Mechanical Engineering Principles

36 MECHANICAL ENGINEERING PRINCIPLES

By Pythagoras’ theorem,

r=

√

1. 5352 + 9. 5302

= 9. 653 ,

and by trigonometry, angle

φ=tan−^1

9. 530

1. 535

= 80. 85 °

Hence the resultant of the two forces shown in
Figure 3.26 is a force of 9.653 N acting at 80.85°
to the horizontal.

Problems 9 and 10 demonstrate the use of resolution
of forces for calculating the resultant of two coplanar
forces acting at a point. However the method may
be used for more than two forces acting at a point,
as shown in Problem 11.

Problem 11. Determine by resolution of

forces the resultant of the following three

coplanar forces acting at a point: 200 N

acting at 20°to the horizontal; 400 N acting

at 165°to the horizontal; 500 N acting at

250 °to the horizontal.

A tabular approach using a calculator may be made
as shown below:

Horizontal component

Force 1 200 cos 20° = 187.94

Force 2 400 cos 165° = −386.37

Force 3 500 cos 250° = −171.01

Total horizontal component = −369.44

Vertical component

Force 1 200 sin 20° = 68.40

Force 2 400 sin 165° = 103.53

Force 3 500 sin 250° = −469.85

Total vertical component = − 297. 92

The total horizontal and vertical components are
shown in Figure 3.28.

Resultant r=

√

369. 442 + 297. 922

= 474. 60 ,

and angle φ=tan−^1

297. 92

369. 44

= 38. 88 °,

from which, α= 180 °− 38. 88 °= 141. 12 °

–369.44

–297.92

φ

r

α

Figure 3.28

Thus the resultant of the three forces given

is 474.6 N acting at an angle of − 141. 12 ° (or

+ 218. 88 °) to the horizontal.

Now try the following exercise

Exercise 17 Further problems on resolu-

tion of forces

1. Resolve a force of 23.0 N at an angle
   of 64° into its horizontal and vertical
   components. [10.08 N, 20.67 N]


2. Forces of 5 N at 21° and 9 N at 126°
   act at a point. By resolving these forces
   into horizontal and vertical components,
   determine their resultant.
   [9.09 N at 93.92°]

In questions 3 and 4, determine the mag-

nitude and direction of the resultant of the

coplanar forces given, which are acting at

a point, by resolution of forces.

3. ForceA, 12 N acting horizontally to the
   right, force B, 20 N acting at 140° to
   force A,forceC, 16 N acting 290° to
   forceA.[3.1Nat− 45 °to forceA]


4. Force 1, 23 kN acting at 80°to the hor-
   izontal, force 2, 30 kN acting at 37° to
   force 1, force 3, 15 kN acting at 70°to
   force 2.
   [53.5 kN at 37°to force 1
   (i.e. 117°to the horizontal)]


5. Determine, by resolution of forces, the
   resultant of the following three coplanar
   forces acting at a point: 10 kN acting at
   32 °to the horizontal, 15 kN acting at 170°
   to the horizontal; 20 kN acting at 240°to
   the horizontal.
   [18.82 kN at 210.03°to the horizontal]