Solution Individual Task

7/24/2019 Solution Individual Task

1/14

2007 The McGraw-Hill Companies, Inc. All rights reserve.

Vector Mechanics for Engineers: StaticsEighth

2 - 1

Sample Problem 2.2

a) the tension in each of the ropes

for = 45o,

b) the value of for which the

tension in rope 2 is a minimum.

A barge is pulled by two tugboats.

If the resultant of the forces

eerted by the tugboats is 5!!! "

directed along the ais of the

barge, determine

#$%&'I$"(

*ind a graphical solution by applying the+arallelogram ule for vector addition. 'he

parallelogram has sides in the directions of

the two ropes and a diagonal in the direction

of the barge ais and length proportional to

5!!! ".

'he angle for minimum tension in rope 2 is

determined by applying the 'riangle ule

and observing the effect of variations in .

*ind a trigonometric solution by applying

the 'riangle ule for vector addition. -ith

the magnitude and direction of the resultant

nown and the directions of the other two

sides parallel to the ropes given, apply the%aw of #ines to find the rope tensions.


2/14



2 - 2

Sample Problem 2.2

/raphical solution 0 +arallelogram ule

with nown resultant direction and

magnitude, nown directions for sides."21!!"3!! 2 == TT

'rigonometric solution 0 'riangle ule

with %aw of #ines

=

=

!5sin

"5!!!

!sin45sin2 TT

"25!"11! 2 == TT


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2 - 3

Sample Problem 2.2

'he angle for minimum tension in rope 2 is

determined by applying the 'riangle uleand observing the effect of variations in .

'he minimum tension in rope 2 occurs when

T1and T

2are perpendicular.

( ) = !sin"5!!!2T "25!!2 =T

( ) = !cos"5!!!T "4!=T

= !! = 1!


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2 - 4

Addition of Forces by Summing Components

SQPR ++=

-ish to find the resultant of or more

concurrent forces,

( ) ( )jSQPiSQPjSiSjQiQjPiPjRiR

yyyxxx

yxyxyxyx

+++++=

+++++=+

esolve each force into rectangular components

=++=

x

xxxxF

SQPR

'he scalar components of the resultant are

e6ual to the sum of the corresponding scalar

components of the given forces.

=++=

y

yyyyF

SQPR

x

yyx

R

RRRR 22 tan=+=

'o find the resultant magnitude and direction,

E


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2 - 5

Sample Problem 2.3

*our forces act on boltAas shown.

7etermine the resultant of the force

on the bolt.

#$%&'I$"(

esolve each force into rectangular

components.

8alculate the magnitude and direction

of the resultant.

7etermine the components of the

resultant by adding the corresponding

force components.

E


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2 - 6

Sample Problem 2.3#$%&'I$"(

esolve each force into rectangularcomponents.

.251.1!!!.!!!

2.354.239!

!.35.25!

4

2

+

+

++

FF

F

F

compycompxmagforce

22 .4. +=R "1.=R

8alculate the magnitude and direction.

".

".4tan = = .4

7etermine the components of the resultant by

adding the corresponding force components.

.+=xR .4+=yR

E


7/14 2007 The McGraw-Hill Companies, Inc. All rights reserve.


2 - 7

Equilibrium of a Particle -hen the resultant of all forces acting on a particle is :ero, the particle is

in equilibrium.

+article acted upon bytwo forces(

0 e6ual magnitude

0 same line of action

0 opposite sense

+article acted upon by three or more forces(0 graphical solution yields a closed polygon

0 algebraic solution

!!

!

==

==

yx FF

FR

Newtons First aw( If the resultant force on a particle is :ero, the particle will

remain at rest or will continue at constant speed in a straight line.

E




2 - 8

Free-Body iagrams

Space !iagram( A setch showing

the physical conditions of the

problem.

Free"#o$y !iagram( A setch showing

only the forces on the selected particle.

E




2 - 9

Sample Problem 2.!

In a ship0unloading operation, a

5!!0" automobile is supported bya cable. A rope is tied to the cable

and pulled to center the automobile

over its intended position. -hat is

the tension in the rope;

#$%&'I$"(

8onstruct a free0body diagram for the

particle at the




2 - 10

Sample Problem 2.!

#$%&'I$"(

8onstruct a free0body diagram for the

particle atA.

Apply the conditions for e6uilibrium.

#olve for the unnown force magnitudes.

=

=

59sin

"5!!

2sin2!sinA%A# TT

"53!=A#T

"44=A%T

V t M h i f E i St tiE




2 - 11

Sample Problem 2."

It is desired to determine the drag force

at a given speed on a prototype sailboat

hull. A model is placed in a test

channel and three cables are used to

align its bow on the channel centerline.

*or a given speed, the tension is 4! "

in cableA#and 1! " in cableA&.

7etermine the drag force eerted on the

hull and the tension in cableA%.

#$%&'I$"(

8hoosing the hull as the free body,

draw a free0body diagram.

press the condition for e6uilibrium

for the hull by writing that the sum of

all forces must be :ero.

esolve the vector e6uilibrium

e6uation into two component

e6uations. #olve for the two unnown

cable tensions.





2 - 12

Sample Problem 2."

#$%&'I$"(

8hoosing the hull as the free body, draw a

free0body diagram.

=

==

25.1!

35.m4

m3tan

=

==

51.2!

35.!m4

m.5tan

press the condition for e6uilibrium

for the hull by writing that the sum of

all forces must be :ero.

!=+++= !A&A%A# FTTTR





2 - 13

Sample Problem 2."

esolve the vector e6uilibrium e6uation into

two component e6uations. #olve for the twounnown cable tensions.

( ) ( )

( ) ( )

( )

( )

( )jT

iFT

R

iFF

iT

jTiTjTiTT

ji

jiT

A%

!A%

!!

A%A%

A%A%A%

A#

1!1.!94.

52.!3.4

!

"!1

1.!52.!51.2!cos51.2!sin

"94."3.4

21.1!cos"4!21.1!sin"4!

++

++=

=

=

=

+=+=

+=

+=



14/14 2007 The McGraw Hill Companies Inc All rights reserve


2 - 14

Sample Problem 2."

( )

( ) jT

iFT

R

A%

!A%

1!1.!94.

52.!3.4

!

++

++=

=

'his e6uation is satisfied only if each component

of the resultant is e6ual to :ero.

( )

( ) 1!1.!94.!!52.!3.4!!

+==

++==

A%y

!A%x

TF

FTF

"11.".42

+=+=

!

A%FT

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