Editing Centripetal force (section)

=== 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.