Rotational Acceleration


We live in a world that is defined by three spatial dimensions and one time dimension. Objects move within this domain in two ways. An object translates, or changes location, from one point to another. And an object rotates, or changes its orientation. In general, the motion of an object involves both translation in all three directions and rotation about three principle axes.

On this page we will only consider the rotation of a solid object about one axis. The rotation of an object is similar to the translation in the number of variables we must consider, but the notation is very confusing because it has traditionally been described using Greek symbols. On the slide at the top of the page we have used the traditional Greek notation. To simplify Article 508 compliance, we will just spell out the names of the variables here in the text, rather than use a symbol font.Theta is the symbol that looks like a 0 with a horizontal line through it. Phi is the symbol that looks like a 0 with a vertical line through it. Omega is the symbol that looks like a curly wAlpha is the symbol that looks like a crossed ribbon.

Because the object rotates about an axis of rotation the simplest way to describe the motion is to use. We can specify the angular orientation of an object at any time t by specifying the angle theta the object has rotated from some reference line. Initially, our object is at orientation "0", specified by angle theta 0 at time t0. We have drawn a red line on the disc indicating the initial orientation. The object rotates until time t1 and the red line rotates to angle theta 1. We can define an angular displacement - phi as the difference in angle from condition "0" to condition "1".

phi = theta 1 - theta 0

Angular displacement is a vector quantity, which means that angular displacement has a size and a direction associated with it. The direction is important for later mathematical processes, but the definition is a bit confusing. As the object rotates from point "0" to point "1", it rotates about an axis, so the direction of the angular displacement is measured along the axis. A positive value for the direction of the axis is defined by the right hand ruleExtend your right hand as if to shake hands with someone. Curve your fingers with the base at point "0" and the tips going to point "1". Your thumb points perpendicular to the plane of rotation in the positive direction along the axis of rotation.

Angular displacement is measured in units of radiansTwo pi radians equals 360 degrees. The angular displacement is not a length (not measured in meters or feet), so an angular displacement is different than a linear displacement. As the solid object rotates about the axis of rotation, all of the points of the object experience the same angular displacement, but points farther away from the axis move farther than points closer to the axis. On the slide we consider two points; one is located at radius ra on the edge of the disk, and the other is located at radius rb which is less than ra. As the object rotates through the angular displacement phi, the point on the edge of the disk moves distance sa along a circular path. The point at rb also moves in a circular path, but the distance sb is shorter than the distance sa. In general, the length of the circular path s is equal to the radius r times the angular displacement phi, expressed in radians.

for angular displacement phi,

s = phi * r

ra > rb

sa > sb