268
Ans:
4–41.
Determine the smallest force Fthat must be applied along
the rope in order to cause the curved rod, which has a radius
of 5 ft, to fail at the support C.This requires a moment of
to be developed at C.
SOLUTION
Position Vector and Force Vector:
Moment of Force FAB About Point C:
M=80 lb #ft
5 ft
5 ft
60
z
x
y
6 ft
60 lb
A
C
B
7 ft
269
Ans:
4–42.
A 20-N horizontal force is applied perpendicular to the
handle of the socket wrench. Determine the magnitude and
the coordinate direction angles of the moment created by
this force about point O.
SOLUTION
MO=rA*F=3ijk
0.05176 0.1932 0.075
200 mm
75 mm
20 N
A
z
270
Ans:
4–43.
SOLUTION
Position Vector And Force Vector:
Moment of Force FAbout Point A: Applying Eq. 4–7, we have
T
h
e p
i
pe assem
bl
y
i
s su
bj
ecte
d
to t
h
e 80-N force.Determ
i
ne
the moment of this force about point A.
400 mm
y
300 mm
200 mm
x
z
B
A
271
*4–44.
T
h
e p
i
pe assem
bl
y
i
s su
bj
ecte
d
to t
h
e 80-N force.Determ
i
ne
the moment of this force about point B.
SOLUTION
Position Vector And Force Vector:
Moment of Force FAbout Point B: Applying Eq. 4–7, we have
400 mm
y
300 mm
x
z
B
A
272
Ans:
4–45.
SOLUTION
Aforce of produces a moment
of about the origin of
coordinates, point O. If the force acts at a point having an x
coordinate of
Note: The figure shows F and MO in an arbitrary position.
determine the yand zcoordinates..x=1m,
MO=54i+5j–14k6kN #m
F=56i–2j+1k6kN
M
O
d
z
x
y
O
1m
z
PF
273
Ans:
4–46.
The force creates a moment about
point Oof If the force
passes through a point having an xcoordinate of 1 m,
determine the yand zcoordinates of the point.Also,realizing
that determine the perpendicular distance dfrom
point
and MO in an arbitrary position.
Oto the line of action of F.
MO=Fd,
MO=5–14i+8j+2k6N#m.
F=56i+8j+10k6N
SOLUTION
M
O
d
z
y
O
y
1m
z
PF
Note: The figure shows F
274
Ans:
4–47.
SOLUTION
A force
F
having a magnitude of acts along the
diagonal of the parallelepiped. Determine the moment of F
about point A, using and MA=rC:F.MA=rB:F
F
=
100 N
F
z
y
C
200 mm
400 mm
r
C
275
Ans:
*4–48.
F
orce
F
acts perpendicular to the inclined plane. Determine
the moment produced by Fabout point A. Express the
result as a Cartesian vector.
SOLUTION
Force Vector: Since force Fis perpendicular to the inclined plane, its unit vector
is equal to the unit vector of the cross product, ,Fig. a. Here
Thus,
Then,
b=rAC *rBC
uF
z
xy
3m
3m
4m
A
BC
F400 N
276
Ans:
4–49.
F
orce
F
acts perpendicular to the inclined plane. Determine
the moment produced by Fabout point B. Express the
result as a Cartesian vector.
z
xy
3m
3m
4m
A
BC
F400 N
SOLUTION
Force Vector: Since force Fis perpendicular to the inclined plane, its unit vector
is equal to the unit vector of the cross product, ,Fig. a. Here
Thus,
Then,
b=rAC *rBC
uF
277
4–50.
Strut AB of the 1-m-diameter hatch door exerts a force of
450 N on point B. Determine the moment of this force
about point O.
SOLUTION
Position Vector And Force Vector:
Moment of Force FAbout Point O: Applying Eq. 4–7, we have
Or
z
y
F= 450 N
0.5 m A
B
O
30°
30°
0.5 m
278
Ans:
4–51.
Using a ring collar the 75-N force can act in the vertical
plane at various angles Determine the magnitude of the
moment it produces about point A, plot the result of M
(ordinate) versus (abscissa) for and
specify the angles that give the maximum and minimum
moment.
0° …u…180°,u
u.
1.5 m
y
2m
z
A
SOLUTION
279
Ans:
*4–52.
The lug nut on the wheel of the automobile is to be removed
using the wrench and applying the vertical force of
at A. Determine if this force is adequate, provided
of torque about the xaxis is initially required to turn the nut.
If the 30-N force can be applied at Ain any other direction,
will it be possible to turn the nut?
14 N #m
F=30 N
SOLUTION
F30 N
A
B
0.25m
0.3 m
z
y
0.5 m
280
Ans:
4–53.
SOLUTION
Solve Prob. 4–52 if the cheater pipe AB is slipped over the
handle of the wrench and the 30-N force can be applied at
any point and in any direction on the assembly.
F30 N
A
B
0.25m
0.3 m
z
y
0.5m
281
SOLUTION
4–54.
The A-frame is being hoisted into an upright position by
the vertical force of
F=80 lb.
Determine the moment of
this force about the
y′
axis passing through points A and B
when the frame is in the position shown.
30
15
6 ft
y
y¿
x¿
C
A
B
F
x
z
6 ft
282
SOLUTION
4–55.
The A-frame is being hoisted into an upright position by the
vertical force of
F=80 lb.
Determine the moment of this
force about the x axis when the frame is in the position
shown.
30
15
6 ft
y
y¿
x¿
C
A
B
F
x
z
6 ft
283
*4–56.
Determine the magnitude of the moments of the force
F
about the x,y,and zaxes.Solve the problem (a) using a
Cartesian vector approach and (b) using a scalar approach.
4ft
3ft
y
z
C
A
B
x
SOLUTION
a) Vector Analysis
Position Vector:
Moment of Force FAbout x,y, and z Axes: The unit vectors along x, y, and zaxes are
i,j, and krespectively. Applying Eq. 4–11, we have
b) Scalar Analysis
My=j#(rAB *F)
284
4–57.
Determine the moment of the force
F
about an axis
extending between Aand C. Express the result as a
Cartesian vector.
4ft
3ft
y
z
A
B
x
SOLUTION
Position Vector:
Unit Vector Along AC Axis:
Moment of Force FAbout AC Axis: With , applying Eq. 4–7,
we have
Or
Expressing MAC as a Cartesian vector yields
F={4i+12j–3k}lb
285
4–58.
The board is used to hold the end of a four-way lug wrench in
the position shown when the man applies a force of F =
N. Determine the magnitude of the moment produced by
this force about the xaxis. Force Flies in a vertical plane.
SOLUTION
Vector Analysis
Moment About the x Axis:The position vector rAB,Fig.a, will be used to determine
the moment of Fabout the xaxis.
Knowing that the unit vector of the xaxis is i, the magnitude of the moment of F
about the xaxis is given by
Scalar Analysis
250 mm
F
250 mm
z
y
x
60
100
286
4–59.
The board is used to hold the end of a four-way lug wrench
in position. If a torque of about the xaxis is
required to tighten the nut, determine the required magni-
tude of the force Fthat the man’s foot must apply on the
end of the wrench in order to turn it. Force Flies in a verti-
cal plane.
SOLUTION
Vector Analysis
Moment About the x Axis:The position vector rAB,Fig.a, will be used to deter-
mine the moment of Fabout the xaxis.
The negative sign indicates that Mxis directed towards the negative xaxis.The
magnitude of Frequired to produce can be determined from
Mx=30 N #m
30 N #m
250 mm
F
250 mm
z
y
x
60
*4–60.
SOLUTION
Using ,,z:
Coordinates of point C:
y¿x¿
T
h
e A-frame
i
s
b
e
i
ng
h
o
i
ste
d
i
nto an upr
i
g
h
t pos
i
t
i
on
b
y t
h
e
vertical force of Determine the moment of this
force about the yaxis when the frame is in the position shown.
F=80 lb.
15
6 ft
C
A
F
z