Kinematics

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In physics, kinematics is the branch of classical mechanics concerned with describing the motions of objects without considering the factors that cause or affect the motion. By contrast, the science of dynamics is concerned with the forces and interactions that produce or affect the motion. The term "kinematics" derives from the Greek word κινειν, meaning "to move".

The basis of kinematics is the choice of coordinates that describe the position(s) and/or orientation(s) of object(s). The time derivatives of these coordinates correspond to velocities and accelerations. An important component of kinematics is differentiating position to obtain the velocity or acceleration, and vice versa, integrating velocity or acceleration to obtain the position. Another key component of kinematics is converting between different sets of coordinates that describe the same motion. Both of these components are fundamentally mathematical methods, and are not based on physical principles.

The simplest form of kinematics is the description of point particle motion (translational kinematics). The description of rotation (rotational kinematics) is more complicated. The state of a generic rigid body may be described by combining both translational and rotational kinematics (rigid-body kinematics). The most complicated case is the kinematics of a system of rigid bodies.

In all of these cases, the most useful choice of coordinates may be determined by constraints on the motion, or by the geometrical nature of the force causing or affecting the motion. For example, it may be convenient to describe the motion of a bead constrained on a circular hoop by its angle on the hoop. Similarly, it may be practical for calculations to describe the motion of a particle acted upon by a central force using polar coordinates.

[edit] Fundamental equations

[edit] Relative motion

To describe the motion of one body, A, with respect to another body, O, when we know how each is moving with respect to another body, B, we use the following equation:

$r_{A/O} = r_{B/O} + r_{A/B} \,\!$

The above relative motion equation states that the motion of A relative to O is equal to the motion of B relative to O plus the motion of A relative to B. The equation is derived from the law of vector addition.

For example, Ann is moving with velocity $V A$ and Bob is moving with velocity $V B$ , each of these velocities being given with respect to the ground. We wish to know how fast Ann is moving relative to Bob; we call this velocity $V A / B$ . From the equation above we have:

$V_{A} = V_{B} + V_{A/B} \,\! .$

To find $V A / B$ we simply rearrange this equation to obtain:

$V_{A/B} = V_{A} -V_{B} \,\! .$

At velocities comparable to the speed of light, these equations of relative motions are found through Einstein's theory of special relativity rather than the above equation of relative motion.

[edit] Rotating frame

One fundamental equation in kinematics is the equation for the derivative of a vector described in a rotating frame of reference. As a sentence, it is: the time derivative of a vector in a fixed frame is equal to the derivative of the vector relative to the rotating frame plus the cross product of the angular velocity of the frame with the vector. In equation form that is:

$\left.\frac{dr(t)}{dt}\right|_{X,Y,Z} = \left.\frac{dr(t)}{dt}\right|_{x,y,z} + \omega \times r(t)$

where:

r(t) is a vector

X,Y,Z is the fixed frame

x,y,z is the rotating frame

ω is the rate of rotation of the frame.

[edit] Algebraic equations

The algebraic equations of linear kinematics with constant acceleration, taught in secondary school physics classes, follow the form:

$\,\!x_f - x_i = v_i t + \frac{1}{2} at^2 \qquad x_f - x_i = \frac{1}{2} (v_f + v_i)t$ $\,\!v_f = v_i + a t \qquad a = \frac{v_f - v_i}{t} \qquad v_f^2 = v_i^2 + 2 a (x_f - x_i)$

Where $\,\!x_i$ and $\,\!x_f$ are respectively the initial and final position, $\,\!v_i$ and $\,\!v_f$ are respectively the initial and final velocity, and $\,\!a$ is the constant acceleration. These equations can be arrived at using serveral different methods, some of which involve calculus, and some that do not.

These equations can easily be extended to planar rotational kinematics with simple variable exchanges:

$\,\!\theta_f - \theta_i = \omega_i t + \frac{1}{2} \alpha t^2 \qquad \theta_f - \theta_i = \frac{1}{2} (\omega_f + \omega_i)t$ $\,\!\omega_f = \omega_i + \alpha t \qquad \alpha = \frac{\omega_f - \omega_i}{t} \qquad \omega_f^2 = \omega_i^2 + 2 \alpha (\theta_f - \theta_i)$

Where $\,\!\theta_i$ and $\,\!\theta_f$ are respectively the initial and final angular position, $\,\!\omega_i$ and $\,\!\omega_f$ are respectively the initial and final angular velocity, and $\,\!\alpha$ is the constant angular acceleration. However, a student must recognize that the equations are not so apparently equivalent, since angular position is not a true vector quantity.

[edit] Coordinate systems

[edit] 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 $\vec i \, \!$ is a unit vector in the x direction, $\vec j \, \!$ is a unit vector in the y direction, and $\vec k \, \!$ is a unit vector in the z direction.

The position vector, $\vec s \, \!$ (or $\vec r \, \!$ ), the velocity vector, $\vec v \, \!$ , and the acceleration vector, $\vec a \, \!$ are expressed using rectangular coordinates in the following way:

$\vec s = x \vec i + y \vec j + z \vec k \, \!$

$\vec v = \dot {s} = \dot {x} \vec {i} + \dot {y} \vec {j} + \dot {z} \vec {k} \, \!$

$\vec a = \ddot {s} = \ddot {x} \vec {i} + \ddot {y} \vec {j} + \ddot {z} \vec {k} \, \!$

Note: $\dot {x} = \frac{dx}{dt}$ , $\ddot {x} = \frac{d^2x}{dt^2}$

Velocity is defined as the rate of displacement of the particle, or in other words Displacement/Time taken. If we shrink the time period to almost 0 we obtain the instantaneous velocity; hence $v = \frac{dr}{dt}$

[edit] Two dimensional rotating reference frame

This coordinate system only expresses planar motion.

This system of coordinates is based on three orthogonal unit vectors: the vector $\vec i$ , and the vector $\vec j$ which form a basis for the plane in which the objects we are considering reside, and $\vec k$ 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.

[edit] 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, we must take the derivatives of the unit vectors into account when taking the derivative of any of these vectors. If the coordinate frame is rotating at a rate of $\vec \omega \, \!$ in the counterclockwise direction (that's $\omega \vec k$ using the right hand rule) then the derivatives of the unit vectors are as follows:

$\dot{\vec i} = \omega \vec k \times \vec i = \omega \vec j$

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

[edit] Position, velocity, and acceleration

Kinematics
← Integrate ... Differentiate → Displacement \| Velocity (Speed) \| Acceleration \| Jerk \| Snap

Given these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this reference frame.

[edit] Position

Position is straightforward:

$\vec s = x \vec i + y \vec j$

It's just the distance from the origin in the direction of each of the unit vectors.

[edit] Velocity

Velocity is the time derivative of position:

$\vec v = \frac{d\vec s}{dt} = \frac{d (x \vec i)}{dt} + \frac{d (y \vec j)}{dt}$

By the chain rule, this is:

$\vec v = \dot x \vec i + x \dot{\vec i} + \dot y \vec j + y \dot{\vec j}$

Which from the identities above we know to be:

$\vec v = \dot x \vec i + x \omega \vec j + \dot y \vec j - y \omega \vec i = (\dot x - y \omega) \vec i + (\dot y + x \omega) \vec j$

or equivalently

$\vec v = (\dot x \vec i + \dot y \vec j) + (y \dot{\vec j} + x \dot{\vec i}) = \vec v_{rel} + \vec \omega \times \vec r$

where $\vec v_{rel}$ is the velocity of the particle relative to the coordinate system.

[edit] Acceleration

Acceleration is the time derivative of velocity.

We know that:

$\vec a = \frac{d \vec v}{dt} = \frac{d \vec v_{rel}}{dt} + \frac{d (\vec \omega \times \vec r)}{dt}$

Consider the $\frac{d \vec v_{rel}}{dt}$ part. $\vec v_{rel}$ has two parts we want to find the derivative of: the relative change in velocity ( $\vec a_{rel}$ ), and the change in the coordinate frame ( $\omega \times \vec v_{rel}$ ).

$\frac{d \vec v_{rel}}{dt} = \vec a_{rel} + \omega \times \vec v_{rel}$

Next, consider $\frac{d (\vec \omega \times \vec r)}{dt}$ . Using the chain rule:

$\frac{d (\vec \omega \times \vec r)}{dt} = \dot{\vec \omega} \times \vec r + \vec \omega \times \dot{\vec r}$

$\dot{\vec r}$ we know from above:

$\frac{d (\vec \omega \times \vec r)}{dt} = \dot{\vec \omega} \times \vec r + \vec \omega \times (\vec \omega \times \vec r) + \vec \omega \times \vec v_{rel}$

So all together:

$\vec a = \vec a_{rel} + \omega \times \vec v_{rel} + \dot{\vec \omega} \times \vec r + \vec \omega \times (\vec \omega \times \vec r) + \vec \omega \times \vec v_{rel}$

And collecting terms:

$\vec a = \vec a_{rel} + 2(\omega \times \vec v_{rel}) + \dot{\vec \omega} \times \vec r + \vec \omega \times (\vec \omega \times \vec r)$

[edit] Three dimensional rotating coordinate frame

(to be written)

This section is a stub. You can help by expanding it.

[edit] Kinematic constraints

A kinematic constraint is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples:

[edit] 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, :

$v_G(t) = \omega \times r_{G/O} \,\!$

For the case of an object that does not tip or turn, this reduces to v = R ω .

[edit] Gears (no slip)

Similar to the case of rolling without slipping, this involves two bodies with the same motion at their contact point. For any bodies 1 and 2 the constraint is:

$r_1 \omega_1 = r_2 \omega_2 \,\!$

where

r is a radius

ω is an angular velocity

[edit] 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.

[edit] Rotational Motion

Rotational motion is the description of the turning of an object and involves the following three quantities, as do linear motion:

[edit] Angular displacement

Angular position θ is the angle that a line from the axis of rotation to a point on an object makes with respect to the positive x-axis, which is measured counterclockwise. $d d f / f d$

[edit] Angular velocity

The magnitude of the angular velocity w is the rate at which the angular position $t h e t a$ changes with respect to time t:

$\mathbf{\omega} = \frac {d\theta}{dt}$

[edit] Angular acceleration

The magnitude of the angular acceleration a is the rate at which the angular velocity $ω$ changes with respect to time t:

$\mathbf{\alpha} = \frac {d\mathbf{\omega}}{dt}$

[edit] See example

[edit] See also

Look up kinematics in
Wiktionary, the free dictionary.

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Categories: Cleanup from December 2006 | Wikipedia articles needing clarification | Articles with sections needing expansion | Classical mechanics | Kinematics