CHAPTER 3
PROBLEM 3.1
A crate of mass 80 kg is held in the position shown. Determine
(a) the moment produced by the weight W of the crate about E,
(b) the smallest force applied at B that creates a moment of equal
magnitude and opposite sense about E.
SOLUTION
(a) By definition,
2
80 kg(9.81 m/s ) 784.8 NW mg= = =
We have
: (784.8 N)(0.25 m)
EE
MMΣ=
196.2 N m
E= ⋅M
(b) For the force at B to be the smallest, resulting in a moment (ME) about E, the line of action of force
PROBLEM 3.2
A crate of mass 80 kg is held in the position shown. Determine
(a) the moment produced by the weight W of the crate about E,
(b) the smallest force applied at A that creates a moment of equal
magnitude and opposite sense about E, (c) the magnitude, sense,
and point of application on the bottom of the crate of the smallest
vertical force that creates a moment of equal magnitude and
opposite sense about E.
SOLUTION
First note. . .
2
mg (80 kg)(9.81 m/s ) 784.8 NW= = =
(a) We have
/
(0.25 m)(784.8 N) 196.2 N m
E HE
M rW= = = ⋅
or
196.2 N m
E
= ⋅M
PROBLEM 3.3
It is known that a vertical force of 200 lb is required to remove the
nail at C from the board. As the nail first starts moving, determine
(a) the moment about B of the force exerted on the nail, (b) the
magnitude of the force P that creates the same moment about B if
α = 10°, (c) the smallest force P that creates the same moment
about B.
SOLUTION
(a) We have
/
(4 in.)(200 lb)
800 lb in.
B CB N
M rF=
=
= ⋅
PROBLEM 3.4
A 300-N force is applied at A as shown. Determine
(a) the moment of the 300-N force about D, (b) the
smallest force applied at B that creates the same moment
about D.
SOLUTION
(a)
PROBLEM 3.5
A 300–N force is applied at A as shown. Determine
(a) the moment of the 300-N force about D, (b) the
magnitude and sense of the horizontal force applied at C
that creates the same moment about D, (c) the smallest
force applied at C that creates the same moment about D.
SOLUTION
(a) See Problem 3.3 for the figure and analysis leading to the determination of MD
41.7 N m
D
= ⋅M
PROBLEM 3.6
A 20-lb force is applied to the control rod AB as shown. Knowing that the length of
the rod is 9 in. and that α = 25°, determine the moment of the force about Point B by
resolving the force into horizontal and vertical components.
SOLUTION
Free–Body Diagram of Rod AB:
PROBLEM 3.7
A 20-lb force is applied to the control rod AB as shown. Knowing that the length of
the rod is 9 in. and that α = 25°, determine the moment of the force about Point B
by resolving the force into components along AB and in a direction perpendicular
to AB.
SOLUTION
Free–Body Diagram of Rod AB:
PROBLEM 3.8
A 20-lb force is applied to the control rod AB as shown. Knowing that the length
of the rod is 9 in. and that the moment of the force about B is 120 lb ∙ in. clockwise,
determine the value of α.
SOLUTION
Free–Body Diagram of Rod AB:
PROBLEM 3.9
Rod AB is held in place by the cord AC. Knowing that the tension in
the cord is 1350 N and that c = 360 mm, determine the moment about
B of the force exerted by the cord at point A by resolving that force
into horizontal and vertical components applied (a) at point A, (b) at
point C.
SOLUTION
Free–Body Diagram of Rod AB:
(a)
22
1350 N (450) (600) 750 mmF AC= = +=
450 600
cos 0.6 sin 0.8
750 750
aa
= = = =
PROBLEM 3.10
Rod AB is held in place by the cord AC. Knowing that c = 840 mm
and that the moment about B of the force exerted by the cord at point
A is 756 N∙m, determine the tension in the cord.
SOLUTION
Free–Body Diagram of Rod AB:
22
(756 N m ) (450) (1080) 1170 mm
BAC=−⋅ = + =Mk
450 1080
cos sin
1170 1170
aa
= =
PROBLEM 3.11
The tailgate of a car is supported by the hydraulic lift BC. If
the lift exerts a 125–lb force directed along its centerline on
the ball and socket at B, determine the moment of the force
about A.
SOLUTION
First note
22
(12.0 in.) (2.33 in.)
12.2241in.
CB
d= +
=
PROBLEM 3.12
The tailgate of a car is supported by the hydraulic lift BC. If
the lift exerts a 125–lb force directed along its centerline on
the ball and socket at B, determine the moment of the force
about A.
SOLUTION
First note
22
(17.2 in.) (7.62 in.)
18.8123 in.
CB
d= +
=
PROBLEM 3.13
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.
SOLUTION
Using (a):
PROBLEM 3.14
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.
SOLUTION
Using (a):
PROBLEM 3.15
Form the vector products B × C and B
′
× C, where B = B′, and use the
results obtained to prove the identity
11
sin cos sin ( ) sin ( ).
22
a β aβ aβ
= ++ −
SOLUTION
Note:
(cos sin )
(cos sin )
(cos sin )
B
B
C
ββ
ββ
aa
= +
′= −
= +
B ij
B ij
C ij
PROBLEM 3.16
The vectors P and Q are two adjacent sides of a parallelogram. Determine the area of the parallelogram
when (a) P = –8i + 4j – 4k and Q = 3i + 3j + 6k, (b) P = 7i – 6j – 3k and Q = –3i + 6j – 2k
SOLUTION
(a) We have
||A= ×PQ
where
844=−+ −P i jk
336=++Qijk
||A= ×PQ
PROBLEM 3.17
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) 2i + 3j – 6k and 5i – 8j – 6k,
(b) 4i – 4j + 3k and –3i + 7j – 5k.
SOLUTION
(a) We have
||
×
=×
AB
λAB
PROBLEM 3.18
A line passes through the points (12 m, 8 m) and (–3 m, –5 m). Determine the perpendicular distance d from
the line to the origin O of the system of coordinates.
SOLUTION
22
[12 m ( 3 m)] [8 m ( 5 m)]
394 m
AB
d= −− + −−
=