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

4.4 Polar coordinates 159
4.4 POLARCOORDINATES
We have seen how the x and y coordinates of a point describe its location in the x--y
plane.Thereisanalternativewaytodescribethelocationofapoint.Figure4.8illustrates
a point Pinthex--yplane.
Phas Cartesian coordinates (x,y). Hence
OA=x,
AP=y
Consider the arm OP. The length of OP is the distance of P from the origin. We denote
thisbyr, that is
length of OP =r
Clearly,risnever negative, thatisr 0.
We note that the angle between the positivexaxis and OP is θ. The value of θ lies
between 0 and 2πradians or 0 ◦ to360 ◦ ifdegrees areused.
The values ofrand θ are known as the polar coordinates of P. Conventionally, the
valueofrisstatedfirst,thenthevalueof θ.Wecommonlywritethesepolarcoordinates
asr̸ θ.
Thevaluesofr and θ specifythepositionofapoint.Conventionally,positivevalues
of θ aremeasured anticlockwise from the positivexaxis.
Thepolarcoordinatesofapoint,P,arer̸ θ.ThevalueofristhedistanceofPfrom
the origin; the value of θ isthe angle between the positivexaxis and the armOP.
r0 0θ<2π(0 ◦ θ<360 ◦ )
Engineeringapplication4.2
Pickandplacerobot
Robotsarenowwidelyusedinfactoriesinordertoreducelabourcosts.Theyvaryin
complexity depending on their function. Almost all printed circuit boards found in
electronicdevicessuchascomputersandmobilesareassembledbyrobots.Figure4.9
shows a simple robot that can pick up a surface-mount electronic component in one
position and placeit inanotherposition.
It consists of a rotating arm the length of which can be extended and contracted.
On the end of the arm is a hand which can be closed to pick up a component and
opened toreleaseit.
r
zero datum
u
Figure4.9
A pickand place robot.
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