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.
