Angular Velocity Vs. Rectilinear Velocity?

Strictly defined, rectilinear or simply linear velocity is the rate-of-travel along a particular straight-line path. Since both magnitude or speed and the line of travel are included in its definition, velocity is a vector or directional quantity. Examples of rectilinear velocity include the eastbound noonday train and the migrating swallows headed from Goya, Argentina to San Juan, Capistrano. Rectilinear velocity is not the only sort of velocity there is, however - there is also angular velocity, which represents rotational motion about a specific axis - examples of which include the spinning electron and the rotating Earth. The mathematics of vectors is of particular importance when considering matters related to angular velocity.

Units of angular velocity, represented by the variable ω (omega) may be given as radians or degrees per unit time. Alternatively, since 2π radians and 360 degrees each equals one complete spherical rotation, angular velocity is sometimes described in units such as cycles per second, rotations per hour or revolutions in a year. Unfortunately, the unit designation cycles per second (cps) is also used for electrical frequency; where it is synonymous with Hertz. Angular velocity and electrical frequency are independent quantities, however, and for this reason angular velocity should not be given the unit designation, Hertz. Although linear velocity is intuitive, since it is experienced every day, angular velocity is not so readily conceptualized.

One of the simplest ways to differentiate between angular velocity and linear velocity, is to imagine a merry-go-round in a school playground. A vertical steel pipe runs through the merry-go-round into the soil beneath it and serves as the axis about which it rotates. There are two children on this merry-go-round - one is located two feet from the center, while the other is sitting at the edge or perimeter of the merry-go-round, four feet from center. Angular velocity represents the rate of rotation of the merry-go-round, that is how much angle it turns through in a given amount of time. For example, if the merry-go-round experiences one rotation every four seconds, it moves through π radians, or 180° of angle in two seconds time, regardless of its radius.

As can be seen, both children experience the same angular velocity, which is independent of where one sits on the merry-go-round. At the perimeter of the merry-go-round, the linear velocity obeys the relationship, v = ωr. For mathematical purposes, the value of r depends on where one sits on the merry-go-round, and can be as little as zero at the center, to the full radius. Thus, even though the radius is four feet, for the child only two feet from the center, r = 2. Since ω = π/2 radians-per-second, the innermost child experiences π radians-per-second linear velocity. Using comparable mathematics, with r = 4 for the outermost child, he experiences 2π radians per second linear velocity - twice the amount of the innermost child.