082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

7.2 Vectors and scalars: basic concepts 227
In considering motion from point A to point B, the vector → AB is called a displacement
vector.
Engineeringapplication7.2
Resultantoftwoforcesactingonabody
Sometimesitisusefultobeabletocalculatetheresultantforcewhenseveraldifferent
forces act on a body. Consider the simplest case when there aretwo forces.
AforceF 1
of2Nactsverticallydownwards,andaforceF 2
of3Nactshorizontally
to the right, upon the body shown in Figure 7.6. Translating F 1
until its tail touches
the head of F 2
we can apply the triangle law to find the combined effect of the two
forces.This isasingle forceRknown as the resultant ofF 2
andF 1
. We write
R=F 2
+F 1
andsaythatRisthevectorsumofF 2
andF 1
.Theresultantforce,R,actsatanangle
of θ to the vertical, where tanθ = 3/2, and has a magnitude given by Pythagoras’s
theorem as √ 2 2 +3 2 = √ 13 N.
F 2
u
F 1
R
u
Figure7.6
Resultant force,R, producedbyavertical force,F 1 ,anda
horizontalforce,F 2 .
Engineeringapplication7.3
Resolvingaforceintotwoperpendiculardirections
Inthepreviousexamplewesawthattwoforcesactinguponabodycanbereplacedby
asingleforcewhichhasthesameeffect.Equivalentlywecanconsiderasingleforce
astwoforcesactingatrightanglestoeachother.ConsidertheforceFinFigure7.7.
It can be replaced by two forces, one of magnitude |F|cosθ and one of magnitude
|F|sin θ asshown.WesaythattheforceFhasbeenresolvedintotwoperpendicular
components.
For example, Figure 7.8 shows a force of 5 N acting at an angle of 30 ◦ to thex
axis. It can be resolved into two components. The first component is directed along
the x axis and has magnitude 5cos30 ◦ N. The second component is perpendicular
to the first and has magnitude 5sin30 ◦ N. These two components combine to have
the sameeffect asthe original5Nforce.
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