chapter 4 general equilibrium

8/13/2019 Chapter 4 General Equilibrium

1/21

General

equilibrium

Conditions of equilibrium

( ) ( ) ( )1 M 0 2 X 0 3 Y 0= = =

( )

( )

( )( )

1 Your drawing must be in a large scale

2 If you didn't find the word hori!ontal "osition# then your body is inclined to the hori!ontal

3 $he reaction of any body is always "er"endicular to the "lane%

( )

&

e are dealing with two "lanes (

)mooth "lanes *ough "lanes

+

& * where Coefficient of friction and * the normal reaction

=

Equilibrium rule :

About to move rule :

( )

$his is called a limiting friction

, 2 2F

Resultant reaction R' = F + R

*

* *

&&

&aceboo- "age ( htt"s(..www/faceboo-/com.Mr)herifYehialMaraghy

Mr . Sherif ehia Al Mara!h" htt#s:$$t%itter.com$Mr&Sherif&"ehia())***, - ())***2, Email : Roosher"hotmail.com

/ntro0uctionIn "reious year# we hae ta-en the equilibrium of a body under the action of three force/

In this cha"ter# we will deal with a body in equilibrium under the action of more than three

forces "laced on smooth or rough "lanes/

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efinition : set of co"lanar forces is said to be in equilibrium if both the ectorsum of

the forces and the ector sum of the moments of the forces about any"oint anish/ lso when a

body is acted on by such a set of co"lanar forces#the body is said to be in equilibrium/

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Some notes %hich %ill ma3e "our %or3 easier

Static - r0secon0ar" 4ha#ter four - General equilibrium1
https://www.facebook.com/MrSherifYehiaAlMaraghyhttps://www.facebook.com/MrSherifYehiaAlMaraghy


2/21

C

o30

o,0

2

3

( )4 5se the following methods to sole your "roblem quic-ly

Co%+

o%+

2

C

o%+

o%+

6

6

107am"le

106

2=

C

C

107am"le

Cos

)in

10)in

10Cos

C C

8 7

If 87 .. C

8 7 87

C C= =



111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111



3/21

uniform rod of 90 cm long of weight 3 -g/wt is hinged at to a ertical wall# lam" of

12 -g/wt is hung from a "oint on the rod 10 cm from the end # this end is tied with a light ro"e

with its end C fied on the wall at "oint ertically aboe and far from it ,0 cm so that the rod

is -e"t hori!ontal# find the tension of the string and the reaction of the hinge and its direction/

( )

( )

( )

:otes 1 $he "roblem mentioned that your rod is hori!ontal

2 :osmooth or rough mentioned in the "roblem

3 hen there is a hinge in the "roblem# then its

reaction has no fied direction# so we must

( ) ( )

( ) ( )

y

2 2

ma-e

a resolution in the ; ais * and the y ; ais *

C ,0 90 100 cm

M 0 ( try to anish the most un-nown forces

;3 %0 12 40 $ )in 90 0

,090$ )in


4/21

( )

( )

( )


2 :o smooth or rough mentioned in the "roblem

3 hen there is a hinge in the "roblem# then its

reaction has no fied direction# so we must

( ) ( )

( ) ( )

y

2 2

ma-e


C %/+ , 4/+ m

&irst ( we hae to -now that $


;+ %/+ 3 , $ )in , 0

,$ )i

= + =

=

=

+ =

Q

( )

( ) ( )

Y Y

%/+n %0/+ $ )in ,/4+ $ ,/4+ $ 11/2+ -g/wt

4/+

,X 0( ( * $ Cos 0 * $ Cos * 11/2+ < -g/wt

4/+

Y 0( ( 8on' t ta-e here as we -now that $

* $ )in 9 * 9 ,/4+ 1/2+ -g/wt

$h

= = = = =

= = = = =

= =

+ = = =

Q

( ) ( )2 22 2

y

1321en the reaction of the hinge is( * * * < 1/2+ -g/wt

%= + = + =

%/+ m

+

1/+ m

3

C

%/+ m

$=

*

*

y*

)in

$ )in

$ Cos

is a rod of , m long of weight + -g/wt and its center of graity is at %/+ m from and hinged

at to a fied hori!ontal hinge# the end is tied to a light string "assing oer a smooth "ulley

lies ertically aboe and at %/+ m away from it# the string carries at its other end a body of

weight / If a body of weight 3 -g/wt is hung from so that the rod became hori!ontal# then

find the magnitude of and find the "ressure of the hinge/



E6am#le 728

Ans%er

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5/21

uniform rod of 1,0 cm long and weight 300 gm/wt is hung by a fied nail C by the mean

of two strings attached to its end and # a weight of ,00 gm/wt is hung to a "oint : on the rod/

If the rod i

o o

s in equilibrium in a hori!ontal "osition and the two strings C and C are inclined

to the rod at angles ,0 #30 res"ectiely# find the length of : and the magnitude of the tension

in the two strings

( )

( )

( )

o1


2 :osmooth or rough mentioned in the "roblem


,00 90 300 90 $ )in,0 1,0 0

%9000 ,00

=

+ =

( )

( )

( ) ( )

( )

1 1

1

o o1 2 1 2 1 2

o o1 2 1 2

2%000 90 3$ 0 90 3$ ,00 4200 %0

2 3 $ 1+ 1900 1

1 3X 0( ( $ Cos,0 $ Cos30 0 $ $ 0 $ 3$ 2

2 2

3 1Y 0( ( $ )in ,0 300 ,00 $ )in30 0 $ $


6/21

uniform rod of 1,0 cm long and weight 30 -g/wt # its lower end rests on a ertical

smooth wall 8# and is tied from "oint = by one end of a string whose other end is attached

to "oint 8 so that t

o

he rod reached its equilibrium state in a "osition inclined to the ertical at

an angle ,0 / If the length = ,0 cm# then find the angle 8= and also find the magnitudes

of the tension of the string and

=the reaction of the wall/

=

:ote when a body is "laced on a smooth "lane# then the

reaction is "er"endicular to the "lane


;3 10 3 * 30 0 30* 30 3 * 3 -g/wt

X 0 ( * $ )in 0 $

= + = = =

= =

( )( )

( ) ( )

( )

o

o

)in * 3 1

Y 0( $ Cos 3 2

3$hen diide 1 by 2 ( $an 30

2

$hen from 1 ( $ )in 30 3 $ 2 3 -g/wt

= =

= =

= =

= =

30 3 cm

*

3

$ Cos

,0 cm

20

90 cm

$ )in

o,0

o30

o30

o30

10

10 3

1+cm

)mooth

8

=

C

uniform ladder of weight %0 -g/wt and 12 m length rests with its end on a smooth

hori!ontal "lane and its end on a smooth ertical wall# the ladder is -e"t in equilibrium by a

string tied to on

o

e of its ends # and its other end is attached to a "oint on the hori!ontal "lane

ertically below # if the ladder is inclined at %+ to the hori!ontal# gien that the string can not

bear a tension of more than +0 -g/wt# "roe that a man of weight the weight of the ladder can

not ascend more than


7/21

uniform rod of lenght 9 m and weight 20 -g/wt# its end is attached to a hinge fied to a

ertical wall# and its end being >oined by a light string , m length whose other end is fied to the

wall at C ertically aboe and 10 m away from it# if a weight of 1+ -g/wt is hung from the end /

&ind the tension in the string# and the magnitude and the direction of the reaction of the hinge

( )

( )

:ote 1 hen a body is "laced on a smooth "lane# then the


2 hen there is a hinge in the "roblem# then its reaction has no fied direction# so we must

( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

y

2 2

2 2 2 2

2 2 2 o

y

y

ma-e


In C( C 10 100

nd C , 9 100

C C m


8/21

uniform rod of weight 1+0 -g/wt and +% cm rests with its lower end on a ertical smooth

wall# and it is fied at "oint = on a hori!ontal smooth su""ort "arallel to the wall# if the length

= equals 12 cm and the rod is in equilibrium when it is "er"endicular to the wall# "roe that

2the sin angle of the inclination between the rod and the wall is # then calculate the magnitude

3

of the reaction of the su""ort and that of the wall/

( ) ( )

( )

o= 1

1

1



M 0 ( * 12 )in


9/21

uniform ladder of length + m and weight 20 -g/wt rests with one end on a smooth ertical wall

and the other end on a smooth hori!ontal floor# if the lower end of the ladder is 3m away from

the wall# the ladder is -e"t from sli""ing by a string attached to a "oint on it and the other end

of the string is fied to a "oint on the line of intersection of the floor and the wall# -nowing that

the direction of the string is "er"endicular to that of the ladder# find the tension of the string and

the reaction of each of the wall and the floor/

( )

( ) ( )2 2

:ote 1 hen a body is "laced on a smooth "lane# then the reaction is "er"endicular to the "lane

7 + 3 % m

18& 7 1/+ m and 7 7

2

nd 7C # then by using 7ucl

= =

= =

Q Q

( )

( )

( ) ( )

( ) ( )

2

y

y y

y

id's theorem

1, 7 C 1, +C C m

+1, 1,

M 0 ( ;20 1/+ $ Cos Cos $ )in )in * 3 0+ +

% 1, % 3 1, 3 1, ;30 $ $ 3* 0 3* $ 30 1

+ + + + + + +

%X 0( ( * $ Cos $ 2

+

3

Y 0( ( * 20 $ )in 20 $ 3+

)ubstit

= = == + =

+ = =

= = =

= = + = +

Q

( ) ( )

( )

( )

y

3 1, < 1, 1+0ute 3 in 1 ( 3 20 $ $ 30 ,0 $ $ 30 $ -g/wt

+ + + + 4

% 1+0 120&rom 2 ( * -g/wt

+ 4 4

3 1+0 230&rom 3 ( * 20 -g/wt

+ 4 4

+ = + = =

= =

= + =

20

C

&

*

y*

$ )in

2/+

$

)mooth

)mooth

1/+

?

8

7

1,)in

+

3 m

2/+$ Cos

1,Cos

+

% m

1,C m

+=



E6am#le 7*8

Ans%er



10/21

uniform ladder rests with its end on a smooth ertical wall and the its other end on a

smooth hori!ontal "lane# the ladder is -e"t from sli""ing by a string whose one of its ends fied

to a "oint

( )

at the bottom of the wall ertically below and the other end is fied to one stair at a

1distance from equals of the ladder length# if the "ressure of the ladder on the wall is @

%

and on the floor ( )is A / If # are at distance ,0 cm# 90 cm res"ectiely from the bottom of the

@ %wall# then "roe that(A +

=

( )

( ) ( )2 2

:ote 1 hen a body is "laced on a smooth "lane# then the


7 + 3 % m

18& 7 %0 cm & 30 cm

2In &8# =C ( &8 .. =C

& =

= =

= = =

=

Q

Q

( ) ( )

( )

( ) ( ) ( ) ( )

2 2

8

8 &8 30 +0 %0

C =C = 4+ =C

= %+ cm &= 1+ cm =C ,0 cm

In C7=( 7C 1+ ,0 1+ 14 cm

e don't need # so we will ta-e M 0 (

; @ 30 $ )in 20 $ Cos 1+ A %0 0

1+ ,0 2 ;30@ $ 20 $ 1+ %0A 0 %0A 30@

1+ 14 1+ 14

= = =

= = =

= + =

=

+ =

+ =

Q

( )

( ) ( )

( ) ( )

0 ,0$ $ 0

14 14

90 %0A 30@ $ 0 1

14

,0 14 X 0( ( @ $ Cos $ $ 2

141+ 14

90 14 )ubstitute 2 in 1 ( %0A 30@ @ 0

%14

@ % %0A 30@ 20@ 0 %0A +0@

A +

=

=

= = = =

=

= = =

C

&

@

A

$ )in

+0

%0 cm

=

8

7

,0 cm

20

2+

$ Cos

30

1+2+

1+ 1+ 14

20%0

$



E6am#le 7)8

Ans%er



11/21

o

is a uniform rod of mass 20 -g rests with its u""er end on a smooth inclined "lane that's

to the hori!ontal at an angle of measure ,0 # and it is resting with its lower end on a smooth

hori!ontal "lane# it is -e"t from sli""ing by a string C which is attached with one end to and

with the other end to C# so that the "lane C is "er"endicular to the line of intersection of the

two "lanes/ &

o

ind the magnitude of the reaction of each of the "lane on the rod# and find the

tension on the string when the angle of inclination of the rod to the hori!ontal is 30

( ) ( )

o



1M 0 ( ;* 3 20 3 $ 0

2

8iide by ; 3* 10 3 $ 0 1

3X 0 ( $ * Cos30 0 $ *

2

= + + =

+ + = = = =

( )

( )

( ) ( ) ( )

( ) ( )

( )

o

2 * $ 2

3

Y 0( * * )in30 20 3

1 2 3)ubstitute 2 in 3 ( * $ 20 * 20 $ %2 33

3)ubstitute % in 1 ( ; 3 20 $ 10 3 $ 0 ;20 3 $ 10 3 $ 0

3

2$ 10 3 $ + 3 -g/wt # then from 2 ( *

=

= + =

+ = =

+ + = + + + =

= =

( )

2+ 3 10 -g/wt

3

3

then from % ( * 20 + 3 1+ -g/wt 3

= =

= =

20

*

$

o30

1

2

o

* Cos30

o,0

13

2

o* )in30

o30

o30

o,0

3

120

o30

o30

1

2

*

C



E6am#le 7(8

Ans%er

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12/21

uniform rod BM of length 1+0 cm and weight + -g/wt tied by its two ends B#M by means of two

strings B: and M= whose lenghts are 1+0# 300 cm res"ectiely# and the two free ends of the two

strings are f

( ) ( )

ied at "oints :# = on a hori!ontal straight line# := 300 cm# then a hori!ontal

force & acted on the rod and its end B so the rod became in equilibrium "osition as the

string B: is ertical and the whol

=

( )

e system of forces lie in the same ertical "lane# determine

& and find the magnitude of the tension in each string/

=

1$

:

& 2$

2$ )in

2

B

?

M2$ Cos

4+


13/21

5roblems of equilibrium on rou!h #lanes

o

uniform ladder of weight 30 -g/wt rests with one of its ends on a smooth ertical wall and with

its other end on a rough hori!ontal floor# so that it is inclined at angle %+ to the hori!ontal# if a

man whose weight is ,0 -g/wt is ascending the ladder slowly# and the ladder was about to sli"

3when the man ascended of the length of the ladder# find the coefficient of friction between

%

the the floor and the ladder# and if the man wanted to continue to ascend the to" of the ladder#

then find the least hori!ontal force which acts at the lower end at this case/( )

( )

:ote 1 when a body is "laced on a "lane# then the


2 when a body is "laced on a "lane# then the

reaction is "er"endicular to the "l

smooth

rou!h

( )

&

st

1

1 1

1

ane and & a""ears

let the length of the ladder be %

31 case when the man ascended the length of the ladder(

%

% 3 2 2M 0 ( ;* ,0 30 0

2 2 2

%* 2%0 * ,0 -g/wt

X 0( ( *

= + + =

= == =

( )

( )

&

2

&&

nd

1

1 1

& ,0 -g/wt

Y 0( ( * ,0 30


14/21

%

2

( )

uniform ladder of length % m rests on a hori!ontal rough floor and a smooth ertical wall/ If

the ladder is about to slide under the action of a weight when it is inclined to the hori!ontal

at an an

o

3gle of tangent # find the coefficient of friction# and if the ladder was inclined to the

%hori!ontal at an angle %+ # find the farthest distance on the ladder a man whose weight is % times

the ladder's weight can ascend before the ladder is about to slide/

( )

( )

:ote 1 when a body is "laced on a "lane# then the reaction is "er"endicular to the "lane



smooth

rou!h

( )

( )

&

st

1

1 1

ane and & a""ears

C 3 C 12)in C m

% + % +

8 % 8 9nd Cos 8 m

2 + 2 +1 case If the ladder is about to slide under weight (

12 9 +M 0 ( ;* 0

+ + %

2 3* 2 * 1

3

X 0(

= = =

= = =

= + =

= =

=

Q

Q

( )

( )

( )

( ) ( )

( ) ( )

1 &

2

& 2

nd o

1

1 1

1 & 2

&

2( * &

3

Y 0( ( *

2 2nd & *

3 3

2 case when %+ (

% 2M 0 ( ;* % 0 2

2 2 2

1 %* % 2 % * 12

1X 0( ( * & nd Y 0( ( * +

2

and & *

= =

= =

= = =

= = + + =

= + = +

= = = + = =

= +

Q

Q ( )1 2 1 10 14

+ m2 3 2 3 ,

= + = =

2

9

+

C

12

+

1*

2*

&&

2

*ough

)mooth

2

2

C

%

2

1*o

%+

2*

&&

,0 %

2

2

2

7

2

&

%

2

8

%

+3

%

o%+



E6am#le 7(8

Ans%er



15/21

uniform ladder of length + m and weight 20 -g/wt rests with its end on a smooth ertical

wall and with its other end on a rough hori!ontal floor# the coefficient of friction between the

ground a1

nd the ladder is / If the "oint is 3 m away from the wall# "roe that the ladder can%

not be in equilibrium in this case# and then find the magnitude of the least weight could be "laced

at to "ree1

nt sliding# gien that the coeficient of friction between this weight and the ground is /+

( )

( )

:ote 1 when a body is "laced on a "lane# then the




smooth

rou!h

( ) ( )

( )

( )1

1

1

&

st 2

2 2

2 &

1

& 1

& 1

ane and & a""ears

let the length of the ladder be %

1 case M 0 ( 20 1/+ * % 0

%* 30 * 4/+ -g/wt

X 0( ( * & 4/+ -g/wt

Y 0( ( * 20 -g/wt

1)o & 4/+ and * 20 + -g/wt

%$hen & * $

= =

= =

= = =

= =

= = =

>

Q

1

1

nd

& 1

& 1

hen the ladder is not in equilibrium

2 case to get the least weight to "reent the ladder from

sliding & must be equal to *

& * + -g/wt

nd the wei

= =

Q

( ) ( )( )

( )

2

1 2 2 2

&

3 3 3

3 & & & &

2 2 2 2

1 2

ght has another friction &

M 0 ( ;* % 20 1/+ 0 %* 30 * 4/+ -g/wt

X 0( ( * & & 0 4/+ + & & 2/+ -g/wt

1 * 2/+ * 2/+ * 12/+ -g/wt

+

Y 0( ( * * 20 20 12/+ 20 12/+ -g/wt

= + = = == = =

= = =

= + = + + = + =

20

C

%

2*

1*

1&&

2/+

2/+

*ough

)mooth

3m

3 m

20

C

%

3*

2*

2&&

2/+

2/+

1&&

1*



E6am#le 7(98

Ans%er



16/21

uniform inclined ladder rests with the end on a rough hori!ontal floor# and the other end

1on a rough ertical wall# if the coefficient of the friction between the ladder and the floor is # a

2nd

1

that between the ladder and the wall is # and the ladder was about to slide# find the tangent of the%

angle of the inclination of the ladder to the hori!ontal/

( ) ( ) ( )

( )

=

+ = + =

2

2

&

st

& 2

& 2

2 2

:ote when a body is "laced on a "lane# then the

reaction is "er"endicular to the "lane and & a""ears

1 case M 0 (

;& 2Cos * 2 )in Cos 0 ;2& Cos 2* )in Cos 0

;2 * Cos

rou!h

( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

+ =

+ =

+ =

= = = =

= + = + = + =

=

+

1

2

2

2 2

2 2

2 & 1 1 1

& 1 2 2 1 2 1

1 2

2 2

2* )in Cos 0

1;2 * Cos 2* )in Cos 0 Cos

%

1 * 2* $an 1

2

1X 0( ( * & * * 22

1Y 0( ( & * * * * * 3

%

&rom 2 ( * 2* %

1$hen substitute % in 3 ( * 2*

%

( )

( ) ( ) ( )

= = =

+ = + =

= = =

2 2

< % * * +

%

chapter 4 general equilibrium

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