Rotational Kinematics

    We have talked about how our position vectors can be represented as groups of numbers, for instance either in individual components or as the magnitude and angles associated with the vector. Then, in introducing centripetal motion, we saw that the acceleration is what causes the object to curve into the circular shape. Later we will see that the equivalent force associated with centripetal motion is not a "physical" force, so in many ways we would like to introduce a new notation in which we no longer have to consider centripetal motion. This is done by considering rotational motion. Rotational motion concerns the motion of an object about some central point associated with the object. We shall see that all of the mechanical relationships that we have found so far have equivalents in rotational motion.

Angular Displacement

    To begin, let us consider the rotation of a ball around its center. How far does a point on the surface move? From geometry, we know that if the ball rotates through an angle q, then the surface will move a distance x = rq. This lets us write
 
formula (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, 2pr. So from (14) we get that theta = 2p. Since this is the same as 360 degrees, we get

formula

    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.

Angular Velocity

    What happens if ask how fast the sphere is rotating? If the sphere rotates through an angle q in a time t, we can write the average angular speed as
 
formula (15)

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

formula (16)
 
Let us rewrite Dt in terms of the velocity at a distance r
formula

Substituting this into (15) yields

formula

or
 

formula (17)
 

Angular Acceleration

    So far we have found rotational analogs for position and velocity. What about acceleration? We define angular acceleration as the change in angular velocity per unit time
 
formula (18)

Substituting (17) for the angular velocity, we get

formula

but, v/t =at, so
 

formula (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
 

formula (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.

formula
formula

Example:
    A ball on a string is swung in such a way that it is experiencing an angular acceleration of 0.05 rad/s2. 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?

drawing

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

formula

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

formula

or
 

formula (21)

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

formula

Angular Equations of Motion

    Before leaving circular motion, we can use the same approach we used to derive (21) to find the other rotational equivalents to our kinematical equations. Thus
v(t) = v0 + at

becomes
 

formula (22)
 
and
formula

becomes
 

formula (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.
formula