*Notes to a video lecture on http://www.unizor.com*

__Rotational Kinematics__

So far we were mostly considering

*translational*motion of point-objects - a motion along a straight line with or without external

*forces*acting upon this object.

*Rotational*motion obeys the rules in many respects analogous to the laws of

*translational*motion, except we have to change linear movement to rotation.

Consider a point-object

**connected by a rigid rod of the length**

*m***to an axis, around which this object can rotate within a plane of rotation that is perpendicular to an axis of rotation.**

*r*The picture below illustrates such a movement and also indicates the position of

*angular velocity*of a rotation, which we will explain later.

**ω**Let's discuss the similarities and differences between

*translational*movement along a straight line and

*rotation*around an axis within a plane perpendicular to this axis (a

*plane of rotation*).

The first main concept of translational motion is

*position*or

*distance from the beginning of motion*(for a straight line movement) as a function of time. In

*rotational*motion its equivalent is

*angle of rotation*from some original position as a function of time.

Translation | Rotation |

Distances(t) | Angleφ(t) |

The next concept is

*speed*or (better)

*velocity*of translational motion. This is a first derivative of

*position*(or

*distance*) by time:

*v(t) = s'(t)*.

Its equivalent for rotational motion is

*angular speed*, which is a first derivative of

*angle of rotation*by time:

*ω(t) = φ'(t)*

While vector character of

*speed*of translational motion is obvious and is reflected in the term

*velocity*, vector character of

*angular speed*is less obvious.

The

*angle of rotation*from the first glance is a scalar function of time. But only from the first glance.

In theory, rotational motion always assumes existence of an axis of

rotation and a plane of rotation. To reflect these characteristics and a

magnitude of angular speed, an angular speed is represented by a vector

from a center of rotation along an axis of rotation perpendicularly to a

plane of rotation with a magnitude equal to a value of angular speed.

This allows to represent the rotation in its full spectrum of

characteristics - magnitude, axis, plane of rotation. The picture above

represents angular speed as a vector

*, which we may call*

**ω(t)***angular velocity*vector.

There is one more characteristic of rotational motion not yet discussed - its direction. It is also reflected in

*angular velocity*

as a vector by its direction. In theory, we can choose two different

directions along the axis of rotation. The direction chosen is such

that, if we look from its end onto a plane of rotation, the rotation is

counterclockwise. Another interpretation of this is the "rule of the

right hand" because if you put you right hand on a plane of rotation

such that your finger go around the axis of rotation pointing to a

direction of rotation, your thumb points to a direction of the

*angular velocity*vector.

So,

*angular velocity*vector represents axis, plane, direction of rotation as well as magnitude of

*angular speed*.

To be more precise, since this vector representation of

*angular velocity*is a little unusual, it is customary to call it "pseudo-vector" instead of "vector".

During infinitesimal time interval

*d*an object rotating around an axis on a radius

**t***turns by an angle*

**r***d*, covering the distance

**φ(t)***d*(this is the length of an arc of radius

**s(t)=r**·d**φ(t)***r*and angle

*dφ*, according to a known formula of geometry).

From this follows:

*d*or

**s(t)/**d**t = r**·d**φ(t)/**d**t**

**v(t) = r·ω(t)**Translation | Rotation |

Speedv(t)=s'(t)v(t)=r·ω(t) | Angular Speedω(t)=φ'(t)ω(t)=v(t)/r |

The next concept is

*acceleration*that needs its rotational analogue. Obviously, it's the first derivative of

*angular velocity*or the second derivative of an

*angle of rotation*by time.

Using the vector interpretation of

*angular velocity*, we can consider

*angular acceleration*as a vector as well. It is also directed along the axis of rotation.

During infinitesimal time interval

*d*an

**t***angular velocity*

*changes by*

**ω(t)***d*.

**ω(t)**From this follows relationship between

*linear acceleration*

*and*

**a***angular acceleration*

*:*

**α***or*

**a(t) =**d**v(t)/**d**t = r**·d**ω(t)/**d**t**

**a(t) = r·α(t)**Translation | Rotation |

Accelerationa(t)=v'(t)a(t)=r·α(t) | Angular Acc.α(t)=ω'(t)α(t)=a(t)/r |

Obviously, integrating the definitions of

*angular velocity*

**and**

*ω**angular acceleration*

**for**

*α*__constant__

*angular acceleration*, we come up with formulas similar to those familiar from

*translational*movement:

*ω(t) = ω(0) + α·t*

*φ(t) = φ(0) + ω(0)·t + α·t²/2**Angular acceleration*as a vector (as a pseudo-vector, to be exact), is colinear to the axis of rotation, because

*angular velocity*is.

If rotation goes as on the above picture and the speed of rotation increases, the

*angular acceleration*would be directed upwards, the same way as the

*angular velocity*.

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