Vectors 271
total horizontal component of the two forces,
H = F 1 cos θ 1 + F 2 cos θ 2
The vertical component of force F 1 is F 1 sin θ 1 and the
vertical component of force F 2 is F 2 sin θ 2 . The total
vertical component of the two forces,
V = F 1 sin θ 1 + F 2 sin θ 2
Since we have H and V , the resultant of F 1 and F 2
is obtained by using the theorem of Pythagoras. From
Figure 29.19,
0b
2
= H
2
+ V
2
i.e.
resultant =
H 2 + V 2 at an angle
given by θ = tan
−1
V
H
V
b
R e s u l t a n t
a
H
0
Figure 29.19
Problem 7. A force of 5 N is inclined at an angle
of 45 ◦ to a second force of 8 N, both forces acting at
a point. Calculate the magnitude of the resultant of
these two forces and the direction of the resultant
with respect to the 8 N force
The two forces are shown in Figure 29.20.
458
8 N
5 N
Figure 29.20
The horizontal component of the 8 N force is 8 cos 0 ◦
and the horizontal component of the 5 N force is
5 cos45 ◦ . The total horizontal component of the two
forces,
H = 8 cos 0
◦
+ 5 cos 45
◦
= 8 + 3.5355 = 11.5355
The vertical component of the 8 N force is 8 sin 0
◦ and
the vertical component of the 5 N force is 5 sin 45 ◦ . The
total vertical component of the two forces,
V = 8 sin0
◦
+ 5 sin45
◦
= 0 + 3.5355 = 3.5355
R e s u l t a n t
H ϭ11.5355 N
V ϭ 3.5355 N
Figure 29.21
From Figure 29.21, magnitude of resultant vector
=
H 2 + V 2
=
11.5355 2 + 3.5355 2 = 12.07 N
The direction of the resultant vector,
θ = tan
−1
V
H
= tan
−1
3.5355
11.5355
= tan
−1 0.30648866 ... = 17.04
◦
Thus, the resultant of the two forces is a single vector
of 12.07 N at 17.04 ◦ to the 8 N vector.
Problem 8. Forces of 15 N and 10 N are at an
angle of 90 ◦ to each other as shown in Figure 29.22.
Calculate the magnitude of the resultant of these
two forces and its direction with respect to the
15 N force
10 N
15 N
Figure 29.22
The horizontal component of the 15 N force is 15 cos0 ◦
and the horizontal component of the 10 N force is
10 cos90 ◦ . The total horizontal component of the two
velocities,
H = 15 cos 0
◦
+ 10 cos 90
◦
= 15 + 0 = 15
total horizontal component of the two forces,
H = F 1 cos θ 1 + F 2 cos θ 2
The vertical component of force F 1 is F 1 sin θ 1 and the
vertical component of force F 2 is F 2 sin θ 2 . The total
vertical component of the two forces,
V = F 1 sin θ 1 + F 2 sin θ 2
Since we have H and V , the resultant of F 1 and F 2
is obtained by using the theorem of Pythagoras. From
Figure 29.19,
0b
2
= H
2
+ V
2
i.e.
resultant =
H 2 + V 2 at an angle
given by θ = tan
−1
V
H
V
b
R e s u l t a n t
a
H
0
Figure 29.19
Problem 7. A force of 5 N is inclined at an angle
of 45 ◦ to a second force of 8 N, both forces acting at
a point. Calculate the magnitude of the resultant of
these two forces and the direction of the resultant
with respect to the 8 N force
The two forces are shown in Figure 29.20.
458
8 N
5 N
Figure 29.20
The horizontal component of the 8 N force is 8 cos 0 ◦
and the horizontal component of the 5 N force is
5 cos45 ◦ . The total horizontal component of the two
forces,
H = 8 cos 0
◦
+ 5 cos 45
◦
= 8 + 3.5355 = 11.5355
The vertical component of the 8 N force is 8 sin 0
◦ and
the vertical component of the 5 N force is 5 sin 45 ◦ . The
total vertical component of the two forces,
V = 8 sin0
◦
+ 5 sin45
◦
= 0 + 3.5355 = 3.5355
R e s u l t a n t
H ϭ11.5355 N
V ϭ 3.5355 N
Figure 29.21
From Figure 29.21, magnitude of resultant vector
=
H 2 + V 2
=
11.5355 2 + 3.5355 2 = 12.07 N
The direction of the resultant vector,
θ = tan
−1
V
H
= tan
−1
3.5355
11.5355
= tan
−1 0.30648866 ... = 17.04
◦
Thus, the resultant of the two forces is a single vector
of 12.07 N at 17.04 ◦ to the 8 N vector.
Problem 8. Forces of 15 N and 10 N are at an
angle of 90 ◦ to each other as shown in Figure 29.22.
Calculate the magnitude of the resultant of these
two forces and its direction with respect to the
15 N force
10 N
15 N
Figure 29.22
The horizontal component of the 15 N force is 15 cos0 ◦
and the horizontal component of the 10 N force is
10 cos90 ◦ . The total horizontal component of the two
velocities,
H = 15 cos 0
◦
+ 10 cos 90
◦
= 15 + 0 = 15