(14) |

Notice that both *x* and *r* are distances,
so that theta is dimensionless. We associate the "unit" **radian** with
*q* as a
pseudo-distance measurement. The relationship between radians and degrees
can be found by considering the rotation of the ball completely about its
axis. Then the ball will have turned through 360 degrees. The distance
*x* is just the circumference of a circle of radius *r*, *2**pr*.
So from (14) we get that theta* = 2**p*.
Since this is the same as 360 degrees, we get

Let us quickly consider how to make this a vector. We wish to denote the direction through which the angle is measured. In order to do this, we note that the angle is measured from the axis of rotation. We define the direction of the angle by the right hand rule: if we curl the fingers of our right hand in the direction of rotation, then our thumb points in the direction of the angular vector.

(15) |

Now take the limit of this as D*t *
®* 0*. This gives the instantaneous
angular velocity

(16) |

Let us rewrite D

Substituting this into (15) yields

or

(17) |

(18) |

Substituting (17) for the angular velocity, we get

but, *v/t =a _{t}*, so

(19) |

Notice that the angular acceleration is defined
in terms of the **tangential component **of the acceleration. It does
not depend on the centripetal acceleration at all. We can write the centripetal
acceleration in terms of these new angular coordinates. It is

(20) |

This is true even if the angular velocity is
changing.

We can summarize our results
so far by the simple rule: **the rotational analog of a standard kinematical
quantity is just the tangential component of the kinematical quantity divided
by its distance from the axis of rotation**.

__Example__:

The radius of the earth is
approximately 6400 km. If it takes 24 hours to rotate once on its axis,
find the angular velocity of the earth.

__Example__:

A ball on a string is swung
in such a way that it is experiencing an angular acceleration of 0.05 rad/s^{2}.
If it already has an angular velocity of 1.2 rad/s when the timer is started,
how many degrees does it swing through after 30 seconds?

We need the rotational equivalent to the equation relating position, acceleration and time. Start with

substitute in the relationships between the angular coordinates and the Cartesian coordinates to get

or

(21) |

This is the equation of motion for a rotating object. Substituting in our given information, we get

becomes

(22) |

and

becomes

(23) |

Thus, we see that the rotational equations of motion are identical to the linear equations with the rotational variables substituted for the linear ones.