Editing Centripetal force (section)

== Analysis of several cases ==

Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration.

=== Uniform circular motion ===
{{See also|Uniform circular motion}}

Uniform circular motion refers to the case of constant rate of rotation. Here are two approaches to describing this case.

==== Calculus derivation ====
In two dimensions, the position vector <math>\textbf{r}</math>, which has magnitude (length) <math>r</math> and directed at an angle <math>\theta</math> above the x-axis, can be expressed in [[Cartesian coordinates]] using the [[unit vectors]] <math alt="x-hat">\hat\mathbf x</math> and <math alt="y-hat">\hat\mathbf y</math>:<ref>
{{cite book
| title = Vectors in physics and engineering
| author = A. V. Durrant
| publisher = CRC Press
| year = 1996
| isbn = 978-0-412-62710-1
| page = 103
| url = https://books.google.com/books?id=cuMLGAO-ii0C&pg=PA103
}}</ref>
<math display="block"> \textbf{r} = r \cos(\theta) \hat\mathbf x + r \sin(\theta) \hat\mathbf y. </math>

The assumption of [[uniform circular motion]] requires three things:
# The object moves only on a circle.
# The radius of the circle <math>r</math> does not change in time.
# The object moves with constant [[angular velocity]] <math>\omega</math> around the circle. Therefore, <math>\theta = \omega t</math> where <math>t</math> is time.

The [[velocity]] <math>\textbf{v}</math> and [[acceleration]] <math>\textbf{a}</math> of the motion are the first and second derivatives of position with respect to time:

<math display="block"> \textbf{r} = r \cos(\omega t) \hat\mathbf x + r \sin(\omega t) \hat\mathbf y, </math>
<math display="block" qid="Q11465"> \textbf{v} = \dot{\textbf{r}} = - r \omega \sin(\omega t) \hat\mathbf x + r \omega \cos(\omega t) \hat\mathbf y,  </math>
<math display="block" qid=Q11376> \textbf{a} = \ddot{\textbf{r}} = - \omega^2 (r \cos(\omega t) \hat\mathbf x + r \sin(\omega t) \hat\mathbf y). </math>

The term in parentheses is the original expression of <math>\textbf{r}</math> in [[Cartesian coordinates]]. Consequently,
<math display="block"> \textbf{a} = - \omega^2 \textbf{r}. </math>
The negative sign shows that the acceleration is pointed towards the center of the circle (opposite the radius), hence it is called "centripetal" (i.e. "center-seeking"). While objects naturally follow a straight path (due to [[inertia]]), this centripetal acceleration describes the circular motion path caused by a centripetal force.

==== Derivation using vectors ====

[[File:Circular motion vectors.svg|right|thumb|Vector relationships for uniform circular motion; vector '''Ω''' representing the rotation is normal to the plane of the orbit with polarity determined by the [[right-hand rule]] and magnitude ''dθ'' /''dt''.]]
The image at right shows the vector relationships for uniform circular motion. The rotation itself is represented by the angular velocity vector '''Ω''', which is normal to the plane of the orbit (using the [[right-hand rule]]) and has magnitude given by:
: <math> |\mathbf{\Omega}| = \frac {\mathrm{d} \theta } {\mathrm{d}t} = \omega \ , </math>

with ''θ'' the angular position at time ''t''. In this subsection, d''θ''/d''t'' is assumed constant, independent of time. The distance traveled '''dℓ''' of the particle in time d''t'' along the circular path is
: <math> \mathrm{d}\boldsymbol{\ell} = \mathbf {\Omega} \times \mathbf{r}(t) \mathrm{d}t \ , </math>

which, by properties of the [[vector cross product]], has magnitude ''r''d''θ'' and is in the direction tangent to the circular path.

Consequently,
: <math>\frac {\mathrm{d} \mathbf{r}}{\mathrm{d}t} = \lim_{{\Delta}t \to 0} \frac {\mathbf{r}(t + {\Delta}t)-\mathbf{r}(t)}{{\Delta}t} = \frac{\mathrm{d} \boldsymbol{\ell}}{\mathrm{d}t} \ .</math>

In other words,
: <math> \mathbf{v}\ \stackrel{\mathrm{def}}{ = }\ \frac {\mathrm{d} \mathbf{r}}{\mathrm{d}t} = \frac {\mathrm{d}\mathbf{\boldsymbol{\ell}}}{\mathrm{d}t} = \mathbf {\Omega} \times \mathbf{r}(t)\ . </math>

Differentiating with respect to time,
<math display="block"> \mathbf{a}\ \stackrel{\mathrm{def}}{ = }\ \frac {\mathrm{d} \mathbf{v}} {d\mathrm{t}} = \mathbf {\Omega} \times \frac{\mathrm{d} \mathbf{r}(t)}{\mathrm{d}t} = \mathbf{\Omega} \times \left[ \mathbf {\Omega} \times \mathbf{r}(t)\right] \ .</math>

[[Triple product#Vector triple product|Lagrange's formula]] states:
<math display="block"> \mathbf{a} \times \left ( \mathbf{b} \times \mathbf{c} \right ) = \mathbf{b} \left ( \mathbf{a} \cdot \mathbf{c} \right ) - \mathbf{c} \left ( \mathbf{a} \cdot \mathbf{b} \right ) \ .</math>

Applying Lagrange's formula with the observation that '''Ω • r'''(''t'') = 0 at all times,
<math display="block"> \mathbf{a} = - {|\mathbf{\Omega|}}^2 \mathbf{r}(t) \ .</math>

In words, the acceleration is pointing directly opposite to the radial displacement '''r''' at all times, and has a magnitude:
<math display="block"> |\mathbf{a}| = |\mathbf{r}(t)| \left ( \frac {\mathrm{d} \theta}{\mathrm{d}t} \right) ^2 = r {\omega}^2 </math>
where vertical bars |...| denote the vector magnitude, which in the case of '''r'''(''t'') is simply the radius ''r'' of the path. This result agrees with the previous section, though the notation is slightly different.

When the rate of rotation is made constant in the analysis of [[#Nonuniform circular motion|nonuniform circular motion]], that analysis agrees with this one.

A merit of the vector approach is that it is manifestly independent of any coordinate system.

==== Example: The banked turn ====
{{Main|Banked turn}}
{{See also|Reactive centrifugal force}}

[[File:Banked turn.svg|thumb|Upper panel: Ball on a banked circular track moving with constant speed ''v''; Lower panel: Forces on the ball]]

The upper panel in the image at right shows a ball in circular motion on a banked curve. The curve is banked at an angle ''θ'' from the horizontal, and the surface of the road is considered to be slippery. The objective is to find what angle the bank must have so the ball does not slide off the road.<ref name = Lerner>{{cite book |title = Physics for Scientists and Engineers |author = Lawrence S. Lerner |page = 128 |url = https://books.google.com/books?id=kJOnAvimS44C&pg=PA129 |isbn = 978-0-86720-479-7 |year = 1997 |location = Boston |publisher = Jones & Bartlett Publishers |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007055705/https://books.google.com/books?id=kJOnAvimS44C&pg=PA129#v=onepage&q&f=false |url-status = live }}</ref> Intuition tells us that, on a flat curve with no banking at all, the ball will simply slide off the road; while with a very steep banking, the ball will slide to the center unless it travels the curve rapidly.

Apart from any acceleration that might occur in the direction of the path, the lower panel of the image above indicates the forces on the ball. There are ''two'' forces; one is the force of gravity vertically downward through the center of mass of the ball ''m'''''g''', where ''m'' is the mass of the ball and '''g''' is the [[gravitational acceleration]]; the second is the upward [[normal force]] exerted by the road at a right angle to the road surface ''m'''''a'''<sub>n</sub>. The centripetal force demanded by the curved motion is also shown above. This centripetal force is not a third force applied to the ball, but rather must be provided by the [[net force]] on the ball resulting from [[vector addition]] of the [[normal force]] and the [[force of gravity]]. The resultant or [[net force]] on the ball found by [[vector addition]] of the [[normal force]] exerted by the road and vertical force due to [[gravity]] must equal the centripetal force dictated by the need to travel a circular path. The curved motion is maintained so long as this net force provides the centripetal force requisite to the motion.

The horizontal net force on the ball is the horizontal component of the force from the road, which has magnitude {{math|1={{!}}'''F'''<sub>h</sub>{{!}} = ''m''{{!}}'''a'''<sub>n</sub>{{!}} sin ''θ''}}. The vertical component of the force from the road must counteract the gravitational force: {{math|1={{!}}'''F'''<sub>v</sub>{{!}} = ''m''{{!}}'''a'''<sub>n</sub>{{!}} cos ''θ'' = ''m''{{!}}'''g'''{{!}}}}, which implies {{math|1={{!}}'''a'''<sub>n</sub>{{!}} = {{!}}'''g'''{{!}} / cos ''θ''}}. Substituting into the above formula for {{math|1={{!}}'''F'''<sub>h</sub>{{!}}}} yields a horizontal force to be:
<math display="block" qid=Q11402> |\mathbf{F}_\mathrm{h}| = m |\mathbf{g}| \frac { \sin \theta}{ \cos \theta} = m|\mathbf{g}| \tan \theta \, . </math>

On the other hand, at velocity |'''v'''| on a circular path of radius ''r'', kinematics says that the force needed to turn the ball continuously into the turn is the radially inward centripetal force '''F'''<sub>c</sub> of magnitude:
<math display="block">|\mathbf{F}_\mathrm{c}| = m |\mathbf{a}_\mathrm{c}| = \frac{m|\mathbf{v}|^2}{r} \, . </math>

Consequently, the ball is in a stable path when the angle of the road is set to satisfy the condition:
<math display="block">m |\mathbf{g}| \tan \theta = \frac{m|\mathbf{v}|^2}{r} \, ,</math>
or,
<math display="block"> \tan \theta = \frac {|\mathbf{v}|^2} {|\mathbf{g}|r} \, .</math>

As the angle of bank ''θ'' approaches 90°, the [[tangent function]] approaches infinity, allowing larger values for |'''v'''|<sup>2</sup>/''r''. In words, this equation states that for greater speeds (bigger |'''v'''|) the road must be banked more steeply (a larger value for ''θ''), and for sharper turns (smaller ''r'') the road also must be banked more steeply, which accords with intuition. When the angle ''θ'' does not satisfy the above condition, the horizontal component of force exerted by the road does not provide the correct centripetal force, and an additional frictional force tangential to the road surface is called upon to provide the difference. If [[friction]] cannot do this (that is, the [[coefficient of friction]] is exceeded), the ball slides to a different radius where the balance can be realized.<ref name = Schaum>{{cite book |title = Schaum's Outline of Applied Physics |author = Arthur Beiser |page = 103 |url = https://books.google.com/books?id=soKguvJDgmsC&pg=PA103 |publisher = McGraw-Hill Professional |year = 2004 |location = New York |isbn = 978-0-07-142611-4 |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007055651/https://books.google.com/books?id=soKguvJDgmsC&pg=PA103 |url-status = live }}</ref><ref name = Darbyshire>{{cite book |title = Mechanical Engineering: BTEC National Option Units |author = Alan Darbyshire |page = 56 |url = https://books.google.com/books?id=fzfXLGpElZ0C&pg=PA57 |isbn = 978-0-7506-5761-7 |publisher = Newnes |year = 2003 |location = Oxford |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007055651/https://books.google.com/books?id=fzfXLGpElZ0C&pg=PA57 |url-status = live }}</ref>

These ideas apply to air flight as well. See the FAA pilot's manual.<ref name = FAA>{{cite book |title = Pilot's Encyclopedia of Aeronautical Knowledge |author = Federal Aviation Administration |page = Figure 3–21 |url = https://books.google.com/books?id=m5V04SXE4zQC&pg=PT33 |isbn = 978-1-60239-034-8 |year = 2007 |publisher = Skyhorse Publishing Inc. |location = Oklahoma City OK |no-pp = true |access-date = 30 March 2021 |archive-date = 7 October 2024 |archive-url = https://web.archive.org/web/20241007060154/https://books.google.co.id/books?id=m5V04SXE4zQC&pg=PT33&redir_esc=y |url-status = live }}</ref>

=== Nonuniform circular motion ===
{{See also|Circular motion|Non-uniform circular motion}}

[[File:Nonuniform circular motion.svg|thumb|{{clarification needed span|text=Velocity and acceleration for nonuniform circular motion: the velocity vector is tangential to the orbit, but the acceleration vector is not radially inward because of its tangential component '''a'''<sub>θ</sub> that increases the rate of rotation: ''dω'' / ''dt'' = {{!}} '''a'''<sub>''θ''</sub>{{!}} / ''R''.|reason=This caption is not displaying properly, and needs to be corrected so that it does.|date=March 2025}}]]

As a generalization of the uniform circular motion case, suppose the angular rate of rotation is not constant. The acceleration now has a tangential component, as shown the image at right. This case is used to demonstrate a derivation strategy based on a [[polar coordinate system]].

Let '''r'''(''t'') be a vector that describes the position of a [[point mass]] as a function of time. Since we are assuming [[circular motion]], let {{math|1='''r'''(''t'') = ''R''·'''u'''<sub>''r''</sub>}}, where ''R'' is a constant (the radius of the circle) and '''u'''<sub>r</sub> is the [[unit vector]] pointing from the origin to the point mass. The direction of '''u'''<sub>''r''</sub> is described by ''θ'', the angle between the x-axis and the unit vector, measured counterclockwise from the x-axis. The other unit vector for polar coordinates, '''u'''<sub>θ</sub> is perpendicular to '''u'''<sub>''r''</sub> and points in the direction of increasing ''θ''. These polar unit vectors can be expressed in terms of [[Cartesian coordinate system|Cartesian]] unit vectors in the ''x'' and ''y'' directions, denoted <math>\hat\mathbf i</math> and <math>\hat\mathbf j</math> respectively:<ref>Note: unlike the Cartesian unit vectors <math>\hat\mathbf i</math> and <math>\hat\mathbf j</math>, which are constant, in [[Polar coordinate system|polar coordinates]] the direction of the unit vectors '''u'''<sub>''r''</sub> and '''u'''<sub>''θ''</sub> depend on ''θ'', and so in general have non-zero time derivatives.</ref>
<math display="block">\mathbf u_r = \cos \theta \ \hat\mathbf i + \sin \theta \ \hat\mathbf j</math> and <math display="block">\mathbf u_\theta = - \sin \theta \ \hat\mathbf i + \cos \theta \ \hat\mathbf j.</math>

One can differentiate to find velocity:
<math display="block">\begin{align}
\mathbf{v} &= r \frac {d \mathbf{u}_r}{dt} \\
&= r \frac {d}{dt} \left( \cos \theta \ \hat\mathbf{i} + \sin \theta \ \hat\mathbf{j}\right) \\
&= r \frac {d \theta}{dt} \frac{d}{d \theta} \left( \cos \theta \ \hat\mathbf{i} + \sin \theta \ \hat\mathbf{j}\right) \\
& = r \frac {d \theta} {dt} \left( -\sin \theta \ \hat\mathbf{i} + \cos \theta \ \hat\mathbf{j}\right)\\
& = r \frac{d\theta}{dt} \mathbf{u}_\theta \\
& = \omega r \mathbf{u}_\theta
\end{align}</math>
where {{mvar|ω}} is the angular velocity {{math|''dθ''/''dt''}}.

This result for the velocity matches expectations that the velocity should be directed tangentially to the circle, and that the magnitude of the velocity should be {{math|''rω''}}. Differentiating again, and noting that
<math display="block">\frac {d\mathbf{u}_\theta}{dt} = -\frac{d\theta}{dt} \mathbf{u}_r = - \omega \mathbf{u}_r \ , </math>
we find that the acceleration, '''a''' is:
<math display="block">\mathbf{a} = r \left( \frac {d\omega}{dt} \mathbf{u}_\theta - \omega^2 \mathbf{u}_r \right) \ . </math>

Thus, the radial and tangential components of the acceleration are:
<math display="block">\mathbf{a}_{r} = - \omega^{2} r \ \mathbf{u}_r = - \frac{|\mathbf{v}|^2}{r} \ \mathbf{u}_r </math> and <math display="block">\mathbf{a}_\theta = r \ \frac {d\omega}{dt} \ \mathbf{u}_\theta = \frac {d | \mathbf{v} | }{dt} \ \mathbf{u}_\theta \ , </math>
where {{math|1={{!}}'''v'''{{!}} = ''r'' ''ω''}} is the magnitude of the velocity (the speed).

These equations express mathematically that, in the case of an object that moves along a circular path with a changing speed, the acceleration of the body may be decomposed into a [[perpendicular component]] that changes the direction of motion (the centripetal acceleration), and a parallel, or [[tangential component]], that changes the speed.

=== 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]].