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Components or resolved forces

We have seen that two forces can be combined into a

single force which is called their resultant.

There is the reverse process which consists of expressinga single force in terms of its components. Thesecomponents are sometimes referred to as the resolvedparts of the force.

F

x

y

O X

Y

OX= F cos

OY= F sin

Learning objectives

Splitting forces into their components

Finding the resultant of two or more forces

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Example

30

Y

X

20 N

30

20 N

X = 20 x Cos 30 = 17.3 N

Y = 20 x Sin 30 = 10.0 N

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Finding the resultant of two forcesusing the component method

30

8 N

5 N

X = 8 + 5 x Cos 30 = 12.33 N

Y = 5 x Sin 30 = 2.5 N

12.33

2.5 N

Resultant = 22 5.233.12 = 12.6 N

8 N 5cos30

5sin3030

tan -1(2.5/12.33) 11.5

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Draw a force diagram

and, where appropriate,

redraw each force into

any two perpendicular

directions.

Find the resultant

in each direction.

Find the resultant of the following forces:

2 N

6 N

30

2 N

6 N

30

6 cos 30N

6 sin 30N

2 N

Horizontal component

X = 6 cos 30+ 2

= 7.196

Vertical component

Y = 6 sin 30

= 3

Find the overall

resultant.22

3...19.7 F

= 7.79 7.1 N

3 N FN

The resultant force has magnitude 7.8 N (2 s.f.)

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Find the resultant of the following forces:

10 N

4 N

120

Draw a force diagram

and, where appropriate,

redraw each force

resolved into any two

perpendicular directions.10 N

4 N

12060

4 sin 60N

10 N

4 cos 60N

Find the resultant

in each direction.

Find the

overall resultant.

Horizontal component

X= 104 cos 60

= 8

Vertical component

Y= 4 sin 60= 3.46

228...46.3 F

= 8.71

The resultant force has magnitude 8.7 N (2 s.f.)

8 N

3.46 N FN

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More than two forces

22YX

Split each force into components in these directions.

Choose two directions at right angles to each other.

For each direction, find the sum of the components.

Find the resultant.

Find the required angle.

Pythagoras:R

Angle with the X direction =

X

Y1tan

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Example

22YX

x

y20 N

70

12 N

8 N

30

Force X Y

20 N 20 cos 70 = 6.8404 20 sin 70 = 18.7939

12 N 12.0000 0

8 N -8 cos 30 =- 6.9282 -8 sin 30 = - 4.0000

Total 11.9122 14.7939

R =

R = 19.0 N

=

X

Y1tan

= 51.2

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Example

x

y

6 N

30 N

50

20 N

355 N

X = 20 cos 35 30 sin 50 5 = - 11.5983

Y = 6 20 sin 35 30 cos 50 = - 24.7552

22 YX R = = 27.3 N

=

X

Y1tan = 64.9

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More exampleTwo forces act at a point. The magnitude of the forces are 3.95 N and

2.5 N, and angle between their direction is 90+, where 0