Editing Centripetal force (section)

=== Derivation ===
[[File:Velocity-acceleration.svg|left|frameless|upright=0.8]]

The centripetal acceleration can be inferred from the diagram of the velocity vectors at two instances. In the case of uniform circular motion the velocities have constant magnitude. Because each one is perpendicular to its respective position vector, simple vector subtraction implies two similar isosceles triangles with congruent angles – one comprising a [[base (geometry)|base]] of <math>\Delta \textbf{v}</math> and a [[isosceles triangle|leg]] length of <math>v</math>, and the other a [[base (geometry)|base]] of <math>\Delta \textbf{r}</math> (position vector [[Euclidean vector#Addition and subtraction|difference]]) and a [[isosceles triangle|leg]] length of <math>r</math>:<ref name="uniform_circular_motion">{{cite web |author=OpenStax CNX |title=Uniform Circular Motion |url=https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/4-4-uniform-circular-motion/ |access-date=25 December 2020 |archive-date=7 October 2024 |archive-url=https://web.archive.org/web/20241007055805/https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/4-4-uniform-circular-motion/ |url-status=live }}</ref>
<math display="block">\frac{|\Delta \textbf{v}|}{v} = \frac{|\Delta \textbf{r}|}{r}</math>
<math display="block">|\Delta \textbf{v}| = \frac{v}{r}|\Delta \textbf{r}|</math>
Therefore, <math>|\Delta\textbf{v}|</math> can be substituted with <math>\frac{v}{r} |\Delta \textbf{r}|</math>:<ref name="uniform_circular_motion" />
<math display="block">a_c = \lim_{\Delta t \to 0} \frac{|\Delta \textbf{v}|}{\Delta t} = \frac{v}{r} \lim_{\Delta t \to 0} \frac{|\Delta \textbf{r}|}{\Delta t} = \frac{v^2}{r}</math>
The direction of the force is toward the center of the circle in which the object is moving, or the [[osculating circle]] (the circle that best fits the local path of the object, if the path is not circular).<ref>{{cite book
| title = Experimental physics
| author1 = Eugene Lommel
| author2 = George William Myers
| publisher = K. Paul, Trench, Trübner & Co
| year = 1900
| page = 63
| url = https://books.google.com/books?id=4BMPAAAAYAAJ&pg=PA63
| access-date = 30 March 2021
| archive-date = 7 October 2024
| archive-url = https://web.archive.org/web/20241007055648/https://books.google.com/books?id=4BMPAAAAYAAJ&pg=PA63#v=onepage&q&f=false
| url-status = live
}}</ref>
The speed in the formula is squared, so twice the speed needs four times the force, at a given radius.

This force is also sometimes written in terms of the [[angular velocity]] ''ω'' of the object about the center of the circle, related to the tangential velocity by the formula
<math display="block">v = \omega r</math>
so that
<math display="block">F_c = m r \omega^2 \,.</math>

Expressed using the [[orbital period]] ''T'' for one revolution of the circle,
<math display="block" qid=Q161635>\omega = \frac{2\pi}{T} </math>
the equation becomes<ref>{{cite web|last=Colwell |first=Catharine H. |title=A Derivation of the Formulas for Centripetal Acceleration |url=http://dev.physicslab.org/Document.aspx?doctype=3&filename=CircularMotion_CentripetalAcceleration.xml |work=PhysicsLAB| access-date=31 July 2011| archive-url=https://web.archive.org/web/20110815162111/http://dev.physicslab.org/Document.aspx?doctype=3&filename=CircularMotion_CentripetalAcceleration.xml| archive-date=15 August 2011|url-status=dead}}</ref>
<math display="block">F_c = m r \left(\frac{2\pi}{T}\right)^2.</math>

In particle accelerators, velocity can be very high (close to the speed of light in vacuum) so the same rest mass now exerts greater inertia (relativistic mass) thereby requiring greater force for the same centripetal acceleration, so the equation becomes:<ref>{{cite book |title=An Introduction to the Physics of Particle Accelerators |first1=Mario |last1=Conte |first2=William W |last2=Mackay |publisher=World Scientific |year=1991 |isbn=978-981-4518-00-0 |page=8 |url=https://books.google.com/books?id=yJrsCgAAQBAJ |access-date=18 May 2020 |archive-date=7 October 2024 |archive-url=https://web.archive.org/web/20241007055648/https://books.google.com/books?id=yJrsCgAAQBAJ |url-status=live }} [https://books.google.com/books?id=yJrsCgAAQBAJ&pg=PA8 Extract of page 8] {{Webarchive|url=https://web.archive.org/web/20241007055804/https://books.google.com/books?id=yJrsCgAAQBAJ&pg=PA8#v=onepage&q&f=false |date=7 October 2024 }}</ref>
<math display="block">F_c = \frac{\gamma m v^2}{r}</math>
where
<math display="block" qid=Q599404>\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}</math>
is the [[Lorentz factor]].

Thus the centripetal force is given by:
<math display="block">F_c = \gamma m v \omega</math>
which is the rate of change of [[relativistic momentum]] <math>\gamma m v</math>.