Hall A.S., Archer F.E., Gilbert R.I. Engineering Statics
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The most common problem is that of finding components of Fwhich lie parallel to and
perpen~icular to a given direction
Ox
(Figure
2.8).
In this case, the axes Oy and
Ox
are at
les
(4
=
30")
and
F
is at an angle
0
to the direction
Ox.
From simple geometry:
F:
=
Fcos
6'
and
Fy
=
F
sin
0
re
It
follows that Fis the resultant of its components
F,
and
f;y.
In the orthogonal system
of Figure
2.8,
where the components
F,
and
Fy
are at right angles, the magnitude and
direction of Fare obtained by substituting
F'
for
F2
and
F,
for
F,
in Equation
2.1:
If the lines of action of a number of forces pass through a common point, the forces are
said to be
co~~~~~e~~.
(4
(b)
4
point
of/
concurrence
C
D
Consider the four concurrent forces
Fl
to
F*
shown in Figure 2.9a. The resultant
of
the group of forces can be found from the graphical construction of Figure 2.9b. Forces
1;;
and
F2
may be combined by the method of vector addition into a resultant
RI
(triangle
Al3C
in Figure 2.9b).
RI
may then be combined with
F3
(triangle
ACD)
to give a
resultant
4.
Finally
i;:
is combined with
4
(triangle
ADE)
to
give
the final resultant
R.
The line
AE
gives the magnitude and direction of
R.
The line of action of the resultant
passes through the point of concurrence (point
0
in Figure 2.9a). By successively adding
n
Figure
29b,
the re~~lt~nt
of
any
number
an
Determine the
x
and y components of the forces shown in Figure P2.1.
Determine the u and v components of the forces shown in Figure P2.2.
A
force of 100
N
acts in the direction of the
x
axis. Resolve this force into
two
components Pand Qwhich make angles of
+30"
and
-45"
respectively with
the
x
axis.
The following forces are concurrent:
50
N
at
0",
80
N
at 25", 10
N
at
85",
and
40
N
at
190",
the angles being measured anticlockwise from the positive
x
direction. Find the resultant analytically. Draw a polygon of forces
to
check
your solution graphically.
Find the resultant of
two
concurrent forces
P
and
Q.
P
has a magnitude of
80
N
and acts along a line which makes an angle of 20" with the
x
direction. The
magnitude of Qis
30
N
and this force acts along a line which makes an angle
of
155"
with the
x
direction. Angles are measured anticlockwise from the
positive
x
direction.
two
water skiers
as
shown
in
F
at at
A.
The
te~sio~ in
e resultant force acting
to the centreline of the boat.
10
N
12
N
50
N
10
N
x
15
N
res~lta~t
of
the
sy~te~
of
co~c~r
8
kN
4
l<N
For
the
force
s~ste~s
shown
in
Fi
(i)
Find
the
value
of
force
A
and
t
(ii)
Find
the
value
of
irectio~
of C,
if
A
is
the
resu~t~~t
of
the
(4
(h)
(C>
Y
A
B=15N
x
X
D
=
12
300
N
250
kg
Two forces
P
and
Q
act through a point, the angle between their lines of
action being
8.
Show that the magnitude of the resultant is:
Iz
=
VP2
+
Q2
+
2PQcos
8
and that its line of action is inclined at the angle
a
to the direction of force
P
where:
Qsin
t3
ran
a=
P+
Qcos
8
Forces of 50
N
and 60
N
act through the point
0,
as shown in Figure P2.13.
Find their resultant in magnitude and direction. Calculate the components of
the resultant in the directions Ou and
Ov.
Check the answers by
ete er mining
separately the components of the
50
N
force and the
60
N
force in the
directions Ou and
Ov
and adding the results.
0
35"
X
A
force
of
4
N
is the resultant of two concurrent forces, one of
5
N
and the
other of
7
N.
What is the angle between these two forces?
Three concurrent forces act in the directions shown in Figure P2.15. If the
magnitude of the resultant is 14.56
kN,
find the angles
a
and
p.
6
kN
0
Three forces act through the point
0
in the directions shown in Figure
P2.16.
The magnitudes of the forces are
15
kN,
10
and
P.
The resultant
R
acts
at an angle of
10"
to
Ox.
Find the magnitudes of
P
and
R
15
kN
P
If a body is acted upon by
a
system of forces which has
a
resultant, this resultant produces
a
change in the state of motion in accordance with Newton’s Second
Law.
When the resultant of
a
group of
cu~e~~~e~t
~~ces
is zero, the motion remains
unchanged and the body is in
e~~~Zi~~i~~.
The system of forces
is
also said to be in
equilibrium, Conversely, any body which remains at rest
(as
a
large number of e~gineering
structures do) must be acted upon by
a
system of forces having
a
zero resultant.
If
the resultant of a system is zero, the forces, when added vectorially
(as
in Figure 2.9b,
page
lo),
must form a closed polygon (i.e. the last point of the force polygon must coincide
with the initial point), When all the forces but one are known, the magnitude and direction
of the unknown force may be found if the system is known to be in equilibrium,
as
the
unknown force will be represented by the vector required to close the force polygon.