Editing Circular motion (section)

== Uniform circular motion ==
[[File:Uniform circular motion.svg|thumb|upright=0.82|Figure 1: Velocity {{math|'''v'''}} and acceleration {{math|'''a'''}} in uniform circular motion at angular rate {{mvar|ω}}; the speed is constant, but the velocity is always tangential to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation.]]
[[File:Velocity-acceleration.svg|thumb|upright=1.14|right|Figure 2: The velocity vectors at time {{mvar|t}} and time {{math|''t'' + ''dt''}} are moved from the orbit on the left to new positions where their tails coincide, on the right. Because the velocity is fixed in magnitude at {{math|1=''v'' = ''r'' ''ω''}}, the velocity vectors also sweep out a circular path at angular rate {{mvar|ω}}. As {{math|''dt'' → 0}}, the acceleration vector {{math|'''a'''}} becomes perpendicular to {{math|'''v'''}}, which means it points toward the center of the orbit in the circle on the left. Angle {{math|''ω'' ''dt''}} is the very small angle between the two velocities and tends to zero as {{math|''dt'' → 0}}.]]
[[File:Breaking String.PNG|thumb|upright=1.36|Figure 3: (Left) Ball in a circular motion – rope provides centripetal force to keep the ball in a circle (Right) Rope is cut and the ball continues in a straight line with the velocity at the time of cutting the rope, in accord with Newton's law of inertia, because centripetal force is no longer there.]]

In [[physics]], '''uniform circular motion''' describes the motion of a body traversing a [[Circle|circular]] path at a constant [[speed]]. Since the body describes circular motion, its [[distance]] from the axis of rotation remains constant at all times. Though the body's speed is constant, its [[velocity]] is not constant: velocity, a [[Euclidean vector|vector]] quantity, depends on both the body's speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this [[centripetal acceleration]] is of constant magnitude and directed at all times toward the axis of rotation. This acceleration is, in turn, produced by a [[centripetal force]] which is also constant in magnitude and directed toward the axis of rotation.

In the case of [[rotation around a fixed axis]] of a [[rigid body]] that is not negligibly small compared to the radius of the path, each particle of the body describes a uniform circular motion with the same angular velocity, but with velocity and acceleration varying with the position with respect to the axis.

=== Formula ===
[[File:Circular motion vectors.svg|right|upright=1.33|thumb|Figure 1: Vector relationships for uniform circular motion; vector {{math|'''Ω'''}} representing the rotation is normal to the plane of the orbit.]]

For motion in a circle of [[radius]] {{mvar|r}}, the circumference of the circle is {{math|1=''C'' = 2''πr''}}. If the period for one rotation is {{mvar|T}}, the angular rate of rotation, also known as [[angular velocity]], {{mvar|ω}} is:
<math display="block" qid="Q161635">\omega = \frac {2 \pi}{T} = 2\pi f = \frac{d\theta}{dt} </math> and the units are radians/second.

The speed of the object traveling the circle is:
<math display="block" qid="Q3711325">v = \frac{2 \pi r}{T} = \omega r</math>

The angle {{mvar|θ}} swept out in a time {{mvar|t}} is:
<math display="block" qid="Q11352">\theta = 2 \pi \frac{t}{T} = \omega t</math>

The [[angular acceleration]], {{mvar|α}}, of the particle is:
<math display="block" qid="Q186300">\alpha = \frac{d\omega}{dt}</math>

In the case of uniform circular motion, {{mvar|α}} will be zero.

The acceleration due to change in the direction is:
<math display="block" qid="Q2248131">a_c = \frac{v^2}{r} = \omega^2 r</math>

The [[Centripetal force|centripetal]] and [[centrifugal force (rotating reference frame)|centrifugal]] force can also be found using acceleration:
<math display="block" qid="Q172881">F_c = \dot{p} \mathrel\overset{\dot{m} = 0}{=} ma_c = \frac{mv^2}{r}</math>

The vector relationships are shown in Figure 1. The axis of rotation is shown as a vector {{math|'''''ω'''''}} perpendicular to the plane of the orbit and with a magnitude {{math|1=''ω'' = ''dθ'' / ''dt''}}. The direction of {{math|'''''ω'''''}} is chosen using the [[right-hand rule]]. With this convention for depicting rotation, the velocity is given by a vector [[cross product]] as
<math display="block">\mathbf{v} = \boldsymbol \omega \times \mathbf r ,</math>
which is a vector perpendicular to both {{math|'''''ω'''''}} and {{math|'''r'''(''t'')}}, tangential to the orbit, and of magnitude {{math|''ω'' ''r''}}. Likewise, the acceleration is given by
<math display="block">\mathbf{a} = \boldsymbol \omega \times \mathbf v = \boldsymbol \omega \times \left( \boldsymbol \omega \times \mathbf r \right) , </math>
which is a vector perpendicular to both {{math|'''''ω'''''}} and {{math|'''v'''(''t'')}} of magnitude {{math|1=''ω'' {{abs|'''v'''}} = ''ω''<sup>2</sup> ''r''}} and directed exactly opposite to {{math|'''r'''(''t'')}}.<ref>{{cite book |last1=Knudsen |first1=Jens M. |url=https://books.google.com/books?id=Urumwws_lWUC&pg=PA96 |title=Elements of Newtonian mechanics: including nonlinear dynamics |last2=Hjorth |first2=Poul G. |publisher=Springer |year=2000 |isbn=3-540-67652-X |edition=3 |page=96}}</ref>

In the simplest case the speed, mass, and radius are constant.

Consider a body of one kilogram, moving in a circle of [[radius]] one metre, with an [[angular velocity]] of one [[radian]] per [[second]].
* The [[speed]] is 1 metre per second.
* The inward [[acceleration]] is 1 metre per square second, {{math|''v''{{i sup|2}}/''r''}}.
* It is subject to a [[centripetal force]] of 1 kilogram metre per square second, which is 1 [[newton (unit)|newton]].
* The [[momentum]] of the body is 1&nbsp;kg·m·s<sup>−1</sup>.
* The [[moment of inertia]] is 1&nbsp;kg·m<sup>2</sup>.
* The [[angular momentum]] is 1&nbsp;kg·m<sup>2</sup>·s<sup>−1</sup>.
* The [[kinetic energy]] is 0.5&nbsp;[[joule]].
* The [[circumference]] of the [[orbit]] is 2[[Pi|{{pi}}]] (~6.283) metres.
* The period of the motion is 2{{pi}}&nbsp;seconds.
* The [[frequency]] is (2{{pi}})<sup>−1</sup>&nbsp;[[hertz]].

==== In polar coordinates ====
{{See also|Velocity#Polar coordinates}}
[[File:Vectors in polar coordinates.PNG|thumb|350px|Figure 4: Polar coordinates for circular trajectory. On the left is a unit circle showing the changes <math>\mathbf{d\hat\mathbf{u}_R} </math> and <math>\mathbf{d\hat\mathbf{u}_\theta}</math> in the unit vectors <math>\mathbf{\hat\mathbf{u}_R} </math> and <math>\mathbf{\hat\mathbf{u}_\theta}</math> for a small increment <math>d \theta</math> in angle <math>\theta</math>.]]
During circular motion, the body moves on a curve that can be described in the [[polar coordinate system]] as a fixed distance {{math|''R''}} from the center of the orbit taken as the origin, oriented at an angle {{math|''θ''(''t'')}} from some reference direction. See Figure 4. The displacement ''vector'' <math>\mathbf{r}</math> is the radial vector from the origin to the particle location:
<math display="block">\mathbf{r}(t) = R \hat\mathbf{u}_R(t)\,,</math>
where <math>\hat\mathbf{u}_R(t)</math> is the [[unit vector]] parallel to the radius vector at time {{mvar|t}} and pointing away from the origin. It is convenient to introduce the unit vector [[Orthogonality (mathematics)#Euclidean vector spaces|orthogonal]] to <math>\hat\mathbf{u}_R(t)</math> as well, namely <math>\hat\mathbf{u}_\theta(t)</math>. It is customary to orient <math>\hat\mathbf{u}_\theta(t)</math> to point in the direction of travel along the orbit.

The velocity is the time derivative of the displacement:
<math display="block">\mathbf{v}(t) = \frac{d}{dt} \mathbf{r}(t) = \frac{d R}{dt} \hat\mathbf{u}_R(t) + R \frac{d \hat\mathbf{u}_R}{dt} \, .</math>

Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector <math>\hat\mathbf{u}_R(t)</math> has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle {{mvar|θ}} the same as the angle of <math>\mathbf{r}(t)</math>. If the particle displacement rotates through an angle {{math|''dθ''}} in time {{math|''dt''}}, so does <math>\hat\mathbf{u}_R(t)</math>, describing an arc on the unit circle of magnitude {{math|''dθ''}}. See the unit circle at the left of Figure 4. Hence: 
<math display="block">\frac{d \hat\mathbf{u}_R}{dt} = \frac{d \theta}{dt} \hat\mathbf{u}_\theta(t) \, ,</math>
where the direction of the change must be perpendicular to <math>\hat\mathbf{u}_R(t)</math> (or, in other words, along <math>\hat\mathbf{u}_\theta(t)</math>) because any change <math>d\hat\mathbf{u}_R(t)</math> in the direction of <math>\hat\mathbf{u}_R(t)</math> would change the size of <math>\hat\mathbf{u}_R(t)</math>. The sign is positive because an increase in {{math|''dθ''}} implies the object and <math>\hat\mathbf{u}_R(t)</math> have moved in the direction of <math>\hat\mathbf{u}_\theta(t)</math>.
Hence the velocity becomes:
<math display="block">\mathbf{v}(t) = \frac{d}{dt} \mathbf{r}(t) = R\frac{d \hat\mathbf{u}_R}{dt} = R \frac{d \theta}{dt} \hat\mathbf{u}_\theta(t) = R \omega \hat\mathbf{u}_\theta(t) \, .</math>

The acceleration of the body can also be broken into radial and tangential components. The acceleration is the time derivative of the velocity:
<math display="block">\begin{align}
\mathbf{a}(t) &= \frac{d}{dt} \mathbf{v}(t) = \frac{d}{dt} \left(R \omega \hat\mathbf{u}_\theta(t) \right) \\
              &= R \left( \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) + \omega \frac{d \hat\mathbf{u}_\theta}{dt} \right) \, .
\end{align}</math>

The time derivative of <math>\hat\mathbf{u}_\theta(t)</math> is found the same way as for <math>\hat\mathbf{u}_R(t)</math>. Again, <math>\hat\mathbf{u}_\theta(t)</math> is a unit vector and its tip traces a unit circle with an angle that is {{math|''π''/2 + ''θ''}}. Hence, an increase in angle {{math|''dθ''}} by <math>\mathbf{r}(t)</math> implies <math>\hat\mathbf{u}_\theta(t)</math> traces an arc of magnitude {{math|''dθ''}}, and as <math>\hat\mathbf{u}_\theta(t)</math> is orthogonal to <math>\hat\mathbf{u}_R(t)</math>, we have:
<math display="block">\frac{d \hat\mathbf{u}_\theta}{dt} = -\frac{d \theta}{dt} \hat\mathbf{u}_R(t) = -\omega \hat\mathbf{u}_R(t) \, ,</math>
where a negative sign is necessary to keep <math>\hat\mathbf{u}_\theta(t)</math> orthogonal to <math>\hat\mathbf{u}_R(t)</math>. (Otherwise, the angle between <math>\hat\mathbf{u}_\theta(t)</math> and <math>\hat\mathbf{u}_R(t)</math> would ''decrease'' with an increase in {{math|''dθ''}}.) See the unit circle at the left of Figure 4. Consequently, the acceleration is:
<math display="block">\begin{align}
\mathbf{a}(t) &= R \left( \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) + \omega \frac{d \hat\mathbf{u}_\theta}{dt} \right) \\
              &= R \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) - \omega^2 R \hat\mathbf{u}_R(t) \,.
\end{align}</math>

The [[centripetal force|centripetal acceleration]] is the radial component, which is directed radially inward:
<math display="block">\mathbf{a}_R(t) = -\omega^2 R \hat\mathbf{u}_R(t) \, ,</math>
while the tangential component changes the [[Vector (geometry)#Length|magnitude]] of the velocity: 
<math display="block">\mathbf{a}_\theta(t) = R \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) = \frac{d R \omega}{dt} \hat\mathbf{u}_\theta(t) = \frac{d \left|\mathbf{v}(t)\right|}{dt} \hat\mathbf{u}_\theta(t) \, .</math>

==== Using complex numbers ====
Circular motion can be described using [[complex number]]s. Let the {{mvar|x}} axis be the real axis and the <math>y</math> axis be the imaginary axis. The position of the body can then be given as <math>z</math>, a complex "vector":
<math display="block">z = x + iy = R\left(\cos[\theta(t)] + i \sin[\theta(t)]\right) = Re^{i\theta(t)}\,,</math>
where {{math|''i''}} is the [[imaginary unit]], and <math>\theta(t)</math> is the argument of the complex number as a function of time, {{mvar|t}}.

Since the radius is constant:
<math display="block">\dot{R} = \ddot R = 0 \, ,</math>
where a ''dot'' indicates differentiation in respect of time.

With this notation, the velocity becomes: 
<math display="block">v = \dot{z}
  = \frac{d}{dt}\left(R e^{i\theta[t]}\right)
  = R \frac{d}{dt}\left(e^{i\theta[t]}\right)
  = R e^{i\theta(t)} \frac{d}{dt} \left(i \theta[t] \right)
  = iR\dot{\theta}(t) e^{i\theta(t)}
  = i\omega R e^{i\theta(t)} = i\omega z
</math>
and the acceleration becomes:
<math display="block">\begin{align}
  a &= \dot{v} = i\dot{\omega} z + i\omega\dot{z} = \left(i\dot{\omega} - \omega^2\right)z \\
    &= \left(i\dot{\omega} - \omega^2 \right) R e^{i\theta(t)} \\
    &= -\omega^2 R e^{i\theta(t)} + \dot{\omega} e^{i\frac{\pi}{2}} R e^{i\theta(t)} \, .
\end{align}</math>

The first term is opposite in direction to the displacement vector and the second is perpendicular to it, just like the earlier results shown before.

==== Velocity ====
Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Because the velocity {{math|'''v'''}} is tangent to the circular path, no two velocities point in the same direction. Although the object has a constant ''speed'', its ''direction'' is always changing. This change in velocity is caused by an acceleration {{math|'''a'''}}, whose magnitude is (like that of the velocity) held constant, but whose direction also is always changing. The [[acceleration]] points radially inwards ([[centripetal]]ly) and is perpendicular to the velocity. This acceleration is known as centripetal acceleration.

For a path of radius {{mvar|r}}, when an angle {{mvar|θ}} is swept out, the distance traveled on the [[wikt:periphery|periphery]] of the orbit is {{math|1=''s'' = ''rθ''}}. Therefore, the speed of travel around the orbit is
<math display="block">v = r \frac{d\theta}{dt} = r\omega ,</math>
where the angular rate of rotation is {{math|''ω''}}. (By rearrangement, {{math|1=''ω'' = ''v''/''r''}}.) Thus, {{math|''v''}} is a constant, and the velocity vector {{math|'''v'''}} also rotates with constant magnitude {{math|''v''}}, at the same angular rate {{math|''ω''}}.

==== Relativistic circular motion ====

In this case, the three-acceleration vector is perpendicular to the three-velocity vector, 
<math display="block">\mathbf{u} \cdot \mathbf{a} = 0. </math>
and the square of proper acceleration, expressed as a scalar invariant, the same in all reference frames,
<math display="block">\alpha^2 = \gamma^4 a^2 + \gamma^6 \left(\mathbf{u} \cdot \mathbf{a}\right)^2/c^2, </math>
becomes the expression for circular motion, 
<math display="block">\alpha^2 = \gamma^4 a^2. </math>
or, taking the positive square root and using the three-acceleration, we arrive at the proper acceleration for circular motion: 
<math display="block">\alpha = \gamma^2 \frac{v^2}{r}. </math>

==== Acceleration ====
{{main|Acceleration}}

The left-hand circle in Figure 2 is the orbit showing the velocity vectors at two adjacent times. On the right, these two velocities are moved so their tails coincide. Because speed is constant, the velocity vectors on the right sweep out a circle as time advances. For a swept angle {{math|1=''dθ'' = ''ω'' ''dt''}} the change in {{math|'''v'''}} is a vector at right angles to {{math|'''v'''}} and of magnitude {{math|''v'' ''dθ''}}, which in turn means that the magnitude of the acceleration is given by 
<math display="block">a_c = v \frac{d\theta}{dt} = v\omega = \frac{v^2}{r}</math>

{| class="wikitable"
|+ Centripetal acceleration for some values of radius and magnitude of velocity
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| bgcolor="#99ffff" |200&nbsp;mm/s<sup>2</sup><br />0.020 ''g''
| bgcolor="#99ffff" |800&nbsp;mm/s<sup>2</sup><br />0.082 ''g''
| bgcolor="#ccffcc" |5.0&nbsp;m/s<sup>2</sup><br />0.51 ''g''
| bgcolor="#ffff99" |20&nbsp;m/s<sup>2</sup><br />2.0 ''g''
|-
! 1&nbsp;km<br />3300&nbsp;ft
! [[High-speed rail|High-speed<br />railway]]
| bgcolor="#ffffff" |1.0&nbsp;mm/s<sup>2</sup><br />0.00010 ''g''
| bgcolor="#ffffff" |4.0&nbsp;mm/s<sup>2</sup><br />0.00041 ''g''
| bgcolor="#ddddff" |25&nbsp;mm/s<sup>2</sup><br />0.0025 ''g''
| bgcolor="#99ffff" |100&nbsp;mm/s<sup>2</sup><br />0.010 ''g''
| bgcolor="#99ffff" |400&nbsp;mm/s<sup>2</sup><br />0.041 ''g''
| bgcolor="#ccffcc" |2.5&nbsp;m/s<sup>2</sup><br />0.25 ''g''
| bgcolor="#ffff99" |10&nbsp;m/s<sup>2</sup><br />1.0 ''g''
|}