228
4–1.
If
A
,
B
,an
d
D
are g
i
ven vectors, prove t
h
e
distributive law for the vector cross product, i.e.,
.A:(B+D)=(A:B)+(A:D)
SOLUTION
Consider the three vectors; with A vertical.
Note obd is perpendicular to A.
The three vector cross products also form a closed triangle which is similar to
triangle obd.Thus from the figure,
Note also,
o¿b¿d¿
229
4–2.
Prove t
h
e tr
i
p
l
e sca
l
ar pro
d
uct
id
ent
i
ty
A # (B : C) = (A : B) # C.
SOLUTION
As shown in the figure
Thus,
Also,
230
4–3.
Given the three nonzero vectors
A
,
B
,and
C
,show that if
, the three vectors must lie in the same
plane.
A#(B:C)=0
SOLUTION
Consider,
231
Ans:
*4–4.
Determine the moment about point Aof each of the three
forces acting on the beam.
SOLUTION
= 500 lbF2
=160 lbF3
4
3
5
= 375 lbF
1
8ft6ft
0.5 ft
30˚
5ft
B
A
232
Ans:
4–5.
SOLUTION
Determine the moment about point Bof each of the three
forces acting on the beam.
= 500 lbF2
4
3
5
= 375 lbF
1
8ft6ft
0.5 ft
30˚
5ft
B
A
233
Ans:
4–6.
The crowbar is subjected to a vertical force of P
=
25 lb at the
grip, whereas it takes a force of F
=
155 lb at the claw to pull
the nail out. Find the moment of each force about point A and
determine if P is sufcient to pull out the nail. The crowbar
contacts the board at point A.
SOLUTION
20
3 in.
1.5 in.
60
O
A
F
P
14 in.
234
Ans:
4–7.
2 m3 m
4 m
60
30
F1 250 N
F2 300 N
A
D
etermine the moment of each of the three forces about
point A.
SOLUTION
The moment arm measured perpendicular to each force from point Ais
235
Ans:
*4–8.
Determine the moment of each of the three forces about point B.
SOLUTION
The forces are resolved into horizontal and vertical component as shown in Fig. a.
For F1,
2 m3 m
4 m
60
30
F1 250 N
F2 300 N
A
236
4–9.
20
2.5 ft
A
FB
FC
0.75 ft
30
B
C
25
Determine the moment of each force about the bolt located
at A.Take FB=40 lb, FC=50 lb.
SOLUTION
237
Ans:
4–10.
20
2.5 ft
A
FB
FC
0.75 ft
30
B
C
25
If and determine the resultant
moment about the bolt located at A.
FC=45 lb,FB=30 lb
SOLUTION
238
Ans:
SOLUTION
4–11.
The towline exerts a force of
P=6 kN
at the end of the
8-m-long crane boom. If u
=30°,
determine the placement
x of the hook at B so that this force creates a maximum
moment about point O. What is this moment?
1 m
O
8 m
A
B
P 6 kN
u
x
239
*4–12.
The towline exerts a force of
P=6 kN
at the end of the
8-m-long crane boom. If
x=10 m,
determine the position
u
of the boom so that this force creates a maximum moment
about point O. What is this moment?
SOLUTION
In order to produce the maximum moment about point O, P must act perpendicular
to the boom’s axis OA as shown in Fig. a. Thus,
From the geometry shown in Fig. b,
1 m
O
8 m
A
B
P 6 kN
u
x
240
SOLUTION
Force Vector And Position Vector. Referring to Fig. a,
Moment of Force F about point B.
4–13.
The 20-N horizontal force acts on the handle of the socket
wrench. What is the moment of this force about point B.
Specify the coordinate direction angles
a,
b
,
g
of the
moment axis.
O
x
z
BA
y
50 mm
200 mm
10 mm
20 N
60
Ans:
241
SOLUTION
Force Vector And Position Vector. Referring to Fig. a,
Moment of F About point O.
The magnitude of
MO
is
Thus, the coordinate direction angles of
MO
are
4–14.
The 20-N horizontal force acts on the handle of the socket
wrench. Determine the moment of this force about point O.
Specify the coordinate direction angles
a,
b
,
g
of the
moment axis.
O
x
z
BA
y
50 mm
200 mm
10 mm
20 N
60
242
Ans:
4–15.
Two men exert forces of and on the
ropes.Determine the moment of each force about A.Which
way will the pole rotate,clockwise or counterclockwise?
P
=
50 lbF
=
80 lb
A
P
F
B
C
6ft
45
12 ft
3
4
5
SOLUTION
243
Ans:
*4–16.
If the man at Bexerts a force of on his rope,
determine the magnitude of the force Fthe man at Cmust
exert to prevent the pole from rotating, i.e., so the resultant
moment about Aof both forces is zero.
P
=
30 lb
P
F
B
6ft
45
12 ft
3
4
5
SOLUTION
244
Ans:
4–17.
SOLUTION
The torque wrench ABC is used to measure the moment or
torque applied to a bolt when the bolt is located at A and a
force is applied to the handle at C. The mechanic reads the
torque on the scale at B. If an extension AO of lengthd is
used on the wrench, determine the required scale reading if
the desired torque on the bolt at O is to be M.
A
F
B
C
dl
O
M
4–18.
The tongs are used to grip the ends of the drilling pipe P.
Determine the torque (moment) that the applied force
exerts on the pipe about point Pas a function of
. Plot this moment versus for .0…u…90°uMP
u
F=150 lb
MP
6 in.
F
P
u
SOLUTION
4–19.
The tongs are used to grip the ends of the drilling pipe P.If
a torque (moment) of is needed at Pto
turn the pipe, determine the cable force Fthat must be
applied to the tongs. Set .u=30°
MP=800 lb #ft
6 in.
F
P
u
SOLUTION
247
Ans:
*4–20.
The handle of the hammer is subjected to the force of
Determine the moment of this force about the
point A.
F=20 lb.
SOLUTION
Resolving the 20-lb force into components parallel and perpendicular to the
F
A
18 in.
5in.
30