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Cylindrical Coordinates

In cylindrical co-ordinates, the position of particle P relative to origin O is described by the two circular polar co-ordinates (ρ, Ø) defined in the plane Z = 0, and the third being the Z-co-ordinates itself. The cylindrical co-ordinates (ρ, Ø, z) are related to Cartesian co-ordinates as,

Conversely, x = ρ cos Ø, y = ρ sin Ø, z = z (ii)

The equation ρ = constant describes a right circular cylinder of radius ρ about Z-axis. The equation Ø = constant describes the plane containing Z-axis and making an angle Ø to the XZ-plane. The general differential displacement is given by

d r = d ρ e_ρ + ρ d Ø e_Ø + dz e_z

where e_ρ and e_Ø are related to (i, j, k). The unit vectors (e_ρ, e_Ø, e_z) constitute an orthogonal (right-handed) co-ordinate system, i.e.

e_ρ × e_Ø = e_z, e_Ø × e_z= e_ρ, e_z × e_ρ = e_Ø

and, e_ρ. e_Ø = e_ρ . e_z = e_Ø . e_z = 0

In situations where there is an axis of symmetry, cylindrical co-ordinates becomes the right choice to describe the system. A typical example is the (planar) rotation of a particle about an axis.

In general, if a particle moves in a plane, we describe its motion in polar co-ordinates (ρ, Ø), where ρ gives the radial position of the particle. (We can call the plane as Z = 0, or XY-plane). In particular, if the particle moves in a circle of radius ρ, we have,

where and The vector represents angular velocity whose magnitude is Ø and direction is about the axis of rotation.

The acceleration of the particle rotating about Z-axis can be rewritten as,

is the tangential acceleration.

Note that if the particle is rotating in the XY-plane, corresponds to position vector of the particle. Hence, one usually denotes by r and hoping no confusion, we write

we shall frequently need above relations when describing the motion in rotating frame, and/or rotating motion of an object.