Post on 21-Feb-2018
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.
7/24/2019 Solution Individual Task
2/14
2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
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
7/24/2019 Solution Individual Task
3/14
2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
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!
7/24/2019 Solution Individual Task
4/14
2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
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
7/24/2019 Solution Individual Task
5/14
2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
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
7/24/2019 Solution Individual Task
6/14
2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
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
7/24/2019 Solution Individual Task
7/14 2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
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
7/24/2019 Solution Individual Task
8/14 2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
2 - 8
Free-Body iagrams
Space !iagram( A setch showing
the physical conditions of the
problem.
Free"#o$y !iagram( A setch showing
only the forces on the selected particle.
E
7/24/2019 Solution Individual Task
9/14 2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
2 - 9
Sample Problem 2.!
In a ship0unloading operation, a
5!!0" automobile is supported bya cable. A rope is tied to the cable
and pulled to center the automobile
over its intended position. -hat is
the tension in the rope;
#$%&'I$"(
8onstruct a free0body diagram for the
particle at the
7/24/2019 Solution Individual Task
10/14 2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
2 - 10
Sample Problem 2.!
#$%&'I$"(
8onstruct a free0body diagram for the
particle atA.
Apply the conditions for e6uilibrium.
#olve for the unnown force magnitudes.
=
=
59sin
"5!!
2sin2!sinA%A# TT
"53!=A#T
"44=A%T
V t M h i f E i St tiE
7/24/2019 Solution Individual Task
11/14 2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
2 - 11
Sample Problem 2."
It is desired to determine the drag force
at a given speed on a prototype sailboat
hull. A model is placed in a test
channel and three cables are used to
align its bow on the channel centerline.
*or a given speed, the tension is 4! "
in cableA#and 1! " in cableA&.
7etermine the drag force eerted on the
hull and the tension in cableA%.
#$%&'I$"(
8hoosing the hull as the free body,
draw a free0body diagram.
press the condition for e6uilibrium
for the hull by writing that the sum of
all forces must be :ero.
esolve the vector e6uilibrium
e6uation into two component
e6uations. #olve for the two unnown
cable tensions.
V t M h i f E i St tiE
7/24/2019 Solution Individual Task
12/14 2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
2 - 12
Sample Problem 2."
#$%&'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
V t M h i f E i St tiE
7/24/2019 Solution Individual Task
13/14 2007 The McGraw-Hill Companies, Inc. All rights reserve.
Vector Mechanics for Engineers: StaticsEighth
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!
++
++=
=
=
=
+=+=
+=
+=
V t M h i f E i St tiE
7/24/2019 Solution Individual Task
14/14 2007 The McGraw Hill Companies Inc All rights reserve
Vector Mechanics for Engineers: StaticsEighth
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