Circular motion
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220559899
2008-06-20T13:12:50Z
The Anome
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Reverted edits by [[Special:Contributions/The Anome|The Anome]] ([[User talk:The Anome|talk]]) to last version by Foxjwill
In [[physics]], '''circular motion''' is [[rotation]] along a [[circle]]: a circular path or a circular [[orbit]]. It can be [[uniform circular motion|uniform]], that is, with constant angular rate of rotation, or [[non-uniform circular motion|non-uniform]], that is, with a changing rate of rotation. The [[rotation around a fixed axis]] of a three-dimensional body involves circular motion of its parts. We can talk about circular motion of an object if we ignore its size, so that we have the motion of a point mass in a plane. For example, the [[center of mass]] of a body can undergo circular motion.
Examples of circular motion are: an artificial satellite orbiting the Earth in [[geosynchronous orbit]], a stone which is tied to a rope and is being swung in circles (cf. [[hammer throw]]), a racecar turning through a curve in a [[racetrack]], an electron moving perpendicular to a uniform [[magnetic field]], a [[gear]] turning inside a mechanism.
Circular motion is accelerated even if the angular rate of rotation is constant, because the object's velocity vector is constantly changing direction. Such change in direction of velocity involves [[acceleration]] of the moving object by a [[centripetal force]], which pulls the moving object towards the center of the circular orbit. Without this acceleration, the object would move in a straight line, according to [[Newton's laws of motion]].
==Formulas for uniform circular motion==
[[Image:Circular motion vectors.PNG|right|293 px|thumb|<center>Figure 1: Vector relationships for uniform circular motion; vector '''Ω''' representing the rotation is normal to the plane of the orbit.</center>]]
For motion in a circle of radius R, the circumference of the circle is ''C'' = 2π ''R''. If the period for one rotation is ''T'', the angular rate of rotation ω is:
*<math> \omega = \frac {2 \pi}{T} \ . </math>
The speed of the object traveling the circle is
*<math> v\, = \frac {2 \pi R } {T} = \omega R </math>
The angle θ swept out in a time ''t'' is:
*<math> \theta = 2 \pi \frac{t}{T} = \omega t\,</math><br>
The acceleration due to change in the direction of the velocity is found by noticing that the velocity completely rotates direction in the same time ''T'' the object takes for one rotation. Thus, the velocity vector sweeps out a path of length 2π ''v'' every ''T'' seconds, or:
*<math> a\, = \frac {2 \pi v }{T} = \omega^2 \ R \ ,</math><br>
and is directed ''radially inward''.
The vector relationships are shown in Figure 1. The axis of rotation is shown as a vector '''Ω''' perpendicular to the plane of the orbit and with a magnitude ω = ''d''θ / ''dt''. The direction of '''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> \mathbf{v} = \boldsymbol \Omega \times \mathbf r \ , </math>
which is a vector perpendicular to both '''Ω''' and '''r''' ( ''t'' ), tangential to the orbit, and of magnitude ω ''R''. Likewise, the acceleration is given by
:<math> \mathbf{a} = \boldsymbol \Omega \times \mathbf v \ , </math>
which is a vector perpendicular to both '''Ω''' and '''v''' ( ''t'' ) of magnitude ω |'''v'''| = ω<sup>2</sup> ''R'' and directed exactly opposite to '''r''' ( ''t'' ).
==Constant speed==
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 one metre per second
*The inward [[acceleration]] is one metre per second per second.
*It is subject to a [[centripetal force]] of one kilogram metre per second per second, which is one [[newton]].
*The [[momentum]] of the body is one kg·m·s<sup>−1</sup>.
*The [[moment of inertia]] is one kg·m<sup>2</sup>.
*The [[angular momentum]] is one kg·m<sup>2</sup>·s<sup>−1</sup>.
*The [[kinetic energy]] is 1/2 [[joule]].
*The [[circumference]] of the [[orbit]] is 2[[Pi|π]] (~ 6.283) metres.
*The [[Periodicity|period]] of the motion is 2π seconds per [[turn (geometry)|turn]].
*The [[frequency]] is (2π)<sup>−1</sup> [[hertz]].
*From the point of view of [[quantum mechanics]], the system is in an excited state having quantum number ~ 9.48×10<sup>35</sup>.
Then consider a body of [[mass]] ''m'', moving in a circle of radius ''r'', with an [[angular velocity]] of [[omega|''ω'']].
*The speed is ''v'' = ''r·ω''.
*The centripetal (inward) acceleration is ''a'' = ''r·ω'' <sup>2</sup> = ''r'' <sup>−1</sup>·''v'' <sup>2</sup>.
*The centripetal force is ''F'' = ''m·a'' = ''r·m·ω'' <sup>2</sup> = ''r''<sup>−1</sup>·''m·v'' <sup>2</sup>.
*The momentum of the body is ''p'' = ''m·v'' = ''r·m·ω''.
*The moment of inertia is ''I'' = ''r <sup>2</sup>·m''.
*The angular momentum is ''L'' = ''r·m·v'' = ''r <sup>2</sup>·m·ω'' = ''I·ω''.
*The kinetic energy is ''E'' = 2<sup>−1</sup>''·m·v'' <sup>2</sup> = 2<sup>−1</sup>·''r'' <sup>2</sup>·''m·ω'' <sup>2</sup> = (2·''m'')<sup>−1</sup>·''p'' <sup>2</sup> = 2<sup>−1</sup>·''I·ω'' <sup>2</sup> = (2·''I'')<sup>−1</sup>·''L'' <sup>2</sup> .
*The circumference of the [[orbit]] is 2·π·''r''.
*The period of the motion is ''T'' = 2·π·''ω'' <sup>−1</sup>.
*The frequency is ''f'' = ''T'' <sup>−1</sup> . (Instead of letter ''f'', the frequency is often denoted by the Greek letter [[Nu (letter)|''ν'']], which however is almost indistinguishable from the letter ''v'' used here for velocity).
*The quantum number is ''J'' = 2·π·''L ''[[Planck's constant|''h'']]<sup>−1</sup>
==Variable speed==
In the general case, circular motion requires that the total force can be decomposed into the centripetal force required to keep the orbit circular, and a force tangent to the circle, causing a change of speed.
The magnitude of the centripetal force depends on the instantaneous speed.
In the case of an object at the end of a rope, subjected to a force, we can decompose the force into a radial and a lateral component. The radial component is either outward or inward.
==Description of circular motion using polar coordinates==
[[Image:Vectors in polar coordinates.PNG|thumb|350px|Figure 2: Polar coordinates for circular trajectory. On the left is a unit circle showing the changes ''d''<math>\stackrel{\hat u_R\hat u_R}{} </math> and ''d''<math>\stackrel{\hat u_\theta}{}</math> in the unit vectors <math>\stackrel{\hat u_R}{} </math> and <math>\stackrel{\hat u_\theta}{}</math> for a small increment in angle ''d''θ]]
During circular motion the body moves on a curve that can be described in [[polar coordinates]] as a fixed distance ''R'' from the center of the orbit taken as origin, oriented at an angle θ (''t'') from some reference direction. See Figure 2. The displacement ''vector'' <math>\stackrel{\vec r}{}</math> is the radial vector from the origin to the particle location:
:<math>\vec r=R \hat u_R (t)\ ,</math>
where <math>\hat u_R (t)</math> is the [[unit vector]] parallel to the radius vector at time ''t'' and pointing away from the origin. It is handy to introduce the unit vector [[Orthogonality#In_Euclidean_vector_spaces|orthogonal]] to <math>\hat u_R</math> as well, namely <math>\hat u_\theta</math>. It is customary to orient <math>\hat u_\theta</math> to point in the direction of travel along the orbit.
The velocity is the time derivative of the displacement:
:<math> \vec v = \frac {d}{dt} \vec r(t) = \frac {d R}{dt} \hat u_R + R\frac {d \hat 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 u_R</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 θ the same as the angle of <math>\vec r (t)</math>. If the particle displacement rotates through an angle ''d''θ in time ''dt'', so does <math>\hat u_R</math>, describing an arc on the unit circle of magnitude ''d''θ. See the unit circle at the left of Figure 2. Hence:
:<math> \frac {d \hat u_R } {dt} = \frac {d \theta } {dt} \hat u_\theta \ , </math>
where the direction of the change must be perpendicular to <math>\hat u_R </math> (or, in other words, along <math>\hat u_\theta</math>) because any change ''d''<math>\hat u_R </math> in the direction of <math>\hat u_R </math> would change the size of <math>\hat u_R </math>. The sign is positive, because an increase in ''d''θ implies the object and <math>\hat u_R </math> have moved in the direction of <math>\hat u_\theta</math>.
Hence the velocity becomes:
:<math> \vec v = \frac {d}{dt} \vec r(t) = R\frac {d \hat u_R } {dt} = R \frac {d \theta } {dt} \hat u_\theta \ = R \omega \hat u_\theta \ . </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> \vec a = \frac {d}{dt} \vec v = \frac {d}{dt} \left(R\ \omega \ \hat u_\theta \ \right) \ . </math>
::<math>=R \left( \frac {d \omega}{dt}\ \hat u_\theta + \omega \ \frac {d \hat u_\theta}{dt} \right) \ . </math>
The time derivative of <math>\hat u_\theta</math> is found the same way as for <math>\hat u_R </math>. Again, <math>\hat u_\theta</math> is a unit vector and its tip traces a unit circle with an angle that is π/2 + θ. Hence, an increase in angle ''d''θ by <math>\vec r (t)</math> implies <math>\hat u_\theta</math> traces an arc of magnitude ''d''θ, and as <math>\hat u_\theta</math> is orthogonal to <math>\hat u_R </math>, we have:
:<math> \frac {d \hat u_\theta } {dt} = -\frac {d \theta } {dt} \hat u_R = -\omega \hat u_R\ , </math>
where a negative sign is necessary to keep <math>\hat u_\theta</math> orthogonal to <math>\hat u_R </math>. (Otherwise, the angle between <math>\hat u_\theta</math> and <math>\hat u_R </math> would ''decrease'' with increase in ''d''θ.) See the unit circle at the left of Figure 2. Consequently the acceleration is:
:<math>\vec a = R \left( \frac {d \omega}{dt}\ \hat u_\theta + \omega \ \frac {d \hat u_\theta}{dt} \right)</math>
::<math>=R \frac {d \omega}{dt}\ \hat u_\theta - \omega^2 R \ \hat u_R \ . </math>
The [[centripetal force |centripetal acceleration]] is the radial component, which is directed radially inward:
:<math>\vec a_R= -\omega ^2R \hat u_R \ , </math>
while the tangential component changes the [[Vector_%28spatial%29#Length_of_a_vector|magnitude]] of the velocity:
:<math>\vec a_{\theta}= R \frac {d \omega}{dt}\ \hat u_\theta = \frac {d R \omega}{dt}\ \hat u_\theta =\frac {d |\vec v|}{dt}\ \hat u_\theta \ .</math>
==Description of circular motion using complex numbers==
Circular motion can be described using [[complex numbers]]. Let the <math>x</math> 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>z=x+iy=R(\cos \theta +i \sin \theta)=Re^{i\theta}\ ,</math>
where <math>i</math> is the [[imaginary unit]], and
:<math>\theta =\theta (t)\ ,</math>
is the angle of the complex vector with the real axis and is a function of time ''t''.
Since the radius is constant:
:<math>\dot R =\ddot R =0 \ ,</math>
where a ''dot'' indicates time differentiation.
With this notation the velocity becomes:
:<math>v=\dot z = R \frac {d}{d \theta}\left( e^{i \theta}\right)\ \frac {d \theta}{dt} = iR\dot \theta e^{i\theta} = i\omega \cdot Re^{i\theta}= i\omega z</math>
and the acceleration becomes:
:<math>a=\dot v =i\dot \omega z +i \omega \dot z =(i\dot \omega z -\omega^2)z</math>
::<math>= \left(i\dot \omega-\omega^2 \right) R e^{i\theta} </math>
::<math>=-\omega^2 R e^{i\theta} + \dot \omega e^{i\frac{\pi}{2}}R e^{i\theta} \ .</math>
The first term is opposite to the direction of the displacement vector and the second is perpendicular to it, just like the earlier results.
==See also==
<div style="-moz-column-count:2; column-count:2;">
* [[Angular momentum]]
* [[Equation of motion#Rotational equations of motion|Rotational equations of motion]]
* [[Pendulum (mathematics)]]
* [[Uniform circular motion]]
* [[Simple harmonic motion]]
* [[Time derivative#Example: circular motion|Example: circular motion]]
* [[Centripetal force]]
* [[Fictitious force]]
* [[Non-uniform circular motion]]
</div>
==External links==
* [http://www.lightandmatter.com/html_books/1np/ch09/ch09.html Circular Motion] - a chapter from an online textbook
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