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A. Bazoune

12.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS

Polar Coordinates

Polar coordinates are particularly suitable for solving problems for which data regarding the angular motion of the radial coordinate r is given to describe the particle’s motion. Figure 1 shows the polar coordinates r and θ that specify the position of the particle P that is moving in the

−xy plane. The origin is established at a fixed point, and the radial line r directed to the particle. The transverse coordinate θ is measured counterclockwise from a fixed reference line to the radial line. In planar motion, the polar coordinate r is equal to the magnitude of the position vector r of the particle.

θ

rP

θ θ

x

y

ruθu

j

iFig. 1

Derivative of the Unit Vectors

The unit vectors ru and θu of the polar coordinate system are also shown in Fig. 1. The vector ru is directed along the radial line, pointing away from O ; θu is perpendicular to ru , in the direction of increasing θ .

From Fig.1, it is clear that the unit vectors ru and θu will rotate as the

particle moves. Therefore, ru and θu are the base vectors of a rotating reference

frame, similar to the path ( )n t− coordinate system. The difference between the

two coordinate systems is that ( )n t− coordinates depend on the path and direction

of the particle, whereas polar coordinates are determined by the position of the particle. Consequently, these base vectors possess nonzero derivatives, even though their magnitudes are constant (equal to one). The time derivative of the unit base

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vectors can be determined by first relating the vectors to the xy − coordinate system. From Fig. 1,

r

θ

= θ + θ

= − θ + θ

i j

i j

u

u

cos sin

sin cos (1)

Differentiating with respect to time while noting that 0d d d dt t= =i j

(the xy − coordinate system is fixed) yields

( ) ( )( ) ( )

d d

d d

r rt

tθ θ

= = θ − θ + θ

= = θ − θ − θ

u i j

u i j

u

u

sin cos

cos sin (2)

Comparing Eqs (1) and (2), one can find

rr θ θ= θ = −θu uu u, (3)

The term θ is called the angular velocity of the radial line. The base vectors and

their derivatives are shown in Fig. 2. Notice that ru and θu are perpendicular to ru

and θu , respectively.

θ

rP

x

y

ru

j

i

r θ= θu u

θ

rP

x

y

θu

rθ = −θu u

i

j

Fig. 2 Unit vectors and their derivatives

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Velocity and Acceleration Vectors

The position vector r of the particle can be written in polar coordinates as

rr=r u (4)

Since the velocity vector is, by definition, d dt= rv , we have

( )d d d d

d d d d

rr r r r

rr r r r

t t t t= = = + = +r u

u u u uv

Substituting for ( )d dr rt θ= = θu u u from Eq.(3) gives

r rrv v r rθ θθ= + = + θv u u u u (5)

where

rv r= , and v rθ = θ (6)

The components rv and vθ are called

the radial and transverse components of the velocity, respectively. These components are mutually perpendicular and the magnitude of the velocity or speed is given by

( ) ( )22v r r= + θ (7)

and the direction of v is always tangent to the path.

θ

rP

x

y

rv r=

j

i

v rθ = θ

v

Fig. 3 Components of the velocity vector

The acceleration vector is computed as follows:

( )( ) ( )

d d

d dr

r r

r rt t

r r r r rθ θ

θ

θ

= = + θ

= + + θ + θ + θu

vu u

u u u u

a

Substituting for r θ= θu u and rθ = −θu u from Eq.(3) gives

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( ) ( )( )( ) ( )2 2

r

rr

r

r

r r

r

r

r a

r

a

r

r r θ

θ θ θ

θ θ

θ −θ= + + θ + θ + θ

= +θ θ + +− θ =

u u u

u u u u

u ua

where

( )( )

2

2

radial component

transverse component

ra r r

a r rθ

= ≡

=

− θ

θ + θ ≡

The term 2 2d dtθ = θ is called the

angular acceleration since it measures the change made in angular velocity during an instant of time. Since ra and aθ are always

perpendicular as shown in Fig. 4, the magnitude of the acceleration is simply the positive value of

( ) ( )

( ) ( )

22

2 22 2

ra a a

a r r r r

θ= +

= − θ + θ + θ

The direction is determined from the vector addition of the two components. In general, a will not be tangent to the path, Fig. 4.

θ

r P

x

y

( )2ra r r= − θ

( )2a r rθ = θ+ θ

i

j

a

Fig. 4 Components of the acceleration vector

Cylindrical Coordinates

Motion in three-dimensions (Fig. 5) requires a simple extension of the above formulae to

Position: r zr z= +r u u

Velocity: r zr r zθ= + θ +v u u u

Acceleration: ( ) ( )2 2r zr r r r zθ= − θ + θ + θ +a u u u

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Fig. 5 Cylindrical coordinates in three dimensional motion

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EXAMPLE 12.17 (Textbook )

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EXAMPLE 12.18 (Textbook )

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EXAMPLE 12.19 (Textbook )

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