Kinematics 65914 226119913 2008-07-16T22:15:36Z BenFrantzDale 41799 /* See also */ Chebychev–Grübler–Kutzbach criterion '''Kinematics''' ([[Greek language|Greek]] ''κινειν'', ''kinein'', to move) is a branch of [[dynamics (physics)|dynamics]] which describes the [[motion (physics)|motion]] of objects without consideration of the circumstances leading to the motion. An example is the prediction of [[centripetal force]] in [[uniform circular motion]], regardless of whether the circular path is due to [[gravitational attraction]], a [[banked curve]] on a highway, or an attached string. In contrast, [[Kinetics (physics)|kinetics]] is concerned with the forces and [[interaction]]s that produce or affect the motion.<ref name=Beggs>{{cite book |title=Kinematics |author=Joseph Stiles Beggs |page=p. 1 |url=http://books.google.com/books?id=y6iJ1NIYSmgC&printsec=frontcover&dq=kinematics&lr=&as_brr=0&sig=brRJKOjqGTavFsydCzhiB3u_8MA#PPA1,M1 |isbn=0891163557 |year=1983 |publisher=Taylor & Francis}}</ref><ref name= Bottema>{{cite book |title=Theoretical Kinematics |Page=Preface |url=http://books.google.com/books?id=f8I4yGVi9ocC&printsec=frontcover&dq=kinematics&lr=&as_brr=0&sig=YfoHn9ImufIzAEp5Kl7rEmtYBKc#PPR7,M1 |author=O. Bottema & B. Roth |isbn=0486663469 |publisher=Dover Publications |year=1990}}</ref><ref name=Wright>{{cite book |title=Elements of Mechanics Including Kinematics, Kinetics and Statics|author=Thomas Wallace Wright |url=http://books.google.com/books?id=-LwLAAAAYAAJ&printsec=frontcover&dq=mechanics+kinetics&lr=&as_brr=0#PPA6,M1 |page=Chapter 1 |year=1896 |publisher=E and FN Spon }}</ref><ref name=Whittaker>{{cite book |title=A Treatise on the Analytical Dynamics of Particles and Rigid Bodies |author=Edmund Taylor Whittaker & William McCrea |url=http://books.google.com/books?id=epH1hCB7N2MC&printsec=frontcover&dq=inauthor:%22E+T+Whittaker%22&lr=&as_brr=0&sig=SN7_oYmNYM4QRSgjULXBU5jeQrA&source=gbs_book_other_versions_r&cad=0_2#PPA1,M1 |page=Chapter 1 |year=1988 |publisher=Cambridge University Press |isbn=0521358833 }}</ref> {{Quotation|It is natural to begin this discussion by considering the various possible types of motion in themselves, leaving out of account for a time the causes to which the initiation of motion may be ascribed; this preliminary enquiry constitutes the science of ''Kinematics''.|Whittaker, E. T. (1988). ''A treatise on the analytical dynamics of particles and rigid bodies: with an introduction to the problem of three bodies.'' Chapter 1, p. 1.}} The simplest application of kinematics is to point particle motion ([[translational kinematics]] or linear kinematics). The description of rotation ([[rotational kinematics]] or angular kinematics) is more complicated. The state of a generic rigid body may be described by combining both translational and rotational kinematics ([[rigid-body kinematics]]). A more complicated case is the kinematics of a ''system'' of rigid bodies, possibly linked together by mechanical [[joints]]. The kinematic description of fluid flow is even more complicated, and not generally thought of in the context of kinematics. == Translational motion== Translational or curvilinear kinematics<ref name=Ogden>{{cite book |title=The Mechanics Problem Solver |author=James R. Ogden & Max Fogiel |url=http://books.google.com/books?id=XVyD9pJpW-cC&pg=PA184&dq=%22curvilinear+kinematics%22&lr=&as_brr=0&sig=WW7us4UJzSWOA19pfdAbwTJvPR4 |page=p. 184 |isbn=0878915192 |year=1980 |publisher=Research and Education Association }}</ref><ref name=Gregory1>{{cite book |title=Classical Mechanics: An Undergraduate Text |url=http://books.google.com/books?id=uAfUQmQbzOkC&printsec=frontcover&dq=%22rigid+body+kinematics%22&lr=&as_brr=0#PRA1-PA25,M1 |author=R. Douglas Gregory |year=2006 |isbn=0521826780 |publisher=Cambridge University Press |location=Cambridge UK |page=Chapter 2}}</ref> is the description of the motion in space of a point along a trajectory. This path can be linear, or curved as seen with projectile motion. There are three basic concepts that are ''required'' for understanding translational motion: '''Displacement''' (denoted by '''''r''''' below) is the shortest distance between two points: the origin and the displaced point. The origin is (0,0) on a [[coordinate system]] that is defined by the observer. Because displacement has both magnitude (length) and direction, it is a [[vector]] whose initial point is the origin and terminal point is the displaced point. '''Velocity''' (denoted by <font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> υ </font> below) is the rate of change in displacement with respect to time; that is the displacement of a point changes with time. Velocity is also a vector. For a constant velocity, every unit of time adds the length of the velocity vector (in the same direction) to the displacement of the moving point. Instantaneous velocity (the velocity at an instant of time) is defined as ::<math> \boldsymbol v = \frac {d \boldsymbol r}{d t} \ ,</math> where ''d'''''r''' is an infinitesimally small displacement and ''dt'' is an infinitesimally small length of time. Because ''d'''''r''' is necessarily the distance between two infinitesimally spaced points along the trajectory of the point, it is the same as an increment in arc length along the path of the point, customarily denoted ''d'''''s'''. Average velocity (velocity over a length of time) is defined as <math> \boldsymbol v = \frac {\Delta \boldsymbol s}{\Delta t} </math>, where ''Δ'''''s''' is the change in displacement and ''Δt'' is the interval of time over which displacement changes. '''Acceleration''' (denoted by <font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> a </font> below) is the rate of change in velocity with respect to time. Acceleration is also a vector. As with velocity if acceleration is constant, for every unit of time the length of the acceleration vector (in the same direction) is added to the velocity. If the change in velocity (a vector) is known, the acceleration is parallel to it. Instantaneous acceleration (the acceleration at an instant of time) is defined as: ::<math> \boldsymbol a = \frac {d \boldsymbol v}{d t} \ , </math> where ''d''<font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> υ </font> is an infinitesimally small change in velocity and ''dt'' is an infinitesimally small length of time. Average acceleration (acceleration over a length of time) is defined as: ::<math> \boldsymbol a = \frac {\Delta \boldsymbol v}{\Delta t} \ ,</math> where ''Δ''<font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> υ </font> is the change in velocity and ''Δt'' is the interval of time over which velocity changes. ===Integral relations=== The above definitions can be inverted by integration to find: :<math>\boldsymbol{v}(t) =\boldsymbol{v}_0+\ \int_0^t \ dt' \ \boldsymbol{a} (t') </math> :<math>\boldsymbol{r}(t) =\boldsymbol{r}_0+\ \int_0^t \ dt' \ \boldsymbol{v} (t') </math> ::<math>=\boldsymbol{r}_0+\boldsymbol{v}_0\ t +\ \int_0^t \ dt' \int_0^{t'} \ dt'' \ \boldsymbol{a} (t'') </math> ::<math>=\boldsymbol{r}_0+\boldsymbol{v}_0\ t +\ \int_0^t \ dt' \left(t-t'\right) \ \boldsymbol{a} (t') \ ,</math> where the double integration is reduced to one integration by [[Integration_by_parts#Interchange_of_the_order_of_integration|interchanging the order of integration]], and subscript ''0'' signifies evaluation at ''t'' = 0 (initial values). ===Constant acceleration=== When acceleration is constant both in direction and in magnitude, the point is said to be undergoing ''uniformly accelerated motion''. In this case, the above equations can be simplified: ;Eq. (1) :<math> \boldsymbol v = \int_0^{t} \boldsymbol a dt = \boldsymbol{ v}_0 + \boldsymbol a t </math> Those who are familiar with calculus may recognize this as an initial value problem. Because acceleration (<font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> a </font>) is a constant, integrating it with respect to time (''t'') gives a change in velocity. Adding this to the initial velocity (<font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> υ</font><sub>0</sub>) gives the final velocity (<font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> υ </font>). ;Eq. (2) :<math> \boldsymbol r = \int_0^t \boldsymbol v dt = \int_0^t \left( \boldsymbol v_0 + \boldsymbol a t \right) dt = \boldsymbol v_0 t + \frac{1}{2} \boldsymbol a t^2 </math> Using the above formula, we can substitute for <font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> υ </font> to arrive at this equation, where '''r''' is displacement. ;Eq. (3) :<math> \boldsymbol r = \frac{\boldsymbol v+ \boldsymbol v_0}{2} t </math> By using the definition of an [[average]], this equation states that average velocity times time equals displacement. Using Eq. (1) to find <font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> υ−υ</font><sub>0</sub> and multiplying by Eq. (3) we find a connection between the final velocity at time ''t'' and the displacement at that time: :<math> \boldsymbol {r \cdot a} t = \left( \boldsymbol v - \boldsymbol {v}_0 \right)\boldsymbol{ \cdot} \frac{\boldsymbol v + \boldsymbol {v}_0}{2} t \ , </math> where the "'''•'''" denotes a vector [[dot product]]. Dividing the ''t'' on both sides and carrying out the dot-products: ;Eq. (4) : <math>2 \boldsymbol{r \cdot a} = v^2 - v_0^2 \ . </math> For the case where '''r''' is parallel to <font style="font-family: Times new roman; font-size:120%; font-style:italic; font-weight:bold;"> a</font> resulting in a straight-line motion, the vector '''r''' has magnitude equal to the path length ''s'' at time ''t'', and this equation becomes: :<math> v^2= v_0^2 + 2 a s\ , </math> which can be a useful result when time is not known explicitly. === Relative velocity === {{main|Relative velocity}} To describe the motion of object ''A'' with respect to object ''B'', when we know how each is moving with respect to a reference object ''O'', we can use vector algebra. Choose an origin for reference, and let the positions of objects ''A'', ''B'', and ''O'' be denoted by '''r'''<sub>A</sub>, '''r'''<sub>B</sub>, and '''r'''<sub>O</sub>. Then the position of ''A'' relative to the reference object ''O'' is :<math>\boldsymbol{r}_{A/O} = \boldsymbol{r}_{B} - \boldsymbol{r}_{O} \,\!</math> Consequently, the position of ''A'' relative to ''B'' is :<math> \boldsymbol{r}_{A/B} = \boldsymbol{r}_A - \boldsymbol{r}_B = \boldsymbol{r}_A - \boldsymbol{r}_O - \left(\boldsymbol{r}_B-\boldsymbol{r}_O\right) = \boldsymbol{r}_{A/O}-\boldsymbol{r}_{B/O} \ . </math> The above relative equation states that the motion of A relative to B is equal to the motion of A relative to O minus the motion of B relative to O. It may be easier to visualize this result if the terms are re-arranged: :<math>\boldsymbol{r}_{A/O} = \boldsymbol{r}_{A/B} + \boldsymbol{r}_{B/O} \ , </math> or, in words, the motion of ''A'' relative to the reference is that of ''B'' plus the relative motion of ''A'' with respect to ''B''. These relation between displacements become relations between velocities by simple time-differentiation, and a second differentiation makes them apply to accelerations. For example, let Ann move with velocity <math>\boldsymbol{V}_{A}</math> relative to the reference (we drop the ''O'' subscript for convenience) and let Bob move with velocity <math>\boldsymbol{V}_{B}</math>, each velocity given with respect to the ground (point ''O''). To find how fast Ann is moving relative to Bob (we call this velocity <math>\boldsymbol{V}_{A/B}</math>), the equation above gives: :<math>\boldsymbol{V}_{A} = \boldsymbol{V}_{B} + \boldsymbol{V}_{A/B} \,\! .</math> To find <math>\boldsymbol{V}_{A/B}</math> we simply rearrange this equation to obtain: :<math>\boldsymbol{V}_{A/B} = \boldsymbol{V}_{A} -\boldsymbol{V}_{B} \,\! .</math> At velocities comparable to the [[speed of light]], these equations are not valid. They are replaced by equations derived from Einstein's [[Special relativity|theory of special relativity]]. :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Example: Rectilinear (1D) motion |- |[[Image:Free-fall with initial velocity.gif|thumb|Figure A: An object is fired upwards, reaches its apex, and then begins its descent under a constant acceleration.]] Consider an object that is fired directly upwards and falls back to the ground so that its trajectory is contained in a straight line. If we adopt the convention that the upward direction is the positive direction, the object experiences a constant acceleration of approximately -9.81 m/s<sup>2</sup>. Therefore, its motion can be modeled with the equations governing uniformly accelerated motion. For the sake of example, assume the object has an initial velocity of +50 m/s. There are several interesting kinematic questions we can ask about the particle's motion: ==== ''How long will it be airborne?'' ==== To answer this question, we apply the formula :<math>x_f - x_i = v_i t + \frac{1}{2} at^2.</math> Since the question asks for the length of time between the object leaving the ground and hitting the ground on its fall, the displacement is zero. :<math>0 = v_i t + \frac{1}{2} at^2 = t(v_i + \frac{1}{2} at)</math> We find two solutions for it. The trivial solution says the time is zero; this is actually also true, it is the first moment the displacement is zero: just when it starts motion. However, the solution of interest is :<math>t = -\frac{2v_i}{a} = -\frac{2*50}{-9.81} = 10.2 \ s</math> ==== ''What altitude will it reach before it begins to fall?'' ==== In this case, we use the fact that the object has a velocity of zero at the apex of its trajectory. Therefore, the applicable equation is: :<math>v_f^2 = v_i^2 + 2 a (x_f - x_i)</math> If the origin of our coordinate system is at the ground, then <math>x_i</math> is zero. Then we solve for <math>x_f</math> and substitute known values: :<math>x_f = \frac{v_f^2 - v_i^2}{2 a} + x_i = \frac{0-50^2}{2*-9.81}+0 = 127.55 \ m</math> ==== ''What will its final velocity be when it reaches the ground?'' ==== To answer this question, we use the fact that the object has an initial velocity of zero at the apex before it begins its descent. We can use the same equation we used for the last question, using the value of 127.55 m for <math>x_i</math>. :<math>v_f = \sqrt{v_i^2 + 2 a (x_f - x_i)} = \sqrt{0^2 + 2 (-9.81) (0 - 127.55)} = 50\ m/s</math> Assuming this experiment were performed in a vacuum (negating drag effects), we find that the final and initial speeds are equal, a result which agrees with [[conservation of energy]]. |} :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Example: Projectile (2D) motion |- |[[Image:Casting obliquely.gif|thumb|Figure B: An object fired at an angle <math>\theta</math> from the ground follows a parabolic trajectory.]] Suppose that an object is not fired vertically but is fired at an angle <math>\theta</math> from the ground. The object will then follow a parabolic trajectory, and its horizontal motion can be modeled independently of its vertical motion. Assume that the object is fired at an initial velocity of 50 m/s and 30 degrees from the horizontal. ==== ''How far will it travel before hitting the ground?'' ==== The object experiences an acceleration of -9.81 ms<sup>-2</sup> in the vertical direction and no acceleration in the horizontal direction. Therefore, the horizontal displacement is :<math>\Delta x = x_f - x_i = v_i \cos \theta \ t + \frac{1}{2} at^2 = v_i \cos \theta \ t</math> In order to solve this equation, we must find t. This can be done by analyzing the motion in the vertical direction. If we impose that the vertical displacement is zero, we can use the same procedure we did for rectilinear motion to find t. :<math>0 = v_i \sin \theta \ t + \frac{1}{2} at^2 = t(v_i \sin \theta + \frac{1}{2} at)</math> We now solve for t and substitute this expression into the original expression for horizontal displacement. (Note the use of the [[trigonometric identity]] <math>2\sin\theta\cos\theta = \sin 2\theta</math>) :<math>\Delta x = v_i \cos \theta \left(\frac{-2 v_i \sin \theta}{a}\right) = -\frac{v_i^2 \sin 2\theta}{a} = 220.93 \ m</math> |} ==Rotational motion== {{main|Circular motion}} [[Image:Rotating body.PNG|thumb|250px|Figure 1: The angular velocity vector '''Ω''' points up for counterclockwise rotation and down for clockwise rotation, as specified by the [[right-hand rule]]. Angular position ''θ(t)'' changes with time at a rate ''ω(t)'' = ''dθ'' / ''dt''.]] Rotational kinematics is the description of the rotation of an object.<ref name=Gregory>{{cite book |title=Chapter 16 |author=R. Douglas Gregory |url=http://books.google.com/books?id=uAfUQmQbzOkC&printsec=frontcover&dq=%22rigid+body+kinematics%22&lr=&as_brr=0#PRA1-PA457,M1 |isbn=0521826780 |year=2006 }}</ref> The description of rotation requires some method for describing orientation, for example, the [[Euler angles]]. In what follows attention is restricted to simple rotation about an axis of fixed orientation for convenience chosen as the ''z''-axis.. Description of rotation then involves these three quantities: '''Angular position''': The oriented distance from a selected origin on the rotational axis to a point of an object is a vector '''r''' ( ''t'' ) locating the point. The vector '''r''' ( ''t'' ) has some projection (or, equivalently, some component) '''r'''<sub><math>\perp</math></sub> ( ''t'' ) on a plane perpendicular to the axis of rotation. Then the ''angular position'' of that point is the angle θ from a reference axis (typically the positive ''x''-axis) to the vector '''r'''<sub><math>\perp</math></sub> ( ''t'' ) in a known rotation sense (typically given by the [[right-hand rule]]). '''Angular velocity''': The angular velocity ω is the rate at which the angular position θ changes with respect to time t: :<math> \mathbf{\omega} = \frac {\mathrm{d}\theta}{\mathrm{d}t}</math> The angular velocity is represented in Figure 1 by a vector '''Ω''' pointing along the axis of rotation with magnitude ω and sense determined by the direction of rotation as given by the right-hand rule. '''Angular acceleration''': The magnitude of the angular acceleration <math>\alpha</math> is the rate at which the angular velocity <math>\omega</math> changes with respect to time t: :<math>\mathbf{\alpha} = \frac {\mathrm{d}\mathbf{\omega}}{\mathrm{d}t}</math> The equations of translational kinematics can easily be extended to planar rotational kinematics with simple variable exchanges: :<math>\,\!\theta_f - \theta_i = \omega_i t + \frac{1}{2} \alpha t^2 \qquad \theta_f - \theta_i = \frac{1}{2} (\omega_f + \omega_i)t</math> :<math>\,\!\omega_f = \omega_i + \alpha t \qquad \alpha = \frac{\omega_f - \omega_i}{t} \qquad \omega_f^2 = \omega_i^2 + 2 \alpha (\theta_f - \theta_i)\ .</math> Here <math>\,\!\theta_i</math> and <math>\,\!\theta_f</math> are, respectively, the initial and final angular positions, <math>\,\!\omega_i</math> and <math>\,\!\omega_f</math> are, respectively, the initial and final angular velocities, and <math>\,\!\alpha</math> is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector. ====Point object in circular motion==== {{seealso|Rigid body|Orientation}} [[Image:Nonuniform circular motion.PNG|thumb|250px|Figure 2: 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: ''d''ω / ''dt'' = <nowiki>|</nowiki> '''a'''<sub>θ</sub> <nowiki>|</nowiki> / ''R''.]] This example deals with a "point" object, by which is meant that complications due to rotation of the body itself about its own center of mass are ignored. '''Displacement'''. An object in circular motion is located at a position '''''r''''' ( ''t'' ) given by: :<math>\boldsymbol{r} (t) = R \mathbf{u}_R (t) \ , </math> where '''u'''<sub>R</sub> is a unit vector pointing outward from the axis of rotation toward the periphery of the circle of motion, located at a radius ''R'' from the axis. '''Linear velocity'''. The velocity of the object is then :<math>\boldsymbol{v} (t) =\frac{d}{dt} \boldsymbol{r} (t) = R \frac{d}{dt}\mathbf{u}_R (t) \ . </math> The magnitude of the unit vector '''u'''<sub>R</sub> (by definition) is fixed, so its time dependence is entirely due to its rotation with the radius to the object, that is, :<math> \frac{d}{dt}\mathbf{u}_R (t) = \boldsymbol{\Omega} \mathbf{\times u}_R = \omega (t) \mathbf{ u}_{\theta} \ , </math> where '''u'''<sub>θ</sub> is a unit vector perpendicular to '''u'''<sub>R</sub> pointing in the direction of rotation, ω ( ''t'' ) is the (possibly time varying) angular rate of rotation, and the symbol '''×''' denotes the [[vector cross product]]. The velocity is then: :<math>\boldsymbol{v} (t) = R\omega (t) \mathbf{ u}_{\theta} \ . </math> The velocity therefore is tangential to the circular orbit of the object, pointing in the direction of rotation, and increasing in time if ω increases in time. '''Linear acceleration'''. In the same manner, the acceleration of the object is defined as: :<math>\boldsymbol{a} (t) = \frac{d}{dt} \boldsymbol{v} (t) = R\frac{d}{dt}\omega\mathbf{ u}_{\theta} \ </math> ::<math>=\mathbf{ u}_{\theta} R\frac{d}{dt}\omega + R\omega\frac{d}{dt}\mathbf{ u}_{\theta} </math> ::<math>=\mathbf{ u}_{\theta} R\frac{d}{dt}\omega + R\omega \boldsymbol{\Omega}\mathbf{ \times u}_{\theta} </math> ::<math>=\mathbf{ u}_{\theta} R\frac{d}{dt}\omega - \mathbf{ u}_{R}\omega^2R\ </math> ::<math>=\mathbf{a}_{\theta} + \mathbf{a}_R \ , </math> which shows a leading term '''a'''<sub>θ</sub> in the acceleration tangential to the orbit related to the angular acceleration of the object (supposing ω to vary in time) and a second term '''a'''<sub>R</sub> directed inward from the object toward the center of rotation, called the [[centripetal force|centripetal acceleration]]. == Coordinate systems == {{seealso|Generalized coordinates|Curvilinear coordinates|Orthogonal coordinates|Frenet-Serret formulas}} In any given situation, the most useful coordinates may be determined by [[constraint]]s on the motion, or by the geometrical nature of the force causing or affecting the motion. Thus, to describe the motion of a bead constrained to move along a circular hoop, the most useful coordinate may be its angle on the hoop. Similarly, to describe the motion of a particle acted upon by a [[central force]], the most useful coordinates may be [[polar coordinate]]s. === Fixed rectangular coordinates === In this coordinate system, vectors are expressed as an addition of vectors in the x, y, and z direction from a non-rotating origin. Usually '''i''', '''j''', '''k''' are [[unit vector]]s in the ''x''-, ''y''-, and ''z''-directions. The position vector, '''r''' (or '''s'''), the velocity vector, '''v''', and the [[acceleration]] vector, '''a''' are expressed using rectangular coordinates in the following way: :<math>\boldsymbol{ r} = x\ \hat {\mathbf{ i}} + y \ \hat {\mathbf{ j}} + z \ \hat {\mathbf{ k}} \, \!</math> :<math>\boldsymbol{ v} = \dot {\boldsymbol{ r}} = \dot {x} \ \hat {\mathbf{ i}} + \dot {y} \ \hat {\mathbf{ j}} + \dot {z} \ \hat {\mathbf{ k}} \, \! </math> :<math> \boldsymbol{ a} = \ddot {\boldsymbol{ r}} = \ddot {x} \ \hat {\mathbf{ i}} + \ddot {y} \ \hat {\mathbf{ j}} + \ddot {z} \ \hat {\mathbf{ k}} \, \! </math> Note: <math> \dot {x} = \frac{\mathrm{d}x}{\mathrm{d}t} </math> , <math> \ddot {x} = \frac{\mathrm{d}^2x}{\mathrm{d}t^2}</math> === Two dimensional rotating reference frame === {{seealso|Centripetal force}} This coordinate system expresses only planar motion. It is based on three [[orthogonal]] unit vectors: the vector '''i''', and the vector '''j''' which form a [[basis]] for the plane in which the objects we are considering reside, and '''k''' about which rotation occurs. Unlike rectangular coordinates, which are measured relative to an origin that is fixed and non-rotating, the origin of these coordinates can rotate and translate - often following a particle on a body that is being studied. ====Derivatives of unit vectors==== The position, velocity, and acceleration vectors of a given point can be expressed using these coordinate systems, but we have to be a bit more careful than we do with fixed frames of reference. Since the frame of reference is rotating, the unit vectors also rotate, and this rotation must be taken into account when taking the derivative of any of these vectors. If the coordinate frame is rotating at angular rate ω in the counterclockwise direction (that is, '''Ω''' = ω '''k''' using the [[right hand rule]]) then the derivatives of the unit vectors are as follows: :<math>\dot{\hat {\mathbf{ i}}} = \omega \hat {\mathbf{ k}} \times \hat {\mathbf{ i}} = \omega\hat {\mathbf{ j}}</math> :<math>\dot{\hat {\mathbf{ j}}} = \omega \hat {\mathbf{ k}}\times \hat {\mathbf{ j}} = - \omega \hat {\mathbf{ i}} </math> ====Position, velocity, and acceleration==== Given these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this [[reference frame]]. =====Position===== Position is straightforward: :<math>\boldsymbol{ r} = x \ \hat {\mathbf{ i}} + y \ \hat {\mathbf{ j}}</math> It is just the distance from the origin in the direction of each of the unit vectors. =====Velocity===== Velocity is the time derivative of position: :<math>\boldsymbol{ v} = \frac{\mathrm{d}\boldsymbol{ v}}{\mathrm{d}t} = \frac{\mathrm{d} (x \ \hat {\mathbf{ i}})}{\mathrm{d}t} + \frac{\mathrm{d} (y \ \hat {\mathbf{ j}})}{\mathrm{d}t}</math> By the [[product rule]], this is: :<math>\boldsymbol{ v} = \dot x \ \hat {\mathbf{ i}} + x \dot{\ \hat {\mathbf{ i}}} + \dot y \ \hat {\mathbf{ j}} + y \dot{ \ \hat {\mathbf{ j}}}</math> Which from the identities above we know to be: :<math>\boldsymbol{ v} = \dot x \ \hat {\mathbf{ i}} + x \omega \ \hat {\mathbf{ j}} + \dot y \ \hat {\mathbf{ j}} - y \omega \ \hat {\mathbf{ i}} = (\dot x - y \omega) \ \hat {\mathbf{ i}} + (\dot y + x \omega) \ \hat {\mathbf{ j}}</math> or equivalently :<math>\boldsymbol{ v}= (\dot x \ \hat {\mathbf{ i}} + \dot y \ \hat {\mathbf{ j}}) + (y \dot{ \hat {\mathbf{ j}}} + x \dot{\hat {\mathbf{ i}}}) = \boldsymbol{ v}_{rel} + \boldsymbol{\Omega} \times \boldsymbol{r}</math> where '''v'''<sub>rel</sub> is the velocity of the particle relative to the rotating coordinate system. =====Acceleration===== Acceleration is the time derivative of velocity. We know that: :<math>\boldsymbol{ a} = \frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{ v} = \frac{\mathrm{d} \boldsymbol{ v}_{rel}}{\mathrm{d}t} + \frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{\Omega} \times \boldsymbol{r}</math> Consider the <math>\stackrel{\frac{ \mathrm{d} } { \mathrm{d} t }}{}</math><math> \boldsymbol{ v}_{rel}</math> part. <math>\boldsymbol{ v}_{rel}</math> has two parts we want to find the derivative of: the relative change in velocity (<math>\boldsymbol{ a}_{rel}</math>), and the change in the coordinate frame (<math>\boldsymbol{\Omega} \times \boldsymbol{ v}_{rel}</math>). :<math>\frac{\mathrm{d} \boldsymbol{ v}_{rel}}{\mathrm{d}t} = \boldsymbol{ a}_{rel} + \boldsymbol{\Omega} \times \boldsymbol{ v}_{rel}</math> Next, consider <math>\stackrel{\frac{\mathrm{d}}{\mathrm{d}t}}{}</math><math> (\boldsymbol{\Omega} \times\boldsymbol{ r})</math>. Using the chain rule: :<math>\frac{\mathrm{d} (\boldsymbol{\Omega} \times \boldsymbol{ r})}{\mathrm{d}t} = \dot{\boldsymbol{\Omega}} \times \boldsymbol{ r} + \boldsymbol{\Omega} \times \dot{\boldsymbol{ r}}</math> :<math>\dot{\boldsymbol{ r}}=\boldsymbol{ v}=\boldsymbol{ v}_{rel} + \boldsymbol{\Omega} \times \boldsymbol{ r}</math> from above: :<math>\frac{\mathrm{d} (\boldsymbol{\Omega} \times \boldsymbol{ r})}{\mathrm{d}t} = \dot{\boldsymbol{\Omega}} \times \boldsymbol{ r} + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times\boldsymbol{ r}) + \boldsymbol{\Omega} \times \boldsymbol{ v}_{rel} </math> So all together: :<math>\boldsymbol{ a} = \boldsymbol{ a}_{rel} + \boldsymbol{\Omega} \times \boldsymbol{ v}_{rel} + \dot{\boldsymbol{\Omega}} \times \boldsymbol{ r} + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \boldsymbol{ r}) + \boldsymbol{\Omega} \times \boldsymbol{ v}_{rel} </math> And collecting terms:<ref name=Gregory3>{{cite book |title=pp. 475-476 |url=http://books.google.com/books?id=uAfUQmQbzOkC&printsec=frontcover&dq=%22rigid+body+kinematics%22&lr=&as_brr=0#PRA1-PA475,M1 |author=R. Douglas Gregory |year=2006 |isbn=0521826780 }}</ref> :<math>\boldsymbol{ a} = \boldsymbol{ a}_{rel} + 2(\boldsymbol{\Omega} \times \boldsymbol{ v}_{rel}) + \dot{\boldsymbol{\Omega}} \times \boldsymbol{ r} + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \boldsymbol{ r})\ .</math> === Three dimensional rotating coordinate frame === (to be written) == Kinematic constraints == {{Expand-section|date=June 2008}} A kinematic constraint is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples: === Rolling without slipping === An object that rolls against a [[surface]] without slipping obeys the condition that the [[velocity]] of its [[center of mass]] is equal to the [[cross product]] of its [[angular velocity]] with a vector from the point of contact to the center of mass, :<math> \boldsymbol{ v}_G(t) = \boldsymbol{\Omega} \times \boldsymbol{ r}_{G/O}</math>. For the case of an object that does not tip or turn, this reduces to v = R ω. === Inextensible cord === This is the case where bodies are connected by some cord that remains in tension and cannot change length. The constraint is that the sum of all components of the cord, however they are defined, is the total length, and the time derivative of this sum is zero. ==References and notes== {{Reflist|2}} == See also == {{wiktionary|kinematics}} <div style="-moz-column-count:2; column-count:2;"> *[[Chebychev–Grübler–Kutzbach criterion]] *[[Statics]] *[[Kinetics (physics)]] *[[Kinematic coupling]] *[[Applied mechanics]] *[[Engineering]] *[[Analytical mechanics]] *[[Dynamics (physics)]] *[[Classical mechanics]] * [[Forward kinematics]] * [[Inverse kinematics]] * [[Motion (physics)|Motion]] * [[Celestial mechanics]] * [[Kepler's laws]] * [[Orbital mechanics]] * [[Centripetal force]] * [[Fictitious force]] </div> ==External links== *[http://www.phy.hk/wiki/englishhtm/Kinematics.htm Java applet of 1D kinematics] *[http://frozenport.com/Movies/1D_Kinematics_Beta.swf Flash animated tutorial for 1D kinematics] * [http://www.physclips.unsw.edu.au/ Physclips: Mechanics with animations and video clips] from the University of New South Wales *[http://physicsfunda.googlepages.com/download KINEMATICS FOR HIGH SCHOOL AND IIT JEE LEVEL] *[http://kmoddl.library.cornell.edu KMODDL: Kinematic Models for Design Digital Library, Cornell University Library ] {{Kinematics}} <!--Categories--> [[Category:Classical mechanics]] [[Category:Kinematics|*]] <!--Interwiki--> [[ar:علم الحركة]] [[bs:Kinematika]] [[ca:Cinemàtica]] [[cs:Kinematika]] [[da:Kinematik]] [[de:Kinematik]] [[el:Κινηματική]] [[es:Cinemática]] [[fr:Cinématique]] [[gl:Cinemática]] [[hr:Kinematika]] [[id:Kinematika]] [[it:Cinematica]] [[he:קינמטיקה]] [[ka:კინემატიკა]] [[lt:Kinematika]] [[hu:Kinematika]] [[ms:Kinematik]] [[nl:Kinematica]] [[ja:運動学]] [[pl:Kinematyka]] [[pt:Cinemática]] [[ru:Кинематика точки]] [[sq:Kinematika]] [[sk:Kinematika]] [[sl:Kinematika]] [[fi:Kinematiikka]] [[ta:அசைவு விபரியல்]] [[tr:Kinematik]] [[uk:Кінематика]] [[ur:جنبشیات]] [[yi:קינעמאטיק]] [[zh:运动学]]