Mechanical Engineering Chapter 2 Homework Ab Determine The length The Cable And Express

122

2–101.

The two mooring cables exert forces on the stern of a ship

as shown. Represent each force as a Cartesian vector and

determine the magnitude and coordinate direction angles

y

z

50 ft

40 ft

C

10 ft

F

B

= 150 lb

F

A

= 200 lb

SOLUTION

Unit Vector:

Force Vector:

Resultant Force:

The magnitude of is

The coordinate direction angles of are

FR

of the resultant.

2–102.

The engine of the lightweight plane is supported by struts that

are connected to the space truss that makes up the structure

of the plane.The anticipated loading in two of the struts is

shown. Express each of these forces as a Cartesian vector.

SOLUTION

3ft

2.5ft

0.5ft

B

C

A

D

F

2

600lb

F

1

400lb

z

124

2–103.

Determine the magnitude and coordinates on angles of the

resultant force.

SOLUTION

z

C

A

40 lb

4 ft

2 ft

20 lb

Ans:

125

*2–104.

SOLUTION

Force Vectors: The unit vectors uA,uB,uCand uDof FA,FB,FCand FDmust be

determined first. From Fig. a,

Thus, the force vectors FA,FB,FCand FDare given by

If the force in each cable tied to the bin is 70 lb, determine

the magnitude and coordinate direction angles of the

resultant force.

z

C

E

D

A

6ft

F

C

F

D

F

A

F

B

Resultant Force:

The magnitude of FRis

The coordinate direction angles of FRare

,,

,

126

2–105.

If the resultant of the four forces is ,

determine the tension developed in each cable.Due to

symmetry,the tension in the four cables is the same.

FR=

5

–360k

6

lb

SOLUTION

Force Vectors: The unit vectors uA,uB,uCand uDof FA,FB,FCand FDmust be

determined first. From Fig. a,

Since the magnitudes of FA,FB,FCand FDare the same and denoted as F, the four

vectors or forces can be written as

Resultant Force: The vector addition of FA,FB,FCand FDis equal to FR.Thus,

z

C

E

D

A

6ft

F

C

F

D

F

A

F

B

,,

,

127

2–106.

Express the force F in Cartesian vector form if it acts at the

midpoint B of the rod.

SOLUTION

x

z

4 m

OF  600 N

B

A

C

128

2–107.

Express force F in Cartesian vector form if point B is

located 3 m along the rod end C.

3 m

x

z

4 m

OF  600 N

B

A

C

SOLUTION

129

*2–108.

SOLUTION

FA=60 (4 cos 30° i–4 sin 30° j–6k)

2(4 cos 30°)2+(–4 sin 30°)2+(–6)2

The chandelier is supported by three chains which are

concurrent at point O. If the force in each chain has a

magnitude of 60 lb, express each force as a Cartesian vector

and determine the magnitude and coordinate direction

angles of the resultant force.

z

B

6ft

O

F

A

F

B

F

C

Ans:

130

2–109.

The chandelier is supported by three chains which are

concurrent at point O.If the resultant force at Ohas a

magnitude of 130 lb and is directed along the negative zaxis,

determine the force in each chain.

SOLUTION

z

O

F

B

F

C

x

Ans:

131

2–110.

The window is held open by chain AB. Determine the

length of the chain, and express the 50-lb force acting at A

along the chain as a Cartesian vector and determine its

coordinate direction angles.

SOLUTION

Unit Vector: The coordinates of point Aare

Then

Force Vector:

5ft

12 ft

8ft

=50lb

A

B

5ft

z

5ft

F

Ans:

132

2–111.

The window is held open by cable AB. Determine the

length of the cable and express the 30-N force acting at A

along the cable as a Cartesian vector.

y

250 mm

B

150 mm

z

SOLUTION

133

*2–112.

SOLUTION

Given the three vectors A,B, and D, show that

.A#(B+D)=(A#B)+(A#D)

B

A

z

2 m

134

2–113.

SOLUTION

Vector Dot Product: The magnitude of the component of Fparallel to segment DE

of the pipe assembly is

Determine the magnitudes of the components of

acting along and perpendicular to segment DE

of the pipe assembly.

F=600 N

B

A

z

2m

Ans:

135

Ans:

2–114.

Determine the angle

u

between the two cables.

SOLUTION

Unit Vectors. Here, the coordinates of points A, B and C are

A(2, –3, 3) m

,

B(0, 3, 0)

and

C(–2, 3, 4) m

respectively. Thus, the unit vectors along AB and AC

are

z

x

A

B

C

y

2 m

3 m

2 m

4 m

F1  70 N

F2  40 N

u

136

Ans:

2–115.

Determine the magnitude of the projection of the force F1

along cable AC.

SOLUTION

Unit Vectors. Here, the coordinates of points A, B and C are

A(2, –3, 3)m

,

B(0, 3, 0)

and

C(–2, 3, 4) m

respectively. Thus, the unit vectors along AB and AC are

F1

Projected Component of

F1

. Along AC, it is

z

x

A

B

C

y

2 m

3 m

2 m

4 m

F1  70 N

F2  40 N

u

137

*2–116.

Determine the angle between the yaxis of the pole and

the wire AB.

u

SOLUTION

Position Vector:

The magnitudes of the position vectors are

The Angles Between Two Vectors The dot product of two vectors must be

determined first.

U:

$

\

]

[IW

IW

IW

e

138

2–117.

Determine the magnitudes of the projected components of

the force F = [60i + 12j – 40k] N along the cables AB

andAC.

1.5 m

3 m

1 m

0.75 m

1 m

B

C

A

z

y

x

u

SOLUTION

Ans:

139

2–118.

Determine the angle

u

between cables AB and AC.

1.5 m

3 m

1 m

0.75 m

1 m

B

C

A

z

y

x

u

SOLUTION

140

2–119.

SOLUTION

Aforce of lb acts at the end of the pipe.

Determine the magnitudes of the components and

which are directed along the pipe’saxis and perpendicular

to it.

F2

F1

F=5–40k6

z

y

3ft

5ft

O

Ans:

141

*2–120.

Two cables exert forces on the pipe. Determine the

magnitude of the projected component of along the line

of action of .F2

F1

SOLUTION

Force Vector:

60

z

60

x

F2 25 lb

u