12.4:
Curvilinear Motion: Polar Coordinates
The curvilinear motion of a particle can be described using the polar coordinates system.
The radial coordinate, denoted as 'r,' extends outward from the fixed origin to the particle. The angular coordinate, 'θ (theta),' measured in radians, is the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the point.
The particle's position can be expressed using a unit vector along the radial direction. Differentiating the position of the object with time gives the velocity.
Here, the first term is linear velocity along the radial direction, and the second term is the transverse velocity component of the object. These two components of velocity are always perpendicular to each other.
The time derivative of the velocity expression gives the acceleration. The rate of change of the angular unit vector equals the negative product of angular velocity with the radial unit vector.
Here, the second derivative of the angular coordinate is the angular acceleration of the object. Substituting the terms gives the expression for acceleration having the components perpendicular to each other.
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.