PROBLEM 3.19
Determine the moment about the origin O of the force F = 4i − 3j + 5k that acts at a Point A. Assume that
the position vector of A is (a) r = 2i + 3j − 4k, (b) r = −8i + 6j − 10k, (c) r = 8i − 6j + 5k.
SOLUTION
O
= ×M rF
(a)
PROBLEM 3.20
Determine the moment about the origin O of the force F = 2i + 3j − 4k that acts at a Point A. Assume that
the position vector of A is (a) r = 3i − 6j + 5k, (b) r = i − 4j − 2k, (c) r = 4i + 6j − 8k.
SOLUTION
O
= ×M rF
(a)
3 65
23 4
O
= −
−
ijk
M
PROBLEM 3.21
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.
SOLUTION
2 22
2 22
( 0.75 m) ( 7 m) (6 m) 9.25 m
(4.25 m) ( 7 m) (1 m) 8.25 m
BA
BC
d
d
= − +− + =
= +− + =
Have
AA
A
555 N ( 0.75 m 7 m 6 m )
9.25 m
BB
B
BA
Td
= = − −+T ijk
PROBLEM 3.22
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.
SOLUTION
First note
2 22
( 12) (4.8) ( 8)
15.2 ft
BC
d= − + +−
=
PROBLEM 3.23
A 200–N force is applied as shown to the bracket ABC.
Determine the moment of the force about A.
SOLUTION
We have
/A CA C
= ×Mr F
PROBLEM 3.24
The wire AE is stretched between the corners A and E of a
bent plate. Knowing that the tension in the wire is 435 N,
determine the moment about O of the force exerted by the
wire (a) on corner A, (b) on corner E.
SOLUTION
2 22
(0.21 m) (0.16 m) (0.12 m)
(0.21 m) ( 0.16 m) (0.12 m) 0.29 m
AE
AE
=−+
= +− + =
i jk
PROBLEM 3.25
A 6–ft–long fishing rod AB is securely anchored in the sand of a beach. After a fish takes the bait, the
resulting force in the line is 6 lb. Determine the moment about A of the force exerted by the line at B.
SOLUTION
We have
(6 lb)cos 8 5.9416 lb
xz
T= °=
Then
PROBLEM 3.26
A precast concrete wall section is temporarily held by two
cables as shown. Knowing that the tension in cable BD is 900
N, determine the moment about Point O of the force exerted by
the cable at B.
SOLUTION
where 900 N
BD
FF
BD
= =F
PROBLEM 3.27
In Prob. 3.22, determine the perpendicular distance from point A to
cable BC.
PROBLEM 3.22 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.
SOLUTION
From the solution to problem 3.22
22
(2400) (1440)
2798.9 lb ft
A
M= +
= ⋅
PROBLEM 3.28
In Prob. 3.24, determine the perpendicular distance from point
O to wire AE.
PROBLEM 3.24 The wire AE is stretched between the
corners A and E of a bent plate. Knowing that the tension in
the wire is 435 N, determine the moment about O of the force
exerted by the wire (a) on corner A, (b) on corner E.
SOLUTION
From the solution to Prob. 3.24
PROBLEM 3.29
In Prob. 3.24, determine the perpendicular distance from point
B to wire AE.
PROBLEM 3.24 The wire AE is stretched between the
corners A and E of a bent plate. Knowing that the tension in
the wire is 435 N, determine the moment about O of the force
exerted by the wire (a) on corner A, (b) on corner E.
SOLUTION
From the solution to Prob. 3.24
PROBLEM 3.30
In Prob. 3.25, determine the perpendicular distance from point A to a line drawn through points B and C.
PROBLEM 3.25 A 6–ft–long fishing rod AB is securely anchored in the sand of a beach. After a fish
takes the bait, the resulting force in the line is 6 lb. Determine the moment about A of the force exerted by
the line at B.
SOLUTION
From the solution to Prob. 3.25:
PROBLEM 3.31
In Prob. 3.25, determine the perpendicular distance from point D to a line drawn through points B and C.
PROBLEM 3.25 A 6–ft–long fishing rod AB is securely anchored in the sand of a beach. After a fish
takes the bait, the resulting force in the line is 6 lb. Determine the moment about A of the force exerted by
the line at B.
SOLUTION
6 ft
6 lb
BC
AB
T
=
=
PROBLEM 3.32
In Prob. 3.26, determine the perpendicular distance from point
O to cable BD.
PROBLEM 3.26 A precast concrete wall section is
temporarily held by two cables as shown. Knowing that the
tension in cable BD is 900 N, determine the moment about
Point O of the force exerted by the cable at B.
SOLUTION
From the solution to Prob. 3.26 we have
PROBLEM 3.33
In Prob. 3.26, determine the perpendicular distance from point C
to cable BD.
PROBLEM 3.26 A precast concrete wall section is
temporarily held by two cables as shown. Knowing that the
tension in cable BD is 900 N, determine the moment about
Point O of the force exerted by the cable at B.
SOLUTION
From the solution to Prob. 3.26 we have
PROBLEM 3.34
Determine the value of a that minimizes the
perpendicular distance from Point C to a
section of pipeline that passes through Points
A and B.
SOLUTION
Assuming a force F acts along AB,
/
| | | | ()
C AC
Fd= ×=Mr F
where
d=
perpendicular distance from C to line AB
PROBLEM 3.35
Given the vectors P = 2i + 3j – k, Q = 5i – 4j + 3k, and S = –3i + 2j – 5k, compute the scalar products
P · Q, P · S, and Q · S.
SOLUTION
(23 )(543)
(2)(5) (3)( 4) ( 1)(3)
10 12 3
⋅= +− ⋅ −+
= + − +−
=−−
PQ i j k i j k
PROBLEM 3.36
Form the scalar product B · C and use the result obtained to prove the
identity
cos (α − β) = cos α cos β + sin α sin β .
SOLUTION
cos sinBB
aa
= +B ij
(1)
PROBLEM 3.37
Three cables are used to support a container as shown.
Determine the angle formed by cables AB and AD.
SOLUTION
First note:
PROBLEM 3.38
Three cables are used to support a container as shown.
Determine the angle formed by cables AC and AD.
SOLUTION
First note: