EN4: Dynamics and Vibrations
                                 

 

  Division of Engineering
   Brown University

 

 

2.6.3 Using polar coordinates for planar motion

Normal and tangential coordinates work well when the path of a particle is known, and we only need to compute its velocity and acceleration.

There are situations where a particle moves along a complicated path, and we are interested in computing its position as well as its velocity. In this case it may be best to use cylindrical polar coordinates.

 

A typical cylindrical polar co-ordinate system is shown in the picture. Begin by introducing a fixed {i,j,k} basis in the usual way. The basis is taken to define an inertial frame with fixed origin O. In cylindrical-polar coordinates, the position of the particle is specified by three numbers:

(1) The projected distance from the origin r
(2) The angle shown in the picture
(3) The height z of the particle above the plane containing the i and j vectors

The three numbers are called polar coordinates.

It is straightforward to convert between the usual rectangular Cartesian (x,y,z) coordinates and polar coordinates, and vice-versa:

The z coordinate is the same in both coordinate systems.

When using polar coordinates, we introduce a special basis (called the `natural basis’ for the coordinate system) to describe vectors. The basis is shown in the picture.

(1) The vector points in the radial direction, and is parallel to the i,j plane.

(2) The vector is parallel to k

(3) The vector .

All three vectors must have unit length.

 

Position vector

In our new basis, the position vector of the particle is

Velocity vector

To determine the velocity, we differentiate the position vector

We need to determine the time derivatives of the basis vectors. For this purpose, we need the angular velocity of the basis. The basis vectors must rotate with the same angular velocity as the radial line OA. The angular velocity of this line has already been found to be

Now, using our general expression for time derivatives of unit vectors

 

Substitute these results in our expression for v to see that

Acceleration vector

We can compute the acceleration in exactly the same way. Differentiate the velocity

Substitute for the time derivatives of the basis vectors and simplify to obtain the final result

This result is complicated, and in many cases it is easier to compute the acceleration of a particle by writing down its velocity (or position) and differentiating the expression directly, rather than by remembering and trying to apply this formula.

 

SUMMARY

Position-velocity-acceleration relations in cylindrical polar coordinates are

We can also write the velocity-acceleration relation as follows. Let denote the angular velocity of the basis. Then, write the velocity and acceleration as