12.4: Curvilinear Motion: Polar Coordinates
In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
The particle's location is described using a unit vector along the radial direction. Deriving the particle's position with respect to time provides its velocity. This velocity is comprised of two components: the first is the linear velocity along the radial direction, and the second is the tangential velocity perpendicular to the radial direction.
The time derivative of the velocity yields the acceleration. The angular unit vector's rate of change is the negative product of the angular velocity and the radial unit vector. The second derivative of the angular coordinate represents the angular acceleration of the particle. Analogous to velocity, both components of acceleration are mutually perpendicular. In summary, the polar coordinate system elegantly captures the intricacies of curvilinear motion, unveiling the interplay between radial and tangential dynamics.
