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Kinematics of a Particle in a Plane - The Intrinsic Description (Normal and Tangential)

Key Concepts: The intrinsic description provides yet another way to examine the motion of a particle (point mass) in the xy-plane with respect to a fixed frame of reference with origin O. Rather than starting with the position vector, r, start with the velocity vector, v, and, as before take its derivative to obtain the acceleration vector, a.

In a Nut Shell: The intrinsic description uses the arc length, s, along the path of motion. Note that the direction of both unit vectors , e_t , in the tangential direction and e_n , in the normal direction change with time. Arc length, s, is a measure of distance along the path. Although the velocity is always tangent to its path, the acceleration in general will have both a tangential and a normal component. The normal component is always directed toward its center of curvature, C. ρ is the radius of curvature measured from center of curvature to the particle. See the figure below.

v = (ds/dt) e_t = v e_t = velocity vector of particle at point P at any time t where ds is the differential displacement of the particle along its path and ds/dt = | v | = v = speed of particle, a scalar.

a = dv /dt = [d²s/dt²] e_t + [ v²/ρ] e_n = dv/dt e_t + [ v²/ρ] e_n

equals the acceleration vector of particle at point P at any time t

NOTE: dv/dt is the tangential component of acceleration and

v²/ρ is the normal component of acceleration

Click here for examples. Click here to return to the rectangular description.

Click here for a discussion on center of curvature and the "osculating" circle.

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