1. 2.4 Addition of a System
of Coplanar Forces
For resultant of two or more forces:
Find the components of the forces in the
specified axes
Add them algebraically
Form the resultant
In this subject, we resolve each force into
rectangular forces along the x and y axes.
F = Fx + Fy

2. 2.4 Addition of a System
of Coplanar Forces
Scalar Notation
- x and y axes are designated positive and
negative
- Components of forces expressed as algebraic
scalars
Eg: F = Fx + Fy
Sense of direction
along positive x and
y axes

3. 2.4 Addition of a System
of Coplanar Forces
Scalar Notation
Eg: F ' = F ' x + F ' y
Sense of direction
along positive x and
negative y axes

4. 2.4 Addition of a System
of Coplanar Forces
Scalar Notation
- Head of a vector arrow = sense of the
vector graphically (algebraic signs not
used)
- Vectors are designated using boldface
notations
- Magnitudes (always a positive quantity)
are designated using italic symbols

5. 2.4 Addition of a System
of Coplanar Forces
Cartesian Vector Notation
- Cartesian unit vectors i and j are used to
designate the x and y directions
- Unit vectors i and j have dimensionless
magnitude of unity ( = 1 )
- Their sense are indicated by a positive or
negative sign (pointing in the positive or
negative x or y axis)
- Magnitude is always a positive quantity,
represented by scalars Fx and Fy

6. 2.4 Addition of a System
of Coplanar Forces
Cartesian Vector Notation
F = Fxi + Fyj F’ = F’xi + F’y(-j)
F’ = F’xi – F’yj

7. 2.4 Addition of a System
of Coplanar Forces
Coplanar Force Resultants
To determine resultant of several
coplanar forces:
- Resolve force into x and y
components
- Addition of the respective
components using scalar algebra
- Resultant force is found using the
parallelogram law

8. 2.4 Addition of a System
of Coplanar Forces
Coplanar Force Resultants
Example: Consider three coplanar
forces
Cartesian vector notation
F1 = F1xi + F1yj
F2 = - F2xi + F2yj
F3 = F3xi – F3yj

9. 2.4 Addition of a System
of Coplanar Forces
Coplanar Force Resultants
Vector resultant is therefore
FR = F1 + F2 + F3
= F1xi + F1yj - F2xi + F2yj + F3xi – F3yj
= (F1x - F2x + F3x)i + (F1y + F2y – F3y)j
= (FRx)i + (FRy)j

10. 2.4 Addition of a System
of Coplanar Forces
Coplanar Force Resultants
If scalar notation are used
FRx = (F1x - F2x + F3x)
FRy = (F1y + F2y – F3y)
In all cases,
FRx = ∑Fx
FRy = ∑Fy
* Take note of sign conventions

11. 2.4 Addition of a System
of Coplanar Forces
Coplanar Force Resultants
- Positive scalars = sense of direction
along the positive coordinate axes
- Negative scalars = sense of direction
along the negative coordinate axes
- Magnitude of FR can be found by
Pythagorean Theorem
FR = F 2
Rx +F 2
Ry

12. 2.4 Addition of a System
of Coplanar Forces
Coplanar Force Resultants
- Direction angle θ (orientation of the
force) can be found by trigonometry
−1
FRy
θ = tan
FRx

13. 2.4 Addition of a System
of Coplanar Forces
Example 2.5
Determine x and y components of F1 and F2
acting on the boom. Express each force as a
Cartesian vector

14. 2.4 Addition of a System
of Coplanar Forces
Solution
Scalar Notation
F1x = −200 sin 30o N = −100 N = 100 N ←
F1 y = 200 cos 30o N = 173 N = 173 N ↑
Hence, from the slope
triangle
− 5
θ =tan  
1
12

15. 2.4 Addition of a System
of Coplanar Forces
Solution
Alt, by similar triangles
F2 x 12
=
260 N 13
 12 
F2 x = 260 N   = 240 N
 13 
Similarly,
5
F2 y = 260 N   = 100 N
 13 

16. 2.4 Addition of a System
of Coplanar Forces
Solution
Scalar Notation
F2 x = 240 N = 240 N →
F2 y = −100 N = 100 N ↓
Cartesian Vector Notation
F1 = {-100i +173j }N
F2 = {240i -100j }N

17. 2.4 Addition of a System
of Coplanar Forces
Example 2.6
The link is subjected to two forces F1 and
F2. Determine the magnitude and
orientation of the resultant force.

18. 2.4 Addition of a System
of Coplanar Forces
Solution
Scalar Notation
FRx = ΣFx :
FRx = 600 cos 30o N − 400 sin 45o N
= 236.8 N →
FRy = ΣFy :
FRy = 600 sin 30o N + 400 cos 45o N
= 582.8 N ↑

19. 2.4 Addition of a System
of Coplanar Forces
Solution
Resultant Force
FR = (236.8 N )2 + (582.8 N )2
= 629 N
From vector addition,
Direction angle θ is
 582.8 N 
−1
θ = tan  
 236.8 N 
= 67.9o

20. 2.4 Addition of a System
of Coplanar Forces
Solution
Cartesian Vector Notation
F1 = { 600cos30°i + 600sin30°j } N
F2 = { -400sin45°i + 400cos45°j } N
Thus,
FR = F1 + F2
= (600cos30°N - 400sin45°N)i +
(600sin30°N + 400cos45°N)j
= {236.8i + 582.8j}N

21. 2.4 Addition of a System
of Coplanar Forces
Example 2.7
The end of the boom O is subjected to three
concurrent and coplanar forces. Determine
the magnitude and orientation of the
resultant force.

22. 2.4 Addition of a System
of Coplanar Forces
Solution
Scalar Notation
FRx = ΣFx :
4
FRx = −400 N + 250 sin 45o N − 200  N
5
= −383.2 N = 383.2 N ←
FRy = ΣFy :
3
FRy = 250 cos 45 N + 200  N
o
5
= 296.8 N ↑

23. 2.4 Addition of a System
of Coplanar Forces
Solution
Resultant Force
FR = (− 383.2 N )2 + (296.8 N )2
= 485 N
From vector addition,
Direction angle θ is
 296.8 N 
−1
θ = tan  
 383.2 N 
= 37.8o