# Kinematics

In physics, kinematics is the branch of mechanics concerned with the motions of objects without being concerned with the forces that cause the motion. In this latter respect it differs from dynamics, which is concerned with the forces that affect motion.

Because of its relative simplicity, kinematics is usually taught before dynamics or the concept of a force is introduced. The equations of motion are generally taught at secondary school level.

## Fundamental equations

### Relative motion

This is a simple equation from vector mathematics that restates vector addition: motion of A relative to O is equal to the motion of B relative to O plus the motion of A relative to B:

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

### 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:

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

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.

## Coordinate systems

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

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

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

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

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

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

### Two dimensional rotating coordinate frame

This coordinate system only expresses planar motion.

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

#### 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 $\displaystyle \vec \omega \, \!$ in the counterclockwise direction (that's $\displaystyle \omega \vec k$ using the right hand rule) then the derivatives of the unit vectors are as follows:

$\displaystyle \dot \vec i = \omega \vec k \times \vec i = \omega \vec j$

$\displaystyle \dot \vec j = \omega \vec k \times \vec j = - \omega \vec i$

#### 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 coordinate system.

##### Position

Position is straightforward:

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

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

##### Velocity

Velocity is the time derivative of position:

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

By the chain rule, this is:

$\displaystyle \vec v = \dot x \vec i + x \dot \vec i + \dot y \vec j + y \dot \vec j$

Which from the identities above we know to be:

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

or equivalently

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

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

##### Acceleration

Acceleration is the time derivative of position.

We know that:

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

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

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

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

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

$\displaystyle \dot \vec r$ we know from above:

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

So all together:

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

And collecting terms:

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

(to be written)

## Kinematic constraints

A kinematic constraint is any condition relating properties of a dynamic system that must hold 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, or:

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

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

### 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:

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

where