Editing Centripetal force (section)

===== Alternative approach =====

Looking at the image above, one might wonder whether adequate account has been taken of the difference in curvature between ''ρ''(''s'') and ''ρ''(''s'' + d''s'') in computing the arc length as d''s'' = ''ρ''(''s'')d''θ''. Reassurance on this point can be found using a more formal approach outlined below. This approach also makes connection with the article on [[Curvature#In terms of a general parametrization|curvature]].

To introduce the unit vectors of the local coordinate system, one approach is to begin in Cartesian coordinates and describe the local coordinates in terms of these Cartesian coordinates. In terms of arc length ''s'', let the path be described as:<ref name = curvature>The article on [[curvature]] treats a more general case where the curve is parametrized by an arbitrary variable (denoted ''t''), rather than by the arc length ''s''.</ref>
<math display="block">\mathbf{r}(s) = \left[ x(s),\ y(s) \right] . </math>

Then an incremental displacement along the path d''s'' is described by:
<math display="block">\mathrm{d}\mathbf{r}(s) = \left[ \mathrm{d}x(s),\ \mathrm{d}y(s) \right] = \left[ x'(s),\ y'(s) \right] \mathrm{d}s \ , </math>

where primes are introduced to denote derivatives with respect to ''s''. The magnitude of this displacement is d''s'', showing that:<ref name = Shabana>{{cite book |title = Railroad Vehicle Dynamics: A Computational Approach |author1 = Ahmed A. Shabana |author2 = Khaled E. Zaazaa |author3 = Hiroyuki Sugiyama |page = 91 |url = https://books.google.com/books?id=YgIDSQT0FaUC&pg=PA207 |isbn = 978-1-4200-4581-9 |publisher = CRC Press |year = 2007 |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060724/https://books.google.com/books?id=YgIDSQT0FaUC&pg=PA207#v=onepage&q&f=false |url-status = live }}</ref>
: <math>\left[ x'(s)^2 + y'(s)^2 \right] = 1 \ . </math> {{anchor|Eq. 1}}(Eq. 1)

This displacement is necessarily a tangent to the curve at ''s'', showing that the unit vector tangent to the curve is:
<math display="block">\mathbf{u}_\mathrm{t}(s) = \left[ x'(s), \ y'(s) \right] , </math>
while the outward unit vector normal to the curve is
<math display="block">\mathbf{u}_\mathrm{n}(s) = \left[ y'(s),\ -x'(s) \right] , </math>

[[Orthogonality]] can be verified by showing that the vector [[dot product]] is zero. The unit magnitude of these vectors is a consequence of [[#Eq. 1|Eq. 1]]. Using the tangent vector, the angle ''θ'' of the tangent to the curve is given by:
<math display="block">\sin \theta = \frac{y'(s)}{\sqrt{x'(s)^2 + y'(s)^2}} = y'(s) \ ;</math> and <math>\cos \theta = \frac{x'(s)}{\sqrt{x'(s)^2 + y'(s)^2}} = x'(s) \ .</math>

The radius of curvature is introduced completely formally (without need for geometric interpretation) as:
<math display="block">\frac{1}{\rho} = \frac{\mathrm{d}\theta}{\mathrm{d}s}\ . </math>

The derivative of ''θ'' can be found from that for sin''θ'':
<math display="block">\frac{\mathrm{d} \sin\theta}{\mathrm{d}s} = \cos \theta \frac {\mathrm{d}\theta}{\mathrm{d}s} = \frac{1}{\rho} \cos \theta \ = \frac{1}{\rho} x'(s)\ . </math>

Now:
<math display="block">\frac{\mathrm{d} \sin \theta }{\mathrm{d}s} = \frac{\mathrm{d}}{\mathrm{d}s} \frac{y'(s)}{\sqrt{x'(s)^2 + y'(s)^2}} = \frac{y''(s)x'(s)^2-y'(s)x'(s)x''(s)} {\left(x'(s)^2 + y'(s)^2\right)^{3/2}}\ , </math>
in which the denominator is unity. With this formula for the derivative of the sine, the radius of curvature becomes:
<math display="block">\frac {\mathrm{d}\theta}{\mathrm{d}s} = \frac{1}{\rho} = y''(s)x'(s) - y'(s)x''(s) = \frac{y''(s)}{x'(s)} = -\frac{x''(s)}{y'(s)} \ ,</math>
where the equivalence of the forms stems from differentiation of [[#Eq. 1|Eq. 1]]:
<math display="block">x'(s)x''(s) + y'(s)y''(s) = 0 \ . </math>
With these results, the acceleration can be found:
<math display="block">\begin{align}
\mathbf{a}(s) &= \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{v}(s) = \frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\mathrm{d}s}{\mathrm{d}t} \left( x'(s), \ y'(s) \right) \right] \\
& = \left(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\right)\mathbf{u}_\mathrm{t}(s) + \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right) ^2 \left(x''(s),\ y''(s) \right) \\
& = \left(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\right)\mathbf{u}_\mathrm{t}(s) - \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right) ^2 \frac{1}{\rho} \mathbf{u}_\mathrm{n}(s)
\end{align}</math>
as can be verified by taking the dot product with the unit vectors '''u'''<sub>t</sub>(''s'') and '''u'''<sub>n</sub>(''s''). This result for acceleration is the same as that for circular motion based on the radius ''ρ''. Using this coordinate system in the inertial frame, it is easy to identify the force normal to the trajectory as the centripetal force and that parallel to the trajectory as the tangential force. From a qualitative standpoint, the path can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius.

This result for acceleration agrees with that found earlier. However, in this approach, the question of the change in radius of curvature with ''s'' is handled completely formally, consistent with a geometric interpretation, but not relying upon it, thereby avoiding any questions the image above might suggest about neglecting the variation in ''ρ''.
{{anchor|circular_motion}}