Centripetal force
Local coordinates

Local coordinates mean a set of coordinates that travel with the particle, and have orientation determined by the path of the particle. Unit vectors are formed as shown in the image at right, both tangential and normal to the path. This coordinate system sometimes is referred to as intrinsic or path coordinates or nt-coordinates, for normal-tangential, referring to these unit vectors. These coordinates are a very special example of a more general concept of local coordinates from the theory of differential forms.

Distance along the path of the particle is the arc length s, considered to be a known function of time.

A center of curvature is defined at each position s located a distance ρ (the radius of curvature) from the curve on a line along the normal u (s). The required distance ρ (s) at arc length s is defined in terms of the rate of rotation of the tangent to the curve, which in turn is determined by the path itself. If the orientation of the tangent relative to some starting position is θ (s), then ρ (s) is defined by the derivative d θ /d s:

The radius of curvature usually is taken as positive (that is, as an absolute value), while the curvature κ is a signed quantity.

A geometric approach to finding the center of curvature and the radius of curvature uses a limiting process leading to the osculating circle. See image above.

Using these coordinates, the motion along the path is viewed as a succession of circular paths of ever-changing center, and at each position s constitutes non-uniform circular motion at that position with radius ρ. The local value of the angular rate of rotation then is given by:

with the local speed v given by:

As for the other examples above, because unit vectors cannot change magnitude, their rate of change is always perpendicular to their direction (see the left-hand insert in the image above):

Consequently, the velocity and acceleration are:

and using the chain-rule of differentiation:

In this local coordinate system, the acceleration resembles the expression for nonuniform circular motion with the local radius ρ (s), and the centripetal acceleration is identified as the second term.

Extending this approach to three dimensional space curves leads to the Frenet–Serret formulas.
combination:two_responses, detectable_content:postscript, oulipo:avoid_letter[a]