Engr Help

Center of Curvature - The Intrinsic Description (Normal and Tangential)

Key Concepts: The intrinsic description centers on breaking the components of acceleration

into normal and tangential components. The velocity vector, v, is always tangent to its path

whereas the acceleration, a, has components normal and tangential to its path. Curvature of its

path enters into its description and leads to radius of curvature, ρ, and center of curvature, C.

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. Acceleration in general will have both tangential and normal components. 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, P, and

is the radius of the "osculating circle" shown below for plane motion. The concept of osculating

circle holds for space curves as well. The center of curvature coincides with the center of the

osculating circle.

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 to continue discussion.

Return to Notes on Dynamics