kinematics: Definition from Answers.com

Classical mechanics

$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
Newton's Second Law

Fundamental concepts
Space · Time · Mass · Force Energy · Momentum

Formulations
Newtonian mechanics Lagrangian mechanics Hamiltonian mechanics

Branches
Statics Dynamics Kinematics Applied mechanics Celestial mechanics Continuum mechanics Statistical mechanics

Scientists
Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson

Kinematics (from Greek κινεῖν, kinein, to move) is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion.^[1]^[2]^[3]^[4]

“

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. — ET Whittaker^[4]

”

Kinematics is not to be confused with another branch of classical mechanics: analytical dynamics (the study of the relationship between the motion of objects and its causes), sometimes subdivided into kinetics (the study of the relation between external forces and motion) and statics (the study of the relations in a system at equilibrium). Kinematics also differs from dynamics as used in modern-day physics to describe time-evolution of a system.

The term kinematics is less common today than in the past, but still has a role in physics.^[5] (See analytical dynamics for more detail on usage). The term "kinematics" also finds use in biomechanics and animal locomotion.^[6]

The simplest application of kinematics is for particle motion, translational or rotational. The next level of complexity is introduced by the introduction of rigid bodies, which are collections of particles having time invariant distances amongst themselves. Rigid bodies might undergo translation and rotation or a combination of both. A more complicated case is the kinematics of a system of rigid bodies, possibly linked together by mechanical joints.

Some kinds of computational fluid dynamics use a (complicated) particle-based kinematic model of the fluid, but fluid flow is generally thought of using a continuum mechanics model rather than kinematics.

Linear motion

Linear or translational kinematics^[7]^[8] is the description of the motion in space of a point along a line, also known as trajectory or path.^{[note 1]} This path can be either straight (rectilinear) or curved (curvilinear).

Position, displacement, and distance

The position of a point in space is its location relative to a chosen origin. It is a vector quantity, expressing both the distance of the point from the origin and its direction from the origin. In three-dimensions, the position of point A can be expressed with a three-dimensional vector

$\mathbf{r}_A = (x_A,y_A,z_A),$

where x_A, y_A, and z_A are the Cartesian coordinates of the point.

Displacement is a vector describing the difference in position between two points. If point A has position r_A = (x_A,y_A,z_A) and point B has position r_B = (x_B,y_B,z_B), the displacement r_AB of B from A is given by

$\mathbf{r}_{AB} = \mathbf{r}_B - \mathbf{r}_A = (x_B-x_A,y_B-y_A,z_B-z_A).$

The distance traveled is always greater than or equal to the displacement.

In physics, distance is a distinct quantity from either position or displacement. It is a scalar quantity, describing the length of the path between two points along which the particle has travelled. The straight-line distance |r| of point A from the origin can be found from the position r by

$|\mathbf{r}| = \sqrt{x_A^{\ 2} + y_A^{\ 2} + z_A^{\ 2}}.$

When considering the motion of a particle over time, distance is the length of the particle's path; displacement is the change from its initial position to its final position. For example, a race car traversing a 10 km closed loop from start to finish travels a distance of 10 km; its displacement, however, is zero because it arrives back at its initial position.

If the position of the particle is known as a function of time (r = r(t)), the distance d it travels from time t₁ to time t₂ can be found through calculus by

$d = \int_{t_1}^{t_2} d\mathbf{r} = \int_{t_1}^{t_2} \sqrt{dx^2 + dy^2 + dz^2} = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}\; dt.$

Velocity and speed

Velocity is the measure of the rate of change in displacement with respect to time; that is, how the displacement of a point changes with time. Velocity also is a vector. Instantaneous velocity (the velocity at an instant of time) is defined through differential calculus as

$\mathbf{v} = \frac {d \mathbf{r}}{d t} \ ,$

where dr is an infinitesimally small displacement and dt is an infinitesimally small length of time.^{[note 2]} As per its definition in the derivative form, velocity can be said to be the time rate of change of displacement. Further, as dr is tangential to the actual path, so is the velocity. Average velocity (velocity over a length of time) is defined as

$\overline{\mathbf{v}} = \frac {\Delta \mathbf{r}}{\Delta t} \ ,$

where Δr is the change in displacement and Δt is the interval of time over which displacement changes. As Δt becomes smaller and smaller, the value of v approaches the value of v.

The speed of an object is the magnitude |v| of its velocity. It is a scalar quantity.

Acceleration

Acceleration is the vector quantity describing the rate of change with time of velocity. Instantaneous acceleration (the acceleration at an instant of time) is defined as:

$\mathbf{a} = \frac {d \mathbf{v}}{d t} \ ,$

where dv 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:

$\overline{\mathbf{a}} = \frac {\Delta \mathbf{v}}{\Delta t} \ ,$

where Δv is the change in velocity and Δt is the interval of time over which velocity changes. As Δt becomes smaller and smaller, a → a.

If acceleration is constant in magnitude and direction, 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.

Types of motion

There are two types of motion in general: uniform and non-uniform. Uniform motion implies constant velocity in a straight line. Non-uniform motion implies acceleration. If the acceleration changes in time, the rate of change of acceleration is called the jerk.

Integral relations

The above definitions can be inverted by mathematical integration to find:

$\mathbf{v}(t) =\mathbf{v}_0 + \int_0^t \mathbf{a}(t') \; dt'$

$\begin{align} \mathbf{r}(t) &=\mathbf{r}_0 + \int_0^t \mathbf{v}(t') \; dt' \\ &= \mathbf{r}_0 + \mathbf{v}_0 t + \int_0^t \left[\int_0^{t'} \mathbf{a}(t'') dt'' \right]\; dt' \\ &= \mathbf{r}_0 + \mathbf{v}_0 t + \int_0^t \left(t-t'\right) \mathbf{a}(t')\; dt',\end{align}$

where the double integration is reduced to one integration by interchanging the order of integration, and subscript 0 signifies evaluation at t = 0 (initial values).

Kinematics of constant acceleration

Many physical situations can be modeled as constant-acceleration processes, such as projectile motion.

Integrating acceleration a with respect to time t gives the change in velocity. When acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly accelerated motion. In this case, the integral relations can be simplified:

$\mathbf{v}(t) = \int_0^{t} \mathbf{a} \; dt' = \mathbf{v}_0 + \mathbf{a}t.$

$\begin{align} \mathbf{r}(t) &= \mathbf{r}_0 + \int_0^t \mathbf{v} \; dt' = \mathbf{r}_0 + \int_0^t (\mathbf{v}_0 + \mathbf{a} t) \; dt' \\ &= \mathbf{r}_0 + \mathbf{v}_0 t + \tfrac{1}{2} \mathbf{a} t^2. \end{align}$

Additional relations between displacement, velocity, acceleration, and time can be derived. Since a = (v − v₀)/t,

$\mathbf{r}(t) = \mathbf{r}_0 + \left(\frac{\mathbf{v}+ \mathbf{v_0}}{2}\right) t .$

By using the definition of an average, this equation states that when the acceleration is constant average velocity times time equals displacement.

A relationship without explicit time dependence may also be derived for one-dimensional motion. By setting r₀ = 0 and noting that at = v − v₀,

$\mathbf{r} \cdot \mathbf{a} t = \left( \mathbf{v} - \mathbf{v}_0 \right) \cdot \frac{\mathbf{v} + \mathbf{v}_0}{2} t \ ,$

where · denotes the dot product. Dividing the t on both sides and carrying out the dot-products:

$2\mathbf{r} \cdot \mathbf{a} = v^2 - v_0^{\ 2}.$

In the case of straight-line motion, r is parallel to a, and r has magnitude equal to the path length s = r − r₀ at time t. Then

$v^2= v_0^2 + 2 a(r-r_0).$

This relation is useful when time is not known explicitly.

Relative velocity

Main article: 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_A, r_B, and r_O. Then the position of A relative to the reference object O is

$\mathbf{r}_{A/O} = \mathbf{r}_{A} - \mathbf{r}_{O} \,\!$

Consequently, the position of A relative to B is

$\mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B = \mathbf{r}_A - \mathbf{r}_O - \left(\mathbf{r}_B-\mathbf{r}_O\right) = \mathbf{r}_{A/O}-\mathbf{r}_{B/O} \ .$

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:

$\mathbf{r}_{A/O} = \mathbf{r}_{A/B} + \mathbf{r}_{B/O} \ ,$

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 relations 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 $\mathbf{V}_{A}$ relative to the reference (we drop the O subscript for convenience) and let Bob move with velocity $\mathbf{V}_{B}$ , each velocity given with respect to the ground (point O). To find how fast Ann is moving relative to Bob (we call this velocity $\mathbf{V}_{A/B}$ ), the equation above gives:

$\mathbf{V}_{A} = \mathbf{V}_{B} + \mathbf{V}_{A/B} \,\! .$

To find $\mathbf{V}_{A/B}$ we simply rearrange this equation to obtain:

$\mathbf{V}_{A/B} = \mathbf{V}_{A} -\mathbf{V}_{B} \,\! .$

At velocities comparable to the speed of light, these equations are not valid. They are replaced by equations derived from Einstein's theory of special relativity.

Example: Rectilinear (1D) motion

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⁻². 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⁻¹. 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

$x_{\mathrm{f}} - x_{\mathrm{i}} = v_{\mathrm{i}} t + \tfrac{1}{2} at^2.$

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.

$0 = v_{\mathrm{i}} t + \tfrac{1}{2} at^2 = t\left(v_{\mathrm{i}} + \tfrac{1}{2} at \right)$

There are two solutions: the first, t = 0, is trivial. The solution of interest is

$t = -\frac{2v_{\mathrm{i}}}{a} = \rm -\frac{2(50\ m\ s^{-1})}{-9.81\ m\ s^{-2}} = 10.2\ s\,.$

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:

$v_{\mathrm{f}}^2 = v_{\mathrm{i}}^2 + 2 a (x_{\mathrm{f}} - x_{\mathrm{i}})$

If the origin of our coordinate system is at the ground, then x_i is zero. Then we solve for x_f and substitute known values:

$x_{\mathrm{f}} = \frac{v_{\mathrm{f}}^2 - v_{\mathrm{i}}^2}{2 a} + x_{\mathrm{i}} = \rm \frac{(0\ m\ s^{-1})^2-(50\ m\ s^{-1})^2}{2(-9.81\ m\ s^{-2})} + 0\ m = 127.55\ m\,.$

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 x_i.

$v_{\mathrm{f}} = \sqrt{v_{\mathrm{i}}^2 + 2 a (x_{\mathrm{f}} - x_{\mathrm{i}})} = \rm \sqrt{(0\ m\ s^{-1})^2 + 2(-9.81\ m\ s^{-2})(0\ m - 127.55\ m)} = 50\ m\ s^{-1}\,.$

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.

Example: Projectile (2D) motion

Figure B: An object fired at an angle

θ

from the ground follows a parabolic trajectory.

Suppose that an object is not fired vertically but is fired at an angle θ 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⁻¹ and 30° from the horizontal.

How far will it travel before hitting the ground?

The object experiences an acceleration of −9.81 m s⁻² in the vertical direction and no acceleration in the horizontal direction. Therefore, the horizontal displacement is

$\Delta x = x_{\mathrm{f}} - x_{\mathrm{i}} = v_{\mathrm{i}} \cos \theta \; t + \tfrac{1}{2} at^2 = v_{\mathrm{i}} \cos \theta \; t$

Solving the equation requires finding 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.

$0 = v_{\mathrm{i}} \sin \theta \; t + \tfrac{1}{2} at^2 = t\left(v_{\mathrm{i}} \sin \theta + \tfrac{1}{2} at \right)$

We now solve for t and substitute this expression into the original expression for horizontal displacement.

$\Delta x = v_{\mathrm{i}} \cos \theta \left(\frac{-2 v_{\mathrm{i}} \sin \theta}{a}\right) = -\frac{v_{\mathrm{i}}^2 \sin 2\theta}{a} = 220.93 \ m\,$

Note the use of the trigonometric identity 2sinθ cosθ = sin 2θ.

Rotational motion

Main article: Circular motion

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 or angular kinematics is the description of the rotation of an object.^[9] 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. The z-axis has been chosen for convenience.

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_⊥(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_⊥(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:

$\mathbf{\omega} = \frac {\mathrm{d}\theta}{\mathrm{d}t}$

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 α is the rate at which the angular velocity ω changes with respect to time t:

$\mathbf{\alpha} = \frac {\mathrm{d}\mathbf{\omega}}{\mathrm{d}t}$

The equations of translational kinematics can easily be extended to planar rotational kinematics with simple variable exchanges:

$\omega_{\mathrm{f}} = \omega_{\mathrm{i}} + \alpha t\!$

$\theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \omega_{\mathrm{i}} t + \tfrac{1}{2} \alpha t^2$

$\theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \tfrac{1}{2} (\omega_{\mathrm{f}} + \omega_{\mathrm{i}})t$

$\omega_{\mathrm{f}}^2 = \omega_{\mathrm{i}}^2 + 2 \alpha (\theta_{\mathrm{f}} - \theta_{\mathrm{i}}).$

Here θ_i and θ_f are, respectively, the initial and final angular positions, ω_i and ω_f are, respectively, the initial and final angular velocities, and α 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

Coordinate systems

In any given situation, the most useful coordinates may be determined by constraints 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 coordinates. Polar coordinates are extended into three dimensions with either the spherical polar or cylindrical polar coordinate systems. These are most useful in systems exhibiting spherical or cylindrical symmetry respectively.

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 vectors 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:

$\boldsymbol{ r} = x\ \hat {\mathbf{ i}} + y \ \hat {\mathbf{ j}} + z \ \hat {\mathbf{ k}} \, \!$

$\boldsymbol{ v} = \dot {\boldsymbol{ r}} = \dot {x} \ \hat {\mathbf{ i}} + \dot {y} \ \hat {\mathbf{ j}} + \dot {z} \ \hat {\mathbf{ k}} \, \!$

$\boldsymbol{ a} = \ddot {\boldsymbol{ r}} = \ddot {x} \ \hat {\mathbf{ i}} + \ddot {y} \ \hat {\mathbf{ j}} + \ddot {z} \ \hat {\mathbf{ k}} \, \!$

Note: $\dot {x} = \frac{\mathrm{d}x}{\mathrm{d}t}$ , $\ddot {x} = \frac{\mathrm{d}^2x}{\mathrm{d}t^2}$

Two dimensional rotating reference frame

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:

$\dot{\hat {\mathbf{ i}}} = \omega \hat {\mathbf{ k}} \times \hat {\mathbf{ i}} = \omega\hat {\mathbf{ j}}$

$\dot{\hat {\mathbf{ j}}} = \omega \hat {\mathbf{ k}}\times \hat {\mathbf{ j}} = - \omega \hat {\mathbf{ i}}$

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:

$\boldsymbol{ r} = x \ \hat {\mathbf{ i}} + y \ \hat {\mathbf{ j}}$

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:

$\boldsymbol{ v} = \frac{\mathrm{d}\boldsymbol{ r}}{\mathrm{d}t} = \frac{\mathrm{d} (x \ \hat {\mathbf{ i}})}{\mathrm{d}t} + \frac{\mathrm{d} (y \ \hat {\mathbf{ j}})}{\mathrm{d}t}$

By the product rule, this is:

$\boldsymbol{ v} = \dot x \ \hat {\mathbf{ i}} + x \dot{\ \hat {\mathbf{ i}}} + \dot y \ \hat {\mathbf{ j}} + y \dot{ \ \hat {\mathbf{ j}}}$

Which from the identities above we know to be:

$\boldsymbol{ v} = \dot x \ \hat {\mathbf{ i}} + x \omega \ \hat {\mathbf{ j}} + \dot y \ \hat {\mathbf{ j}} - y \omega \ \hat {\mathbf{ i}} = (\dot x - y \omega) \ \hat {\mathbf{ i}} + (\dot y + x \omega) \ \hat {\mathbf{ j}}$

or equivalently

$\boldsymbol{ v}= (\dot x \ \hat {\mathbf{ i}} + \dot y \ \hat {\mathbf{ j}}) + (y \dot{ \hat {\mathbf{ j}}} + x \dot{\hat {\mathbf{ i}}}) = \boldsymbol{ v}_{rel} + \boldsymbol{\Omega} \times \boldsymbol{r}$

where v_rel is the velocity of the particle relative to the rotating coordinate system.

Acceleration

First to fourth derivatives of position

Acceleration is the time derivative of velocity.

We know that:

$\boldsymbol{ a} = \frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{ v} = \frac{\mathrm{d} \boldsymbol{ v}_{rel}}{\mathrm{d}t} + \frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{\Omega} \times \boldsymbol{r}$

Consider the $\stackrel{\frac{ \mathrm{d} } { \mathrm{d} t }}{}$ $\boldsymbol{ v}_{rel}$ part. $\boldsymbol{ v}_{rel}$ has two parts we want to find the derivative of: the relative change in velocity ( $\boldsymbol{ a}_{rel}$ ), and the change in the coordinate frame

( $\boldsymbol{\Omega} \times \boldsymbol{ v}_{rel}$ ).

$\frac{\mathrm{d} \boldsymbol{ v}_{rel}}{\mathrm{d}t} = \boldsymbol{ a}_{rel} + \boldsymbol{\Omega} \times \boldsymbol{ v}_{rel}$

Next, consider $\stackrel{\frac{\mathrm{d}}{\mathrm{d}t}}{}$ $(\boldsymbol{\Omega} \times\boldsymbol{ r})$ . Using the chain rule:

$\frac{\mathrm{d} (\boldsymbol{\Omega} \times \boldsymbol{ r})}{\mathrm{d}t} = \dot{\boldsymbol{\Omega}} \times \boldsymbol{ r} + \boldsymbol{\Omega} \times \dot{\boldsymbol{ r}}$

$\dot{\boldsymbol{ r}}=\boldsymbol{ v}=\boldsymbol{ v}_{rel} + \boldsymbol{\Omega} \times \boldsymbol{ r}$ from above:

$\frac{\mathrm{d} (\boldsymbol{\Omega} \times \boldsymbol{ r})}{\mathrm{d}t} = \dot{\boldsymbol{\Omega}} \times \boldsymbol{ r} + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times\boldsymbol{ r}) + \boldsymbol{\Omega} \times \boldsymbol{ v}_{rel}$

So all together:

$\boldsymbol{ a} = \boldsymbol{ a}_{rel} + \boldsymbol{\Omega} \times \boldsymbol{ v}_{rel} + \dot{\boldsymbol{\Omega}} \times \boldsymbol{ r} + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \boldsymbol{ r}) + \boldsymbol{\Omega} \times \boldsymbol{ v}_{rel}$

And collecting terms:^[10]

$\boldsymbol{ a} = \boldsymbol{ a}_{rel} + 2(\boldsymbol{\Omega} \times \boldsymbol{ v}_{rel}) + \dot{\boldsymbol{\Omega}} \times \boldsymbol{ r} + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \boldsymbol{ r})\ .$

Kinematic constraints

This section requires expansion.

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,

$\boldsymbol{ v}_G(t) = \boldsymbol{\Omega} \times \boldsymbol{ r}_{G/O}$ .

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 is the total length, and accordingly the time derivative of this sum is zero. See Kelvin and Tait^[11]^[12] and Fogiel.^[13] A dynamic problem of this type is the pendulum. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord.^[14] An equilibrium problem (not kinematic) of this type is the catenary.^[15]

Notes

^ In mathematics, a line refers to a straight trajectory, and a curve to a trajectory which may have curvature. In mechanics and kinematics, "line' and "curve" both refer to any trajectory, in particular a line may be a complex curve in space. Any position along a specified trajectory can be described by a single coordinate, the distance traversed along the path, or arc length. The motion of a particle along a trajectory can be described by specifying the time dependence of its position, for example by specification of the arc length locating the particle at each time t. The following words refer to curves and lines:
- "linear" (= along a straight or curved line;
- "rectilinear" (= along a straight line, from Latin rectus = straight, and linere = spread),
- "curvilinear" (=along a curved line, from Latin curvus = curved, and linere = spread).
^ Because magnitude of dr 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 ds.

References

Moon, Francis C. (2007). The Machines of Leonardo Da Vinci and Franz Reuleaux, Kinematics of Machines from the Renaissance to the 20th Century. Springer. ISBN 9781402055980.

^ Joseph Stiles Beggs (1983). Kinematics. Taylor & Francis. p. 1. ISBN 0891163557. http://books.google.com/books?id=y6iJ1NIYSmgC&printsec=frontcover&dq=kinematics&lr=&as_brr=0&sig=brRJKOjqGTavFsydCzhiB3u_8MA#PPA1,M1.
^ O. Bottema & B. Roth (1990). Theoretical Kinematics. Dover Publications. reface. ISBN 0486663469. http://books.google.com/books?id=f8I4yGVi9ocC&printsec=frontcover&dq=kinematics&lr=&as_brr=0&sig=YfoHn9ImufIzAEp5Kl7rEmtYBKc#PPR7,M1.
^ Thomas Wallace Wright (1896). Elements of Mechanics Including Kinematics, Kinetics and Statics. E and FN Spon. Chapter 1. http://books.google.com/books?id=-LwLAAAAYAAJ&printsec=frontcover&dq=mechanics+kinetics&lr=&as_brr=0#PPA6,M1.
^ ^a ^b Edmund Taylor Whittaker & William McCrea (1988). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press. Chapter 1. ISBN 0521358833. 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.
^ See, for example: Russell C. Hibbeler (2009). "Kinematics and kinetics of a particle". Engineering Mechanics: Dynamics (12th ed.). Prentice Hall. p. 298. ISBN 0136077919. http://books.google.com/books?id=tOFRjXB-XvMC&pg=PA298. , Ahmed A. Shabana (2003). "Reference kinematics". Dynamics of Multibody Systems (2nd ed.). Cambridge University Press. ISBN 0521544114. http://books.google.com/books?id=zxuG-l7J5rgC&pg=PA28. , P. P. Teodorescu (2007). "Kinematics". Mechanical Systems, Classical Models: Particle Mechanics. Springer. p. 287. ISBN 1402054416. http://books.google.com/books?id=k4H2AjWh9qQC&pg=PA287.
^ A. Biewener (2003). Animal Locomotion. Oxford University Press. ISBN 19850022X. http://books.google.com/books?id=yMaN9pk8QJAC.
^ James R. Ogden & Max Fogiel (1980). The Mechanics Problem Solver. Research and Education Association. p. 184. ISBN 0878915192. http://books.google.com/books?id=XVyD9pJpW-cC&pg=PA184&dq=%22curvilinear+kinematics%22&lr=&as_brr=0&sig=WW7us4UJzSWOA19pfdAbwTJvPR4.
^ R. Douglas Gregory (2006). Classical Mechanics: An Undergraduate Text. Cambridge UK: Cambridge University Press. Chapter 2. ISBN 0521826780. http://books.google.com/books?id=uAfUQmQbzOkC&printsec=frontcover&dq=%22rigid+body+kinematics%22&lr=&as_brr=0#PRA1-PA25,M1.
^ R. Douglas Gregory (2006). Chapter 16. Cambridge: Cambridge University. ISBN 0521826780. http://books.google.com/books?id=uAfUQmQbzOkC&printsec=frontcover&dq=%22rigid+body+kinematics%22&lr=&as_brr=0#PRA1-PA457,M1.
^ R. Douglas Gregory (2006). pp. 475-476. Cambridge: Cambridge University. ISBN 0521826780. http://books.google.com/books?id=uAfUQmQbzOkC&printsec=frontcover&dq=%22rigid+body+kinematics%22&lr=&as_brr=0#PRA1-PA475,M1.
^ William Thomson Kelvin & Peter Guthrie Tait (1894). Elements of Natural Philosophy. Cambridge University Press. p. 4. ISBN 1573929840. http://books.google.com/books?id=dHASAAAAIAAJ&pg=PA4&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.
^ William Thomson Kelvin & Peter Guthrie Tait (1894). op. cit.. p. 296. http://books.google.com/books?id=ahtWAAAAMAAJ&pg=PA296&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES#PPA296,M1.
^ M. Fogiel (1980). "Problem 17-11". The Mechanics Problem Solver. Research & Education Assoc.. p. 613. ISBN 0878915192. http://books.google.com/books?id=XVyD9pJpW-cC&pg=PA613&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.
^ Irving Porter Church (1908). Mechanics of Engineering. Wiley. p. 111. ISBN 1110365276. http://books.google.com/books?id=7-40AAAAMAAJ&pg=PA111&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.
^ Morris Kline (1990). Mathematical Thought from Ancient to Modern Times. Oxford University Press. p. 472. ISBN 0195061365. http://books.google.com/books?id=aO-v3gvY-I8C&pg=PA472&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.

External links

Java applet of 1D kinematics
Flash animated tutorial for 1D kinematics
Physclips: Mechanics with animations and video clips from the University of New South Wales
KINEMATICS FOR HIGH SCHOOL AND IIT JEE LEVEL
Kinematic Models for Design Digital Library (KMODDL)
Movies and photos of hundreds of working mechanical-systems models at Cornell University. Also includes an e-book library of classic texts on mechanical design and engineering.

Look up kinematics in Wiktionary, the free dictionary.

v • d • e Kinematics

← Integrate … Differentiate → Displacement (Distance) \| Velocity (Speed) \| Acceleration \| Jerk \| Snap \| Crackle and pop

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Read more
		Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 2006 Encyclopædia Britannica, Inc. All rights reserved. Read more
		Sports Science and Medicine. The Oxford Dictionary of Sports Science & Medicine. Copyright © Michael Kent 1998, 2006, 2007. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Kinematics". Read more

kinematics

Contents

Linear motion

Position, displacement, and distance

Velocity and speed

Acceleration

Types of motion

Integral relations

Kinematics of constant acceleration

Relative velocity

Rotational motion

Point object in circular motion

Coordinate systems

Fixed rectangular coordinates

Two dimensional rotating reference frame

Derivatives of unit vectors

Position, velocity, and acceleration

Position

Velocity

Acceleration

Kinematic constraints

Rolling without slipping

Inextensible cord

See also

Notes

References

External links

	What is kinematic in physics? Read answer...
	What is rotational kinematics? Read answer...
	What are the kinematic equations? Read answer...

	What is Kinematic similarity?
	Is brushing teeth an open kinematic or close kinematic chain?
	Why basketball have deliberate rotational kinematics?