Editing Centripetal force (section)

=== General planar motion ===
{{See also|Generalized forces|Generalized force|Curvilinear coordinates|Generalized coordinates|Orthogonal coordinates}}
{{multiple image
|align = vertical
|width1 = 100
|image1 = Position vector plane polar coords.svg
|caption1 = Position vector '''r''', always points radially from the origin.
|width2 = 150
|image2 = Velocity vector plane polar coords.svg
|caption2 = Velocity vector '''v''', always tangent to the path of motion.
|width3 = 200
|image3 = Acceleration vector plane polar coords.svg
|caption3 = Acceleration vector '''a''', not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
|footer = Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.}}
[[File:Polar unit vectors.PNG|thumb|Polar unit vectors at two times ''t'' and ''t'' + ''dt'' for a particle with trajectory '''r''' ( ''t'' ); on the left the unit vectors '''u'''<sub>ρ</sub> and '''u'''<sub>θ</sub> at the two times are moved so their tails all meet, and are shown to trace an arc of a unit radius circle. Their rotation in time ''dt'' is ''d''θ, just the same angle as the rotation of the trajectory '''r''' ( ''t'' ).]]

==== Polar coordinates ====

The above results can be derived perhaps more simply in [[Polar coordinate system|polar coordinates]], and at the same time extended to general motion within a plane, as shown next. Polar coordinates in the plane employ a radial unit vector '''u'''<sub>ρ</sub> and an angular unit vector '''u'''<sub>θ</sub>, as shown above.<ref name = polar>Although the polar coordinate system moves with the particle, the observer does not. The description of the particle motion remains a description from the stationary observer's point of view.</ref> A particle at position '''r''' is described by:

<math display="block">\mathbf{r} = \rho \mathbf{u}_{\rho} \ , </math>

where the notation ''ρ'' is used to describe the distance of the path from the origin instead of ''R'' to emphasize that this distance is not fixed, but varies with time. The unit vector '''u'''<sub>ρ</sub> travels with the particle and always points in the same direction as '''r'''(''t''). Unit vector '''u'''<sub>θ</sub> also travels with the particle and stays orthogonal to '''u'''<sub>ρ</sub>. Thus, '''u'''<sub>ρ</sub> and '''u'''<sub>θ</sub> form a local Cartesian coordinate system attached to the particle, and tied to the path travelled by the particle.<ref>Notice that this local coordinate system is not autonomous; for example, its rotation in time is dictated by the trajectory traced by the particle. The radial vector '''r'''(''t'') does not represent the [[Osculating circle|radius of curvature]] of the path.</ref> By moving the unit vectors so their tails coincide, as seen in the circle at the left of the image above, it is seen that '''u'''<sub>ρ</sub> and '''u'''<sub>θ</sub> form a right-angled pair with tips on the unit circle that trace back and forth on the perimeter of this circle with the same angle ''θ''(''t'') as '''r'''(''t'').

When the particle moves, its velocity is

: <math> \mathbf{v} = \frac {\mathrm{d} \rho }{\mathrm{d}t} \mathbf{u}_{\rho} + \rho \frac {\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} \, . </math>

To evaluate the velocity, the derivative of the unit vector '''u'''<sub>ρ</sub> is needed. Because '''u'''<sub>ρ</sub> is a unit vector, its magnitude is fixed, and it can change only in direction, that is, its change d'''u'''<sub>ρ</sub> has a component only perpendicular to '''u'''<sub>ρ</sub>. When the trajectory '''r'''(''t'') rotates an amount d''θ'', '''u'''<sub>ρ</sub>, which points in the same direction as '''r'''(''t''), also rotates by d''θ''. See image above. Therefore, the change in '''u'''<sub>ρ</sub> is

: <math> \mathrm{d} \mathbf{u}_{\rho} = \mathbf{u}_{\theta} \mathrm{d}\theta \, , </math>

or

: <math> \frac {\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} = \mathbf{u}_{\theta} \frac {\mathrm{d}\theta}{\mathrm{d}t} \, . </math>

In a similar fashion, the rate of change of '''u'''<sub>θ</sub> is found. As with '''u'''<sub>ρ</sub>, '''u'''<sub>θ</sub> is a unit vector and can only rotate without changing size. To remain orthogonal to '''u'''<sub>ρ</sub> while the trajectory '''r'''(''t'') rotates an amount d''θ'', '''u'''<sub>θ</sub>, which is orthogonal to '''r'''(''t''), also rotates by d''θ''. See image above. Therefore, the change d'''u'''<sub>θ</sub> is orthogonal to '''u'''<sub>θ</sub> and proportional to d''θ'' (see image above):

: <math> \frac{\mathrm{d} \mathbf{u}_{\theta}}{\mathrm{d}t} = -\frac {\mathrm{d} \theta} {\mathrm{d}t} \mathbf{u}_{\rho} \, . </math>

The equation above shows the sign to be negative: to maintain orthogonality, if d'''u'''<sub>ρ</sub> is positive with d''θ'', then d'''u'''<sub>θ</sub> must decrease.

Substituting the derivative of '''u'''<sub>ρ</sub> into the expression for velocity:

: <math> \mathbf{v} = \frac {\mathrm{d} \rho }{\mathrm{d}t} \mathbf{u}_{\rho} + \rho \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t} = v_{\rho} \mathbf{u}_{\rho} + v_{\theta} \mathbf{u}_{\theta} = \mathbf{v}_{\rho} + \mathbf{v}_{\theta} \, . </math>

To obtain the acceleration, another time differentiation is done:
: <math> \mathbf{a} = \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2} \mathbf{u}_{\rho} + \frac {\mathrm{d} \rho }{\mathrm{d}t} \frac{\mathrm{d} \mathbf{u}_{\rho}}{\mathrm{d}t} + \frac {\mathrm{d} \rho}{\mathrm{d}t} \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \frac{\mathrm{d} \mathbf{u}_{\theta}}{\mathrm{d}t} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \mathbf{u}_{\theta} \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2} \, . </math>

Substituting the derivatives of '''u'''<sub>ρ</sub> and '''u'''<sub>θ</sub>, the acceleration of the particle is:<ref name = Taylor>{{cite book |title = Classical Mechanics |author = John Robert Taylor |pages = 28–29 |url = https://books.google.com/books?id=P1kCtNr-pJsC |year = 2005 |isbn = 978-1-891389-22-1 |publisher = University Science Books |location = Sausalito CA |access-date = 4 November 2020 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060320/https://books.google.com/books?id=P1kCtNr-pJsC |url-status = live }}</ref>
: <math>\begin{align}
\mathbf{a} & = \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2} \mathbf{u}_{\rho} + 2\frac {\mathrm{d} \rho}{\mathrm{d}t} \mathbf{u}_{\theta} \frac {\mathrm{d} \theta} {\mathrm{d}t} - \rho \mathbf{u}_{\rho} \left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 + \rho \mathbf{u}_{\theta} \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2} \ , \\
& = \mathbf{u}_{\rho} \left[ \frac {\mathrm{d}^2 \rho }{\mathrm{d}t^2}-\rho\left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 \right] + \mathbf{u}_{\theta}\left[ 2\frac {\mathrm{d} \rho}{\mathrm{d}t} \frac {\mathrm{d} \theta} {\mathrm{d}t} + \rho \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2}\right] \\
& = \mathbf{u}_{\rho} \left[ \frac {\mathrm{d}v_{\rho}}{\mathrm{d}t}-\frac{v_{\theta}^2}{\rho}\right] + \mathbf{u}_{\theta}\left[ \frac{2}{\rho}v_{\rho} v_{\theta} + \rho\frac{\mathrm{d}}{\mathrm{d}t}\frac{v_{\theta}}{\rho}\right] \, .
\end{align}</math>

As a particular example, if the particle moves in a circle of constant radius ''R'', then d''ρ''/d''t'' = 0, '''v''' = '''v'''<sub>θ</sub>, and:

<math display="block">\mathbf{a} = \mathbf{u}_{\rho} \left[ -\rho\left( \frac {\mathrm{d} \theta} {\mathrm{d}t}\right)^2 \right] + \mathbf{u}_{\theta}\left[ \rho \frac {\mathrm{d}^2 \theta} {\mathrm{d}t^2}\right] = \mathbf{u}_{\rho} \left[ -\frac{v^2}{r}\right] + \mathbf{u}_{\theta}\left[ \frac {\mathrm{d} v} {\mathrm{d}t}\right] \ </math>

where <math> v = v_{\theta}. </math>

These results agree with those above for [[#Nonuniform circular motion|nonuniform circular motion]]. See also the article on [[non-uniform circular motion]]. If this acceleration is multiplied by the particle mass, the leading term is the centripetal force and the negative of the second term related to angular acceleration is sometimes called the [[Euler force]].<ref name = Lanczos>{{cite book |author = Cornelius Lanczos |title = The Variational Principles of Mechanics |url = https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA103|page = 103|year = 1986|publisher = Courier Dover Publications|isbn = 978-0-486-65067-8|location = New York}}</ref>

For trajectories other than circular motion, for example, the more general trajectory envisioned in the image above, the instantaneous center of rotation and radius of curvature of the trajectory are related only indirectly to the coordinate system defined by '''u<sub>ρ</sub>''' and '''u<sub>θ</sub>''' and to the length |'''r'''(''t'')| = ''ρ''. Consequently, in the general case, it is not straightforward to disentangle the centripetal and Euler terms from the above general acceleration equation.<ref name = Curtis>See, for example, {{cite book |title = Orbital Mechanics for Engineering Students |author = Howard D. Curtis |isbn = 978-0-7506-6169-0 |publisher = Butterworth-Heinemann |year = 2005 |page = [https://archive.org/details/orbitalmechanics00curt_535/page/n21 5] |title-link = Orbital Mechanics for Engineering Students }}</ref><ref name = Lee>{{cite book |title = Accelerator physics |author = S. Y. Lee |page = 37 |url = https://books.google.com/books?id=VTc8Sdld5S8C&pg=PA37 |isbn = 978-981-256-182-4 |publisher = World Scientific |location = Hackensack NJ |edition = 2nd |year = 2004 |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060154/https://books.google.com/books?id=VTc8Sdld5S8C&pg=PA37#v=onepage&q&f=false |url-status = live }}</ref> To deal directly with this issue, local coordinates are preferable, as discussed next.

==== Local coordinates ====

[[File:Local unit vectors.PNG|thumb |Local coordinate system for planar motion on a curve. Two different positions are shown for distances ''s'' and ''s'' + ''ds'' along the curve. At each position ''s'', unit vector '''u'''<sub>n</sub> points along the outward normal to the curve and unit vector '''u'''<sub>t</sub> is tangential to the path. The radius of curvature of the path is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the [[osculating circle]] at position ''s''. The unit circle on the left shows the rotation of the unit vectors with ''s''.]]

Local coordinates mean a set of coordinates that travel with the particle,<ref name = observer>The ''observer'' of the motion along the curve is using these local coordinates to describe the motion from the observer's ''frame of reference'', that is, from a stationary point of view. In other words, although the local coordinate system moves with the particle, the observer does not. A change in coordinate system used by the observer is only a change in their ''description'' of observations, and does not mean that the observer has changed their state of motion, and ''vice versa''.</ref> and have orientation determined by the path of the particle.<ref name = Ito>{{cite book |title = The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains |author1 = Zhilin Li |author2 = Kazufumi Ito |page = 16 |url = https://books.google.com/books?id=_E084AX-iO8C&pg=PA16 |isbn = 978-0-89871-609-2 |year = 2006 |publisher = Society for Industrial and Applied Mathematics |location = Philadelphia |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060154/https://books.google.com/books?id=_E084AX-iO8C&pg=PA16 |url-status = live }}</ref> 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''<ref name = Kumar>{{cite book |title = Engineering Mechanics |author = K L Kumar |page = 339 |url = https://books.google.com/books?id=QabMJsCf2zgC&pg=PA339 |isbn = 978-0-07-049473-2 |publisher = Tata McGraw-Hill |location = New Delhi |year = 2003 |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060155/https://books.google.com/books?id=QabMJsCf2zgC&pg=PA339 |url-status = live }}</ref><ref name = Rao>{{cite book |title = Engineering Dynamics: Statics and Dynamics |author1 = Lakshmana C. Rao |author2 = J. Lakshminarasimhan |author3 = Raju Sethuraman |author4 = SM Sivakuma |page = 133 |url = https://books.google.com/books?id=F7gaa1ShPKIC&pg=PA134 |isbn = 978-81-203-2189-2 |publisher = Prentice Hall of India |year = 2004 |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060156/https://books.google.com/books?id=F7gaa1ShPKIC&pg=PA134#v=onepage&q&f=false |url-status = live }}</ref> 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.<ref name = Morita>{{cite book |title = Geometry of Differential Forms |author = Shigeyuki Morita |page = [https://archive.org/details/geometryofdiffer00mori/page/1 1] |url = https://archive.org/details/geometryofdiffer00mori|url-access = registration |quote = local coordinates. |isbn = 978-0-8218-1045-3 |year = 2001 |publisher = American Mathematical Society }}</ref>

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

: <math> s = s(t) \ . </math>

A center of curvature is defined at each position ''s'' located a distance ''ρ'' (the [[Osculating circle#Mathematical description|radius of curvature]]) from the curve on a line along the normal '''u'''<sub>n</sub> (''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'':

: <math>\frac{1} {\rho (s)} = \kappa (s) = \frac {\mathrm{d}\theta}{\mathrm{d}s}\ . </math>

The radius of curvature usually is taken as positive (that is, as an absolute value), while the ''[[Curvature#In terms of a general parametrization|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]].<ref name = osculating>The osculating circle at a given point ''P'' on a curve is the limiting circle of a sequence of circles that pass through ''P'' and two other points on the curve, ''Q'' and ''R'', on either side of ''P'', as ''Q'' and ''R'' approach ''P''. See the online text by Lamb: {{cite book |title = An Elementary Course of Infinitesimal Calculus|author = Horace Lamb |page = [https://archive.org/details/anelementarycou01lambgoog/page/n429 406] |url = https://archive.org/details/anelementarycou01lambgoog |quote = osculating circle.|publisher = University Press |year = 1897 |isbn = 978-1-108-00534-0 }}</ref><ref name = Chen0>{{cite book|title = An Introduction to Planar Dynamics|author1 = Guang Chen|author2 = Fook Fah Yap|edition = 3rd|url = https://books.google.com/books?id=xt09XiZBzPEC&pg=PA34|page = 34|isbn = 978-981-243-568-2|year = 2003|publisher = Central Learning Asia/Thomson Learning Asia|access-date = 30 March 2021|archive-date = 7 October 2024|archive-url = https://web.archive.org/web/20241007060232/https://books.google.com/books?id=xt09XiZBzPEC&pg=PA34|url-status = live}}</ref> 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:

: <math> \omega(s) = \frac{\mathrm{d}\theta}{\mathrm{d}t} = \frac{\mathrm{d}\theta}{\mathrm{d}s} \frac {\mathrm{d}s}{\mathrm{d}t} = \frac{1}{\rho(s)}\ \frac {\mathrm{d}s}{\mathrm{d}t} = \frac{v(s)}{\rho(s)}\ ,</math>
with the local speed ''v'' given by:

: <math> v(s) = \frac {\mathrm{d}s}{\mathrm{d}t}\ . </math>

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):<ref name = Gregory>{{cite book |title = Classical Mechanics: An Undergraduate Text |author = R. Douglas Gregory |page = 20 |url = https://books.google.com/books?id=uAfUQmQbzOkC&pg=PA18 |isbn = 978-0-521-82678-5 |publisher = Cambridge University Press |year = 2006 |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060158/https://books.google.com/books?id=uAfUQmQbzOkC&pg=PA18#v=onepage&q&f=false |url-status = live }}</ref>

: <math>\frac{d\mathbf{u}_\mathrm{n}(s)}{ds} = \mathbf{u}_\mathrm{t}(s)\frac{d\theta}{ds} = \mathbf{u}_\mathrm{t}(s)\frac{1}{\rho} \ ; </math> <math>\frac{d\mathbf{u}_\mathrm{t}(s)}{\mathrm{d}s} = -\mathbf{u}_\mathrm{n}(s)\frac{\mathrm{d}\theta}{\mathrm{d}s} = - \mathbf{u}_\mathrm{n}(s)\frac{1}{\rho} \ . </math>
Consequently, the velocity and acceleration are:<ref name = Chen0/><ref name = Whittaker>{{cite book |title = [[Analytical Dynamics of Particles and Rigid Bodies|A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: with an introduction to the problem of three bodies]] |author1=Edmund Taylor Whittaker|author-link1=E. T. Whittaker |author2=William McCrea|author-link2=William Hunter McCrea|page = 20 |edition = 4th |isbn = 978-0-521-35883-5 |publisher = Cambridge University Press |year = 1988 }}</ref><ref name = Ginsberg>{{cite book |title = Engineering Dynamics |author = Jerry H. Ginsberg |page = 33 |url = https://books.google.com/books?id=je0W8N5oXd4C&pg=PA723 |isbn = 978-0-521-88303-0 |year = 2007 |publisher = Cambridge University Press }}</ref>
: <math> \mathbf{v}(t) = v \mathbf{u}_\mathrm{t}(s)\ ; </math>
and using the [[Chain rule|chain-rule of differentiation]]:

: <math> \mathbf{a}(t) = \frac{\mathrm{d}v}{\mathrm{d}t} \mathbf{u}_\mathrm{t}(s) - \frac{v^2}{\rho}\mathbf{u}_\mathrm{n}(s) \ ; </math> with the tangential acceleration <math>\frac{\mathrm{\mathrm{d}}v}{\mathrm{\mathrm{d}}t} = \frac{\mathrm{d}v}{\mathrm{d}s}\ \frac{\mathrm{d}s}{\mathrm{d}t} = \frac{\mathrm{d}v}{\mathrm{d}s}\ v \ . </math>

In this local coordinate system, the acceleration resembles the expression for [[#Nonuniform circular motion|nonuniform circular motion]] with the local radius ''ρ''(''s''), and the centripetal acceleration is identified as the second term.<ref name = Shelley>{{cite book |title = 800 solved problems in vector mechanics for engineers: Dynamics |author = Joseph F. Shelley |page = 47 |url = https://books.google.com/books?id=ByNrVgf041MC&pg=PA46 |isbn = 978-0-07-056687-3 |publisher = McGraw-Hill Professional |year = 1990 |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060744/https://books.google.com/books?id=ByNrVgf041MC&pg=PA46 |url-status = live }}</ref>

Extending this approach to three dimensional space curves leads to the [[Frenet–Serret formulas]].<ref name = Andrews>{{cite book |title = Mathematical Techniques for Engineers and Scientists |author1 = Larry C. Andrews |author2 = Ronald L. Phillips |page = 164 |url = https://books.google.com/books?id=MwrDfvrQyWYC&pg=PA164 |isbn = 978-0-8194-4506-3 |publisher = SPIE Press |year = 2003 |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060709/https://books.google.com/books?id=MwrDfvrQyWYC&pg=PA164#v=onepage&q&f=false |url-status = live }}</ref><ref name = Chand>{{cite book |title = Applied Mathematics |page = 337 |author1 = Ch V Ramana Murthy |author2 = NC Srinivas |isbn = 978-81-219-2082-7 |url = https://books.google.com/books?id=Q0Pvv4vWOlQC&pg=PA337 |publisher = S. Chand & Co. |year = 2001 |location = New Delhi |access-date = 4 November 2020 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060710/https://books.google.com/books?id=Q0Pvv4vWOlQC&pg=PA337 |url-status = live }}</ref>

===== 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}}

===== Example: circular motion =====

To illustrate the above formulas, let ''x'', ''y'' be given as:
: <math>x = \alpha \cos \frac{s}{\alpha} \ ; \ y = \alpha \sin\frac{s}{\alpha} \ .</math>

Then:
: <math>x^2 + y^2 = \alpha^2 \ , </math>

which can be recognized as a circular path around the origin with radius ''α''. The position ''s'' = 0 corresponds to [''α'', 0], or 3 o'clock. To use the above formalism, the derivatives are needed:

: <math>y^{\prime}(s) = \cos \frac{s}{\alpha} \ ; \ x^{\prime}(s) = -\sin \frac{s}{\alpha} \ , </math>
: <math>y^{\prime\prime}(s) = -\frac{1}{\alpha}\sin\frac{s}{\alpha} \ ; \ x^{\prime\prime}(s) = -\frac{1}{\alpha}\cos \frac{s}{\alpha} \ . </math>

With these results, one can verify that:
: <math> x^{\prime}(s)^2 + y^{\prime}(s)^2 = 1 \ ; \ \frac{1}{\rho} = y^{\prime\prime}(s)x^{\prime}(s)-y^{\prime}(s)x^{\prime\prime}(s) = \frac{1}{\alpha} \ . </math>

The unit vectors can also be found:
: <math>\mathbf{u}_\mathrm{t}(s) = \left[-\sin\frac{s}{\alpha} \ , \ \cos\frac{s}{\alpha} \right] \ ; \ \mathbf{u}_\mathrm{n}(s) = \left[\cos\frac{s}{\alpha} \ , \ \sin\frac{s}{\alpha} \right] \ , </math>

which serve to show that ''s'' = 0 is located at position [''ρ'', 0] and ''s'' = ''ρ''π/2 at [0, ''ρ''], which agrees with the original expressions for ''x'' and ''y''. In other words, ''s'' is measured counterclockwise around the circle from 3 o'clock. Also, the derivatives of these vectors can be found:
: <math>\frac{\mathrm{d}}{\mathrm{d}s}\mathbf{u}_\mathrm{t}(s) = -\frac{1}{\alpha} \left[\cos\frac{s}{\alpha} \ , \ \sin\frac{s}{\alpha} \right] = -\frac{1}{\alpha}\mathbf{u}_\mathrm{n}(s) \ ; </math>
: <math> \ \frac{\mathrm{d}}{\mathrm{d}s}\mathbf{u}_\mathrm{n}(s) = \frac{1}{\alpha} \left[-\sin\frac{s}{\alpha} \ , \ \cos\frac{s}{\alpha} \right] = \frac{1}{\alpha}\mathbf{u}_\mathrm{t}(s) \ . </math>

To obtain velocity and acceleration, a time-dependence for ''s'' is necessary. For counterclockwise motion at variable speed ''v''(''t''):
: <math>s(t) = \int_0^t \ dt^{\prime} \ v(t^{\prime}) \ , </math>
where ''v''(''t'') is the speed and ''t'' is time, and ''s''(''t'' = 0) = 0. Then:
: <math>\mathbf{v} = v(t)\mathbf{u}_\mathrm{t}(s) \ ,</math>
: <math>\mathbf{a} = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s) + v\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s) = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s)-v\frac{1}{\alpha}\mathbf{u}_\mathrm{n}(s)\frac{\mathrm{d}s}{\mathrm{d}t} </math>
: <math>\mathbf{a} = \frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_\mathrm{t}(s)-\frac{v^2}{\alpha}\mathbf{u}_\mathrm{n}(s) \ , </math>
where it already is established that α = ρ. This acceleration is the standard result for [[non-uniform circular motion]].