iSoul In the beginning is reality

# Time and circular motion

Consider two circular motions, one a wheel and the other a clock (click for animations):

The wheel is the target motion to be measured. The clock is the reference motion, which maintains a constant angular velocity.

Comparison can be made of either the motion of a radius (e.g., a spoke of the wheel or hand of the clock) or the motion of a point along the circumference (which equals the motion of the wheel along the ground). If the length of the radius or circumference is known, it doesn’t matter which is used because there is a known proportionality between the motion of the two (C = 2πR).

If the length of the radius or circumference are not known or not specified, then there is a choice of whether to use a radius or circumferential point. For the clock, the length of the radius or circumference are not specified, so only the angular velocity is known. For the wheel, it depends on whether the wheel is moving along the ground or not. If it is moving along the ground, a circumferential point is used (case 1), otherwise the motion of a radius would be simpler since it has the same units as the clock (case 2).

The motion of the clock corresponds to time (duration). The motion of the wheel corresponds to space (length) in case 1 and time (duration) in case 2. The rate of motion is the ratio of a measure of the wheel to a measure of the clock. The rate of motion in case 1 is the ratio of the distance traversed by a point on the circumference to the angle traversed by a hand of the clock (e.g., kilometres per hour). The rate of motion in case 2 is the ratio of the angle traversed by a spoke on the wheel to the angle traversed by a hand of the clock (e.g., revolutions per minute).

What does this tell us about space and time? It shows how time is an angular measure and space is a linear (circumferential) measure. A body (e.g., vehicle) with an internal clock has its own measure of time by the motion of the clock at hand and its own measure of space by the wheel(s) traversing the ground (e.g., odometer). A body with an external clock (e.g., the motion of the sun) depends on external observation for knowledge of time (duration). A body that moves without wheels (e.g., a boat) depends on external observation for knowledge of space (length).

Is it possible that the clock could be traversing the ground and so its movement measured by the travel distance? That would be an unusual clock but it is possible. Would such a clock be measuring time or space? It would measure space in that its units would be units of distance but time in that it would be a measure of the reference motion or independent flow. Is time any measure of the reference motion or is time duration?

time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration (Isaac Newton)

Time for Newton, and to this day, is considered both flow and duration but these are different concepts, and should be distinguished.