Example 1: If the resultant force of the two tugboats is required to be directed
towards the positive x axis, and FB is to be a minimum, determine the magnitude
of FR and FB and the angle θ.
Fig 1 : Example 1
Solution:
The parallelogram law of addition and triangular rule are shown in Figs. a and b,
respectively.
FB = 2 sin 30° = 1 kN
FR = 2 cos 30° = 1.73 kN
For FB to be minimum, it has to be directed perpendicular to FR. Thus,
θ= 90°
1
Example 2: Determine the magnitude and orientation, measured
counterclockwise from the positive y axis, of the resultant force acting on the
bracket, if FB = 600 N and θ = 20°.
Fig 2 : Example 2
Solution:
Scalar Notation: Summing the force components algebraically, we have
+
+
FRx = ΣFx;
FRx = 700 sin 30° - 600 cos 20°
= -213.8 N = 213.8 N
FRy = ΣFy;
FRy = 700 cos 30° + 600 sin 20°
= 811.4 N
2
The magnitude of the resultant force FR is
FR = √𝑓𝑥 2 + 𝑓𝑦 2
=√−213.8 2 + 811.42
= 839 N
The directional angle u measured counterclockwise from positive y axis is
θ = tan-1
𝐹𝑥
𝐹𝑦
= tan-1 (
213.8
)
811.4
= 14.8°
Example 3: The beam is to be hoisted using two chains. Determine the
magnitudes of forces FA and FB acting on each chain in order to develop a
resultant force T directed along the positive y axis
Given:
T = 600 N
θ1= 30°
θ = 45°
3
Fig 3 : Example 3
Solution:
𝐹𝐴
𝑇
=
𝑆𝑖𝑛 (θ)
𝑆𝑖𝑛 [180° − (θ − θ1)
FA =
600∗𝑠𝑖𝑚(45)
𝑆𝑖𝑛 [180°−(45−30)
FA = 439 N
𝐹𝐵
𝑆𝑖𝑛 (θ1)
FB =
=
𝑇
𝑆𝑖𝑛 [180°−(θ−θ1)
600∗𝑠𝑖𝑚(30)
𝑆𝑖𝑛 [180°−(45−30)
FA = 311 N
4