# Forest UCM NLM Ch1 CoordSys

Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.

### Cartesian

Vector Notation convention:

Position:

#### Velocity and Acceleration vector in cartesian coordinates

= =

cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)

= =

Similarly Acceleration is given by

= =

### Polar

Vector Notation convention:

Position:

Because points in a unique direction given by we can write the position vector as

: does not have the units of length

The unit vectors ( and ) are changing in time. You could express the position vector in terms of the cartesian unit vectors in order to avoid this

The dependence of position with can be seen if you look at how the position changes with time.

#### Velocity in Polar Coordinates

Consider the motion of a particle in a circle. At time the particle is at and at time the particle is at

If we take the limit ( or ) then we can write the velocity of this particle traveling in a circle as

or

Thus for circular motion at a constant radius we get the familiar expression

If the particle is not constrained to circular motion ( i.e.: can change with time) then the velocity vector in polar coordinates is

=
or in more compact form

linear velocity Angular velocity

Finding the derivative directly

Cast the unit vector in terms of cartesian coordinates and take the derivative.

#### Acceleration in Polar Coordinates

Taking the derivative of velocity with time gives the acceleration

We need to find the derivative of the unit vector with time.

Consider the position change below in terms of only the unit vector

Using the same arguments used to calculate the rate of change in :

If we take the limit ( or ) then we can write the velocity of this particle traveling in a circle as

or

Finding the derivative of directly

Cast the unit vector in terms of cartesian coordinates and take the derivative.

Substuting the above into our calculation for acceleration:

For the case of circular motion at constant

angular (tangential) acceleration

If

Then there are two additional terms

= Coriolis acceleration (to be described later)

### Cylindrical

Cylindrical coordinates are polar coordinates with a third dimension usually labeled

change picture so angle is  not


We just need to add to all the vectors (remember )

### Spherical

Position:

#### Velocity vector in Spherical coordinates

= =

another way to determine the unit vector derivative is to cast them in terms of cartesian coordinate.

The derivative of the above unit vectors are

substituting the above into the definition of velocity

=
=

=
=
=