Begin2.DVI

Here the radial component of acceleration is ( ̈r−r(θ ̇)^2 )and the transverse component

of acceleration or tangential component is (rθ ̈+2 ̇rθ ̇). The magnitude of the acceleration

is given by

a=|
a |=

√

( ̈r−r(θ ̇)^2 )^2 + (rθ ̈+ 2 ̇rθ ̇)^2

Velocity and Acceleration in Cylindrical Coordinates

In rectangular (x, y, z)coordinates the position vector, velocity vector and accel-

eration vector of a moving particle are given by

v =
r =xˆe 1 +yˆe 2 +zˆe 3

v =d
rdt =dxdt ˆe 1 +dydt ˆe 2 +dzdt ˆe 3

a =d
r

dt

=d

(^2)
r
dt^2
=d
(^2) x
dt^2
ˆe 1 +d
(^2) y
dt^2
ˆe 2 +d
(^2) z
dt^2
ˆe 3

Upon changing to a cylindrical coordinates (r, θ, z)using the transformation equations

x=rcos θ, y =rsin θ, z =z

one can represent the position vector of the particle as

r =rcos θˆe 1 +rsin θˆe 2 +zˆe 3 (9 .26)

Using the orthogonal unit vectors

ˆer=∂
r

∂r

= cos θˆe 1 + sin θeˆ 2

ˆeθ=^1

r

∂
r

∂θ =−sin θ

ˆe 1 + cos θˆe 2

ˆez=∂
r

∂z =

ˆe 3

(9 .27)

obtained from equations (7.107), the position vector of a moving particle can be

expressed in cylindrical coordinates as

r =rˆer+zˆez (9 .28)

To obtain the velocity vector in cylindrical coordinates one must differentiate equa-

tion (9.28) with respect to time tto obtain

v =

d
r

dt =

dr

dt ˆer+r

dˆer

dt +

dz

dt ˆez (9 .29)