Kinematics

In Physics , kinematics is the branch of Mechanics concerned with the Motion s of objects without being concerned with the Force s 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

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} \,\!

This is derived from the law of vector addition (an equation from Vector Space ) and states that motion of A relative to O is equal to the motion of B relative to O plus the motion of A relative to B.

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} \,\! .

Note that this is valid only for a limited range of velocities - when any of the bodies are moving at a velocity comparable to the Speed Of Light , Einstein's special theory of relativity is required; see more in the article Special Relativity .

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:

Consider the

rac{d ec v_{rel}}{dt}

part.

ec v_{rel}

has two parts we want to find the derivative of: the relative change in velocity (

ec a_{rel}

), and the change in the coordinate frame (

\omega 	imes ec v_{rel}

rac{d ec v_{rel}}{dt} = ec a_{rel} + \omega 	imes ec v_{rel}

Next, consider

rac{d (ec \omega 	imes ec r)}{dt}

. Using the chain rule:

rac{d (ec \omega 	imes ec r)}{dt} = \dot{ec \omega} 	imes ec r + ec \omega 	imes \dot{ec r}

\dot{ec r}

we know from above:

rac{d (ec \omega 	imes ec r)}{dt} = 
\dot{ec \omega} 	imes ec r + 
ec \omega 	imes (ec \omega 	imes ec r) +
ec \omega 	imes ec v_{rel}

So all together:

ec a =  ec a_{rel} + \omega 	imes ec v_{rel} + 
\dot{ec \omega} 	imes ec r + 
ec \omega 	imes (ec \omega 	imes ec r) +
ec \omega 	imes ec v_{rel}

And collecting terms:

ec a =  ec a_{rel} + 2(\omega 	imes ec v_{rel}) + 
\dot{ec \omega} 	imes ec r + 
ec \omega 	imes (ec \omega 	imes ec r)

Three dimensional rotating coordinate frame

(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, :

v_G(t) = \omega 	imes 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:

r_1 \omega_1 = r_2 \omega_2 \,\!

where

r is a Radius

ω is an angular velocity

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 derivative of this sum is zero.

ROTATIONAL MOTION

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

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.

Angular velocity

The magnitude of the angular velocity w is the rate at which the angular position

theta

changes with respect to time t:

\mathbf{\omega} = rac {d	heta}{dt}

Angular acceleration

The magnitude of the angular acceleration a is the rate at which the angular velocity

\omega

changes with respect to time t:

\mathbf{\alpha} = rac {d\mathbf{\omega}}{dt}

SEE EXAMPLE

Point Mass

Rigid Body

Information About

Kinematics