PROBLEM 3.39
Knowing that the tension in cable AC is 280 lb, determine
(a) the angle between cable AC and the boom AB, (b) the
projection on AB of the force exerted by cable AC at point
A.
SOLUTION
(a) First note
PROBLEM 3.40
Knowing that the tension in cable AD is 180 lb, determine
(a) the angle between cable AD and the boom AB, (b) the
projection on AB of the force exerted by cable AD at point
A.
SOLUTION
(a) First note
PROBLEM 3.41
Ropes AB and BC are two of the ropes used to
support a tent. The two ropes are attached to a
stake at B. If the tension in rope AB is 540 N,
determine (a) the angle between rope AB and
the stake, (b) the projection on the stake of the
force exerted by rope AB at Point B.
SOLUTION
First note:
22 2
222
( 3) (3) ( 1.5) 4.5 m
( 0.08) (0.38) (0.16) 0.42 m
BA
BD
= − + +− =
=−+ + =
PROBLEM 3.42
Ropes AB and BC are two of the ropes used to
support a tent. The two ropes are attached to a
stake at B. If the tension in rope BC is 490 N,
determine (a) the angle between rope BC and
the stake, (b) the projection on the stake of the
force exerted by rope BC at Point B.
SOLUTION
First note:
22 2
222
(1) (3) ( 1.5) 3.5 m
( 0.08) (0.38) (0.16) 0.42 m
BC
BD
= + +− =
=−+ + =
PROBLEM 3.43
The 20-in. tube AB can slide along a horizontal rod. The ends A
and B of the tube are connected by elastic cords to the fixed point
C. For the position corresponding to x = 11 in., determine the
angle formed by the two cords (a) using Eq. (3.32), (b) applying
the law of cosines to triangle ABC.
SOLUTION
(a) Using Eq. (3.32):
PROBLEM 3.44
Solve Prob. 3.43 for the position corresponding to x = 4 in.
PROBLEM 3.43 The 20-in. tube AB can slide along a horizontal
rod. The ends A and B of the tube are connected by elastic cords
to the fixed point C. For the position corresponding to x = 11 in.,
determine the angle formed by the two cords (a) using Eq. (3.32),
(b) applying the law of cosines to triangle ABC.
SOLUTION
(a) Using Eq. (3.32):
PROBLEM 3.45
Determine the volume of the parallelepiped of Fig. 3.25 when
(a) P = 4i − 3j + 2k, Q = −2i − 5j + k, and S = 7i + j − k,
(b) P = 5i − j + 6k, Q = 2i + 3j + k, and S = −3i − 2j + 4k.
SOLUTION
Volume of a parallelepiped is found using the mixed triple product.
(a)
PROBLEM 3.46
Given the vectors P = 3i – j + k, Q = 4i + Qyj – 2k, and S = 2i – 2j + 2k, determine the
value of Qy for which the three vectors are coplanar.
SOLUTION
If P, Q, and S are coplanar, then P must be perpendicular to
( ).×QS
PROBLEM 3.47
A crane is oriented so that the end of the 25–m boom AO lies in
the yz plane. At the instant shown, the tension in cable AB is
4 kN. Determine the moment about each of the coordinate axes
of the force exerted on A by cable AB.
SOLUTION
22
22
()( )
(25 m) (15 m)
20 m
OC OA AC= −
= −
=
PROBLEM 3.48
The 25–m crane boom AO lies in the yz plane. Determine the
maximum permissible tension in cable AB if the absolute value
of moments about the coordinate axes of the force exerted on A
by cable AB must be
|Mx| ≤ 60 kN∙m, |My| ≤ 12 kN∙m, |Mz| ≤ 8 kN∙m
22
22
()( )
(25 m) (15 m)
20 m
OC OA AC= −
= −
=
PROBLEM 3.49
To loosen a frozen valve, a force F of magnitude 70 lb is
applied to the handle of the valve. Knowing that
25 ,
θ
= °
Mx
61 lb ft,=−⋅
and
43 lb ft,
z
M=−⋅
determine
φ
and d.
SOLUTION
We have
/
:
O AO O
Σ ×=M r FM
where
/
(4 in.) (11in.) ( )
(cos cos sin cos sin )
AO
d
F
θφ θ θφ
=−+ −
= −+
r i jk
F ij k
φ
PROBLEM 3.50
When a force F is applied to the handle of the valve
shown, its moments about the x and z axes are,
respectively,
77 lb ft
x
M=−⋅
and
81 lb ft.
z
M=−⋅
For
27d=
in., determine the moment My of F about the y axis.
SOLUTION
We have
/
:
O AO O
Σ ×=M r FM
Where
/
(4 in.) (11in.) (27 in.)
(cos cos sin cos sin )
AO
F
θφ θ θφ
=−+ −
= −+
r ijk
F ij k
PROBLEM 3.50 (Continued)
PROBLEM 3.51
To lift a heavy crate, a man uses a block and tackle attached to the
bottom of an I-beam at hook B. Knowing that the moments about
the y and the z axes of the force exerted at B by portion AB of the
rope are, respectively, 120 N ⋅ m and −460 N ⋅ m, determine the
distance a.
SOLUTION
First note
(2.2 m) (3.2 m) ( m)BA a=−−i jk
Now
/D A D BA
= ×Mr T
where
/(2.2 m) (1.6 m)
(2.2 3.2 ) (N)
AD
BA
BA
BA
Ta
d
= +
= −−
r ij
T i jk
PROBLEM 3.52
To lift a heavy crate, a man uses a block and tackle attached to the
bottom of an I-beam at hook B. Knowing that the man applies a
195-N force to end A of the rope and that the moment of that force
about the y axis is 132 N ⋅ m, determine the distance a.
SOLUTION
First note
2 22
2
(2.2) ( 3.2) ( )
15.08 m
BA
da
a
= +− +−
= +
and
195 N (2.2 3.2 )
BA
BA
a
d
= −−T i jk
PROBLEM 3.53
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.
SOLUTION
The moment about the x–axis due to the two cable forces can be found using the z components of each
force acting at their intersection with the xy plane (A and D). The x components of the forces are parallel
to the x–axis, and the y components of the forces intersect the x–axis. Therefore, neither the x or y
components produce a moment about the x–axis.
We have
: ( )( ) ( )( )
x AB z A DE z D x
MT y T yMΣ +=
PROBLEM 3.54
Solve Problem 3.53 when the tension in cable AB is 306
lb.
PROBLEM 3.53 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.
SOLUTION
The moment about the x–axis due to the two cable forces can be found using the z components of each
force acting at the intersection with the xy plane (A and D). The x components of the forces are parallel to
the x–axis, and the y components of the forces intersect the x–axis. Therefore, neither the x or y
components produce a moment about the x–axis.
We have
: ( )( ) ( )( )
x AB z A DE z D x
MT y T yMΣ +=
PROBLEM 3.55
The 23-in. vertical rod CD is welded to the midpoint C of
the 50-in. rod AB. Determine the moment about AB of the
235-lb force P.
SOLUTION
(32 in.) (30 in.) (24 in.)AB =−−i jk
222
(32) ( 30) ( 24) 50 in.AB = +− +− =
PROBLEM 3.56
The 23-in. vertical rod CD is welded to the midpoint C of
the 50-in. rod AB. Determine the moment about AB of the
174-lb force Q.
SOLUTION
(32 in.) (30 in.) (24 in.)AB =−−i jk
222
(32) ( 30) ( 24) 50 in.AB = +− +− =
PROBLEM 3.57
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.
SOLUTION
/
()
AD AD B A BH
M=⋅×rTλ