Editing Centripetal force (section)

==== 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.