1. ENGINEERING MECHANICS CHAPTER 2 FORCES & RESULTANTS
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By Jeff Lamoon
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2. CHAPTER 2 - Forces & Resultants
2.1
Introduction
This chapter introduces the concept of vectors.
Upon completion of this chapter, the student will be able 1
Distinguish between vector and scalar quantities. 2
Determine the resultants of two force vectors by using
vector triangle, parallelogram law and vector polygon
addition. 3
Resolving a force vector into its components.
Determine the resultant of several force vectors using
the method of perpendicular components. 4
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3. Scalar quantity has only magnitude.
2.2 Vector Quantities
Scalar quantity has only magnitude.
Examples: 10 kg mass, 8 m long, 20 m2 area, etc.
Vector quantity has magnitude + direction.
5 cm
E.g. 1) 5 km south.
8 cm
E.g. 2) 8 kN horizontal force.
9 cm
E.g. 3) 9 m/s north.
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4. Vectors may be in any direction:
600
1500 4 3
Scalar quantities can be added algebraically:
3 cm + 5 cm = 8 cm
6 kg + 8 kg = 14 kg
However, vector quantities cannot be added
or subtracted because vector has the extra
property of direction.
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5. 2.3 Vector Triangle Methody
8 kN kN 60 40 x
a) Graphical Method (Tip To Tail)
Choose a scale like 1 cm to 1 kN, then
begin by drawing any force vector.
6 cm
400 x
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6. ii) Draw the other force taking care to join them tip-to-tail.y
8 kN kN 60 40 x
600
Tip of first arrow join to tail of next
arrow.
8 cm x
400
6 cm
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7. iii) Complete the triangle by drawing the resultant R
from the start point of first arrow to end point of
last arrow (force vector). y
8 kN kN 60 40 x x
400
6 cm
600
8 cm
End point R
Start point
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8. Lets draw the diagram another way : start with the 8 kN force instead
8 kN kN 60 40 x
Step 1
Step 2
Step 3 x
400
6 cm
8 cm
1500 x
400
6 cm
8 cm R x
1500
8 cm
The same resultant R is obtained and it makes no difference which force you start to draw first, as long as you follow ‘Tip to Tail, join start point to end point’ rule.
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9. Start by sketching the triangle ABC.
b) Analytical method
Start by sketching the triangle ABC. A
8 cm
Angle ABC = ( ) = 700
600 R
As triangle is not a right-angle, B x
400
6 cm
Cosine Law is used here. C
R = 8.20 kN
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10. To find direction of R, use sine rule to find , then .x
400
6 kN
600
8 kN C B
8.2 kN
8.2 kN
73.50
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11. Use scale of 1 cm to 1 kN, draw the forces
2.4 Parallelogram Method y
8 kN kN 60 40 x
a) Graphical Method
Use scale of 1 cm to 1 kN, draw the forces
from the same point as in the diagram.
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12. ii) Construct a parallelogram by drawing parallel lines as shown.
8 kN kN 60 40 x
8 kN
600
6 kN
400 x
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13. iii) Draw the diagonal R from the point where
the tails of the two arrows meet. y
8 kN kN 60 40 x
400
6 kN
8 kN x R
Note that half of the parallelogram is
exactly the same triangle as that obtained by
the Vector Triangle Method in 2.3.
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14. b) Analytical Method
We can select any half of the parallelogram to solve for the resultant. The right half here is exactly the same as the one in 2.3 b) above and the analytical method use is also exactly the same. R
400 x
8 kN
6 kN
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15. This method is used to find the resultant of
2.5 Vector Polygon Method
This method is used to find the resultant of
three or more forces. Example below illustrates this principles
4 kN
3 kN
6 kN
200
400
8 kN
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16. a) Graphical Method
4 kN
3 kN
6 kN
200
400
8 kN
Start with any force drawn to scale, and follow tip-to-tail rule , draw the other 3 forces.
8 kN
end point R
4 kN
start point
3 kN
6 kN
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17. * Check sections 2.6 and 2.7 for an easier method
b) Analytical Method
For 3 or more forces, this method involves more calculation steps. The polygon is divided into several triangles and cosine and sine rules are then applied. Using the example above:
The polygon is divided into three triangles 1, 2 and 3 as shown. Starting with triangle 1, using cosine and sine rules, calculate R1. Then the process is repeated for triangle 2 to find R2, and finally triangle 3 to find R, the resultant of the four forces. This is very tedious and not the preferred method.
8 kN R
4 kN
3 kN
6 kN R2 R1
40 0
20 0 3 2 1
* Check sections 2.6 and 2.7 for an easier method
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18. 2.6 Resolution of A Force Into Components
We know that resultant is the sum of 2 or more forces and it produces the same effect on a body.
Hence a system of forces can be replaced by a single resultant force which have the same effect on the body.
Conversely, a single force F (like a resultant), can be replaced by 2 forces or components of F, which produce the same effect on the body.
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19. The method of resolving a single force vector into two components is the reverse of vector addition.
Any single force can be resolved into 2 components in any direction, but in order for us to develop an easier analytical method, we would choose to resolve a single force into its perpendicular components based on the x- and y-axis.
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20. 2.7 Resolution of A Force Into Perpendicular Components
One of the most important and useful concept is the resolution of a force vector into two perpendicular components, in the x and y axis as denoted by the Fx (x) and Fy (y) components. y F x 
Fy = F sin 
Fx = F cos 
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21. Resolve the 300 N force into components along the x and y axes.
Example 2.1
Resolve the 300 N force into components along the x and y axes.
300 N
300
Sketch the parallelogram with Fy perpendicular to Fx. y
300 N Fy
300 x Fx
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22. Select any half of the triangle, a vector triangle, for analysis.Fx Fy x
300
As this is a right-angle triangle,
cos 300 = Fx / 300
i.e Fx = 300 x cos 300
= N
Also since sin 300 = Fy / 300
i.e Fy = 300 sin 300
= 150 N
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23. 2.8 Resolution of A Force By Fx And Fy Components
For a number of forces, this section introduces an easier
analytical method of finding the resultant.
Students should master this method as it would form the basis of all your calculations for resultant forces.
Let us use the same simple example in section 2.3 above to demonstrate the method: y
8 kN kN 60 40 x
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24. We first resolved the 2 forces into perpendicular i.e.
Fx and Fy components.
Fy = ( 8 x 103 sin 30o)
Fy = ( 6 x103 sin 40o)
8 kN
6 kN
30 0
40 0
Fx = ( 8 x 103 cos 30o)
Fx = ( 6 x 103 cos 40o)
2. The 2 forces can now be replaced by the above components as shown below.
Fx = ( 6.93 kN)
Fy = ( 4.0 kN)
Fy = ( 3.86 kN)
Fx = ( 4.6 kN) A
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25. 3. Now we can add the two Fx forces to form a single Rx, as they are both horizontal forces but only opposite in direction, if we adopt a sign convention. Similarly, we can also add the two Fy forces to form a single Ry.
Fx = ( 6.93 kN)
Fy = ( 4.0 kN)
Fy = ( 3.86 kN)
Fx = ( 4.6 kN) A
 Fx = Rx, is the sum of all the Fx forces,
 Fy = Ry , is the sum of all Fy forces.
 Fx = (4.6x103) - (6.93x103) {Fx pointing left is –ve} Rx = kN
 Fy = (3.86x103) + (4.0x103) {Fy pointing up is +ve} Ry = kN
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26. 4. These two forces, Rx and Ry can now replace the previous four component forces in (2) above, and we can use the vector triangle method to add them back to form a single force, which is the resultant R, as shown by the vector triangle below.
Rx = kN
Ry = kN R 
From the right-angle triangle, we then use the Pythagoras Theorem to find the magnitude of R.
The angle  of R is:
= tan -1 (Ry/Rx)
= tan -1 ((7.86x103)/(2.33x103)) =
Same answers as before.
The magnitude of R is:
R =  Rx2 + Ry2 
= (2.33x103)2 + (7.86x103)2
= kN
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27. Resultant By Method of Perpendicular Components:
-ve Fy
+ ve Fx
- ve
Use the Sign convention for
the force components:
Resolve all (but not horizontal or vertical) forces into Fx and Fy components. Ry Rx R
4. Direction of R:
Use  Fx = Rx,
 Fy = Ry
3. Magnitude: R =  Rx2 + Ry2 
Angle:  = tan -1 (Ry/Rx)
(from x axis)
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28. Four forces act on a bolt as shown. Determine the resultant force.
Example 2.2
Four forces act on a bolt as shown. Determine the resultant force.
Solution: y
Fx components in the x direction
80 N
150 N 30 x 15
100 N
110 N
F1x = +150 cos 30 0 = N
F2x = +100 cos 15 0 = N
F3x = - 80 cos 70 0 = N
F4x = 0 N, (for 110 N force)
i.e. Fx = Rx = 150 cos cos cos 70 0
= –
Rx = 199.1N
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29. Fy components in the y direction
150 N 30 x 15
100 N
110 N
Fy components in the y direction
F1y = +150 sin 30 0 = 75 N
F2y = sin 15 0 = N
F3y = + 80 sin 70 0 = N
F4y = N (it is a Fy or
vertical force)
i.e. Fy = Ry = 150 sin sin sin 70 0
= 75 –
Ry = 14.3 N
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30. The angle , R makes with the horizontal isy
80 N
150 N 30 x 15
100 N
110 N
The magnitude of R is
R =  ( Rx2 + Ry2)
R =  ( )
= N
The angle , R makes with the horizontal is
 = tan-1 Ry / Rx
= tan / 199.1
= 4.10
Since Rx and Ry are positive values, Rx R
R = N and  = 
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31. Example 2.3
The rocket’s main engine exerts a total thrust of 900,000 N parallel to the y axis. Each of its two small side engines exerts a thrust of 22,250 N in the direction as shown.
Determine the magnitude and direction of the resultant force exerted on the rocket by the engines.
22250 N
16 0
31 0 y
900,000 N
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32. Solution: Fx = Rx = 22,250 sin 16 – 22,250 sin 31 = - 5326.67 N
Fy = Ry = -22,250 cos 16 – 22,250 cos 31 – 900,000
= – – 900,000
= N
  R = (Rx2 + Ry2) = ( ) = N
tan  = Ry / Rx = / =
 = R
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33. Special situations:
1. Resultant’s direction is along the x-axis (Horizontal)
Fy = Ry = Fx = Rx = R
2. Resultant’s direction is along the y-axis (Vertical)
Fx = Rx = Fy = Ry = R
Example 2.4
A bracket is being pulled by 3 forces as shown. Determine the angle  and the magnitude of force P if the resultant’s direction is along the y-axis and its magnitude is 20 kN. y
10 kN
P kN a
8 kN x
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34. Since the resultant’s direction is along the y-axis.
i.e Fx = Rx = (No Rx component)
and Fy = Ry = 20x103 N ( R must be = Ry) y
10 kN
P kN a
8 kN x
Fx = Rx = 0
(10x103)(cos 60o) + (8x103) – (Px103)(cos ) = (1)
and Fy = Ry = 20x103 N
(10x103)(sin 60o) + (Px103)(sin ) = (2)
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35. From equation (1), P cos  = 13 (3)
From equation (2), P sin  = (4)
Equation (4)/(3), P sin / P cos  = (11.34 / 13) y
10 kN
P kN a
8 kN x
i.e. tan  = / 13 =
 = tan -1 (0.8723)
 =
To find magnitude of P, substitute  = into equations (1) or (2).
P cos  = 13
P = 13 / cos  = (P kN = kN)
End of Chapter 2
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