--
$.!$ and !."# It is known that the connecting rod AB exerts on the crank BC
a 2.5-kN force directed down and to the left along the centerline of
AB. Determine the moment of the force about C.
A
A
144 mm
B
88 mm
C
56 mm
56 mm
B
C
I
42 mm
42 mm
Fig. P$.!$
Fig. P!."#
$.!% Form the vector product P1!" P2 and use the result obtained to prove
the identity
sin (!1 # !2) = sin !1 cos !2 # cos !1 sin !2
y
P1
P2
θ1
θ2
x
Fig. P$.!%
!."$ The vectors P and Q are two adjacent sides of a parallelogram.
Determine the area of the parallelogram when (a) P = –2i + 2j – 6k
and Q = 8i + 2j + 4k, (b) P = 3i – 9j – 7k and Q = –4i + 2j – 5k.
$.!& A plane contains the vectors A and B. Determine the unit vector normal to the plane when A and B are equal to, respectively, (a) 7i + 8j
– 2k and 9i – 4j – 5k, (b) 6i – 3j + 9k and –5i + 4j – 3k.
$.!' A line passes through the points (–4 m, –3 m) and (2 m, 7 m).
Determine the perpendicular distance d from the line to the origin O
of the system of coordinates.
!."% Determine the moment about the origin O of the force F = –2i – 3j +
5k that acts at a point A. Assume that the position vector of A is
(a) r = i + j + k, (b) r = 4i + 6j – 10k, (c) r = 4i + 3j – 5k.
!"#
~
$.#" Determine the moment about the origin O of the force F = 4i + 10j +
6k that acts at a point A. Assume that the position vector of A is
(a) r = 2i – 3j + 4k, (b) r = 2i + 6j + 3k, (c) r = 2i + 5j + 6k.
y
B
!.&" Before the trunk of a large tree is felled, cables AB and BC are attached
as shown. Knowing that the tensions in cables AB and BC are 555 N
and 660 N, respectively, determine the moment about O of the resultant force exerted on the tree by the cables at B.
7m
~
$.## The 12-ft boom AB has a fixed end A. A steel cable is stretched from
the free end B of the boom to a point C located on the vertical wall.
If the tension in the cable is 380 lb, determine the moment about A of
the force exerted by the cable at B.
4.25 m
6m
O
A
1m
C
0.75 m
x
z
y
Fig. P!.&"
C
8 ft
4.8 ft
A
z
B
12 ft
x
Fig. P$.##
$.#$ A 200-N force is applied as shown to the bracket ABC. Determine the
moment of the force about A.
y
200 N
30°
60 mm
B
25 mm
60°
C
y
B
x
A
A
z
P
50 mm
O
x
225 mm
300 mm
~
Fig. P$.#$
!.&# A force P of magnitude 200 N acts along the diagonal BC of the bent
plate shown. Determine the moment of P about point E.
E
C
200 mm
D
z
Fig. P!.&#
!"$
$.)* To loosen a frozen valve, a force F with a magnitude of 70 lb is applied
to the handle of the valve. Knowing that ! = 25°, Mx = –61 lb·ft, and
Mz = –43 lb·ft, determine " and d.!
F
ϕ
θ
A
4 in.
d
11 in.
y
B
x
z
Fig. P$.)* and P!.'(
!.'( When a force F is applied to the handle of the valve shown, its moments
about the x and z axes are Mx = #77 lb·ft and Mz = #81 lb·ft, respectively. For d = 27 in., determine the moment My of F about the y axis.
0.61 " 1.00-m lid ABCD of a storage bin is hinged along side AB
-$.%! The
and is held open by looping cord DEC over a frictionless hook at E. If
the tension in the cord is 66 N, determine the moment about each of
the coordinate axes of the force exerted by the cord at D.!
y
0.3 m
E
0.7 m
A
D
0.11 m
z
0.71 m
C
B
x
Fig. P$.%! and P$.%#
$.%# The 0.61 " 1.00-m lid ABCD of a storage bin is hinged along side AB
and is held open by looping cord DEC over a frictionless hook at E. If
the tension in the cord is 66 N, determine the moment about each of
the coordinate axes of the force exerted by the cord at C.!
!!(
$.%$ A farmer uses cables and winch pullers B and E to plumb one side
of a small barn. If it is known that the sum of the moments about
the x axis of the forces exerted by the cables on the barn at points
A and D is equal to 4728 lb·ft, determine the magnitude of TDE
when TAB = 255 lb.!
y
A
!.'# Solve Prob. 3.53 when the tension in cable AB is 306 lb.
$.%% A force P of magnitude 520 lb acts on the frame shown at point E.
Determine the moment of P about a line joining points O and D.!
12 ft
B
C
14 ft
z
1 ft
E
y
F
10 in.
1.5 ft
30 in.
A
C
x
D
O
P
7.5 in.
z
12 ft
Fig. P$.%$
B
E
7.5 in.
D
F
G
10 in. x
10 in.
H
Fig. P$.%% and P!.'$
!.'$ A force P acts on the frame shown at point E. Knowing that the absolute value of the moment of P about a line joining points F and B is
300 lb·ft, determine the magnitude of the force P.
-
$.%& The frame ACD is hinged at A and D and is supported by a cable that
passes through a ring at B and is attached to hooks at G and H. Knowing
that the tension in the cable is 450 N, determine the moment about the
diagonal AD of the force exerted on the frame by portion BH of the cable.!
y
0.35 m
0.875 m
G
H
O
0.925 m
0.75 m
D
A
z
0.5 m
0.75 m
B
x
C
0.5 m
P
Fig. P$.%&
$.%' In Prob. 3.57, determine the moment about the diagonal AD of the
force exerted on the frame by portion BG of the cable.!
!!&
~
$.%* The triangular plate ABC is supported by ball-and-socket joints at B
and D and is held in the position shown by cables AE and CF. If the
force exerted by cable AE at A is 55 N, determine the moment of that
force about the line joining points D and B.!
y
0.4 m
0.2 m
C
D
0.9 m
$.(" The triangular plate ABC is supported by ball-and-socket joints at B
and D and is held in the position shown by cables AE and CF. If the
force exerted by cable CF at C is 33 N, determine the moment of that
force about the line joining points D and B.!
0.7 m
0.6 m
A
0.6 m
z
F
E
0.9 m
0.3 m
B
0.35 m
0.4 m
0.6 m
$.(! A regular tetrahedron has six edges of length a. A force P is directed
as shown along edge BC. Determine the moment of P about edge
OA.!
x
y
Fig. P$.%* and P$.("
A
O
C
z
B
P
x
Fig. P$.(! and P!.$&
!.$& A regular tetrahedron has six edges of length a. (a) Show that two
opposite edges, such as OA and BC, are perpendicular to each other.
(b) Use this property and the result obtained in Prob. 3.61 to determine the perpendicular distance between edges OA and BC.
$.($ Two forces F1 and F2 in space have the same magnitude F. Prove that
the moment of F1 about the line of action of F2 is equal to the moment
of F2 about the line of action of F1.
*$.() In Prob. 3.55, determine the perpendicular distance between a line
joining points O and D and the line of action of P.!
*$.(% In Prob. 3.56, determine the perpendicular distance between a line
joining points F and B and the line of action of P.!
*!.$$ In Prob. 3.57, determine the perpendicular distance between portion
BH of the cable and the diagonal AD.
*$.(& In Prob. 3.58, determine the perpendicular distance between portion
BG of the cable and the diagonal AD.!
*$.(' In Prob. 3.59, determine the perpendicular distance between cable AE
and the line joining points D and B.!
*!.$% In Prob. 3.60, determine the perpendicular distance between cable CF
and the line joining points D and B.
!!'