To calculate the point of average velocity for any rotating line. In Figure D-1 dividing the radius in an odd number of equal parts corresponding to an odd number of arcs, and taking the middle arc (such that there are an equal number of arcs above and below it.)
For each arc equidistant above and below this arc there will be an equal absolute value of change for each arc. So, if the arc CD = length X, and the arc A = zero. The arc below CD added to the arc above A will also equal X. Stated in terms of the middle arc (BE) the arc above BE + the arc below will = X and so on for addition of each successive arc above and below BE. BE is paired with itself and = X. So that adding each pair together and dividing by 2 (for 2 arcs added) = the average length of all arcs or average linear velocity, this being equal to length of the middle arc or BE in this case. Graphically portrayed if each arc is stretched out;
As BE = average velocity of rotation in terms of a linear velocity (distance) then with BE straightened out;
BE (as circumference)
Since circumference BE = ½2π r [r as C-A] then the area of the square above would =
½ x 2 x π x r x r or πr2 (which is what we want).
Motion of a volume
If a mathematical point travels some distance per unit time (d/t) it has a velocity of such.
For a volume of an immutable solid traveling in linear motion all points must have an equal velocity.
As a ratio between various volumes the number of points in a volume is proportional to the volume.
Therefore the velocity of all points added up is equal to the volume x velocity of any point or volume x average velocity of all points.
Area swept by identical volumes (various shapes) is equal, but to calculate that accurately one must calculate the distance traveled by every point else elongated shapes appear to sweep greater area than robust shapes, etc. However area swept: to velocity. Furthermore all PP are of identical volume.
For a rotating volume things are more complicated, as not all points travel the same velocity. Therefore, one cannot just take the volume and multiply it by the velocity of any point, however, by definition, if we could get the average velocity that could be multiplied by the volume to get the total momentum of the object.
To calculate the velocity for every point is difficult if possible. But for every radius drawn to every point the midpoint of all the radii averaged equals the average velocity by canceling out as was done with a straight line.
Now this would be the same as the mid-mass point for any figure because with homologous figures from the rotation point on an edge and the opposite point reciprocal lines can be drawn to the mid-mass point (so each cancels) and the same for every other point. So that mid-mass point is always the average point.
The same holds true for irregular figures although the reciprocal line would be less symmetrical.
Therefore the mid-mass point (center of mass) for any figure has a rotational velocity equal to the average velocity of the entire figure. So length of arc of mid-mass point x volume / t = p.