2.4: Polar and Cylindrical Coordinates
The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
In the polar coordinate system, as shown in the above figure, the location of a point in a plane is given by two polar coordinates. The first polar coordinate is the radial coordinate, which is the distance to the point from the origin. The second polar coordinate is the angle that the radial vector makes with some chosen direction, usually the positive x-direction. In polar coordinates, angles are measured in radians, or rads.
The radial vector is attached at the origin and points away from the origin to the point. This radial direction is described by a unit radial vector, which can be written as the magnitude times the unit vector in that direction. The second unit vector is a vector orthogonal to the radial vector. The positive direction between the unit vectors indicates how the polar angle changes in the counterclockwise direction.
The transformation equation relates the polar and cartesian coordinates.
Cylindrical-coordinate systems are preferred over Cartesian or polar coordinates for systems with cylindrical symmetry. For example, to describe the surface of a cylinder, Cartesian coordinates require all three coordinates. On the other hand, the cylindrical coordinate system requires only one parameter—the cylinder's radius. As a result, the complicated mathematical calculations become simple.
Cylindrical coordinates belong to the family of curvilinear coordinates. These are an extension of polar coordinates and describe a vector's position in three-dimensional space, as shown in the above figure. A vector in a cylindrical coordinate system is defined using the radial, polar, and z coordinate scalar components. The radial component is the same as the one used in the polar coordinates. It is the distance from the origin to point Q. Here, Q is the projection of point P in the xy plane. The azimuthal angle is again similar to the one used in polar coordinates and represents the angle between the x-axis and the line segment drawn from the origin to point Q. The third cylindrical coordinate, z, is the same as the z cartesian coordinate and denotes the distance of point P to the xy plane.
The transformation equations convert a vector in cylindrical coordinates to Cartesian coordinates.