (or cords", in %eneral,

are assumed to ha&e the

#ollo-in% properties

>ei%htless

Supports onl$ tension in the

direction o# the cable (cannotbe pushed"

Cannot stretch (i.e. increase

in len%th under load"

) cable passin% o&er a#rictionless pulle$ has a constant

ma%nitude

12ECOR1101–Mechanics I


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Cables, ulle$s and Sprin%s

Sprin%s, -hen de#ormed, e9ert

a #orce proportional to the

amount o# de#ormation.

Sprin%s are o#ten de#ined b$ the

sprin% constant or sti##ness k

1he ma%nitude o# #orce e9ertedon a linearl$ elastic sprin% -ith

sti##ness k is %i&en b$* F = ks

s = l − l o ,

l o = unstretched len%th,l = stretched len%th



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Coplanar Force S$stems ('D"


F1

F2

x

y

F 1x

F 2x

F 2y

F 1y

1he t-o equations o# equilibrium can

be sol&ed #or at most t-o un2no-ns.

)pplication o# the equation must ta2e

into account direction o# components

o# the #orce

+# a particle sub?ected

to a s$stem o# coplanar#orces (9!$ plane", then

the #orces can be

resol&ed and

equilibrium equations

applied.


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Establish x-y a9es

Dra- a #ree!bod$ dia%ram

Dra- and label all #orces (2no-n and un2no-n" -ith

ma%nitudes, sense and direction

Choose an arbitrar$ direction #or un2no-n #orces

6esol&e #orces in x-y a9es

)ppl$ equations o# equilibrium

)ssume a @&e direction #or the purpose o# -ritin% $our

equation o# equilibrium

Sol&e #or un2no-n #orces Compare $our ans-ers to $our ori%inal assumption (not to

the @&e direction -hen -ritin% $our equations"

6edra- $our FBD -ith all Forces as positi&e numbers

15

rocedure For )nal$sis o# Coplanar ('D" Force Equilibrium

ECOR1101–Mechanics I


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1hree!Dimensional (3D" Force S$stems

Conditions #or equilibrium

6esol&e #orces into respecti&e

Cartesian components, i, j, k


x

z

F1

F1zF1x

F1y

F2

F2y

F2z

F2x

0 F =∑

00

0

0

0

=+ + =

=

=

=

∑∑ ∑ ∑

∑

∑

∑

F

i j k x y z

x

y

z

F F F

F

F

F 1he three equation o# equilibrium are al%ebraic sums

o# #orce components and can be used to #ind at most

three un2no-ns (coordinate direction an%les or

ma%nitudes o# #orces actin% on a particle"


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Establish x-y-z a9es

Dra- a #ree!bod$ dia%ram

Dra- and label all #orces (2no-n and un2no-n" -ith

ma%nitudes, sense and direction

Choose an arbitrar$ direction #or un2no-n #orces

6esol&e #orces in x-y-z a9es

)ppl$ equations o# equilibrium

)ssume a @&e direction #or the purpose o# -ritin% $our

equation o# equilibrium

Sol&e #or un2no-n #orces Compare $our ans-ers to $our ori%inal assumption (not to

the @&e direction -hen -ritin% $our equations"

17

rocedure For )nal$sis o# 3D Force S$stems

ECOR1101–Mechanics I


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Sample roblem

1he shear le% derric2 is

used to haul the '00!2% net

o# #ish onto the doc2.

Determine the compressi&e

#orce alon% each o# the

le%s )B and CB and the

tension in the -inch cable

DB. )ssume the #orce in

each le% acts alon% its a9is.



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W!"# "$# %&&d!na"#' (& )&!n"' A, B, C, and

*, )&'!"!&n +#%"&', n!" +#%"&', F&%#

-#%"&'

A2m, 0, 0/

B0, 4m, 4m/

C2m, 0, 0/

*0, 5.6m, 0/

W

FBD

x

y

z

FBA

FBC

B

A

C

D

"

-

-./

"


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rBA= 2mi 4m j 4mk

rBC= 2mi 4m j 4mk

rBD= 0mi 9.6m j 4mk

W = 0i 0 j 200×9.81/k

= 0i 0 j 1962Nk

,BA = BA

r BA

=2!− 4− 4k

2/2

+ −4( )2

+ −4( )2

= 0.333!−0.667−0.667k

( ) ( ) , m

r BC k ji

k jiru BCBC 667.0667.0333.0

44/2

442

222−−−=

−+−+−

−−−==

( ) ( ) , m

r BDk ji

k jiru BDBD 385.0923.00

46.9/0

46.90

222−−=

−+−+

−−==


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kN

kN

kN N

WuuuF BDBCBA

65.3

52.2

52.285.2521

01962385.0667.0667.00

445.1

0923.0667.0667.00

0333.0333.00

00

=

−=

−=−=

=−−−−⇒=

−=∴

=−−−⇒=

=∴

=−⇒=

=+++⇒=

∑

∑

∑∑

DB

BC

BA

BD BC BA z

BA BD

BD BC BA y

BC BA

BC BA x

BD BC BA

F

F

F

F F F F

F F

F F F F

F F

F F F

F F F

W

FBD

x

z

FBA

FBC

B

A

C

D


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roblem F3!<

22

Deter!ine the tension deeloed in cables AB2 AC2 and AD.


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roblem 3!A;

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Deter!ine the !a3i!*! wei4ht of the crate that can be s*orted

fro! cables AB2 AC2 and AD so that the tension deeloed in an)one of the cables does not e3ceed "0 lb.


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roblem 3!=

24

If cable AD is ti4htened b) a t*rnb*c5le and deelos a tension of

12600 lb2 deter!ine the tension deeloed in cables AB and ACand the force deeloed alon4 the antenna tower AE at oint A.


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roblem 3!

25

7he joint of a sace fra!e is s*bjected to fo*r !e!ber forces.

Me!ber OA lies in the 3%) lane and !e!ber OB lies in the )%8lane. Deter!ine the forces actin4 in each of the !e!bers re+*ired

for e+*ilibri*! at the joint.

L05 - Equilibrium

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