Simple Nature - Light and Matter

axis. We adopt a coordinate system, d, with an inward (radial) axisand a tangential axis. The length of the infinitesimal circular arcds traveled by the point in a time interval dt is related to dθ by thedefinition of radian measure, dθ = ds/r, where positive and negativevalues of ds represent the two possible directions of motion alongthe tangential axis. We then have v t = ds/ dt = r dθ/ dt = ωr, orv t = ωr . [tangential velocity of a point at adistance r from the axis of rotation]The radial component is zero, since the point is not moving inwardor outward,v r = 0 . [radial velocity of a point at adistance r from the axis of rotation]Note that we had to use the definition of radian measure in thisderivation. Suppose instead we had used units of degrees for our anglesand degrees per second for angular velocities. The relationshipbetween dθ degrees and ds is dθ degrees = (360/2π)s/r, where the extraconversion factor of (360/2π) comes from that fact that there are 360degrees in a full circle, which is equivalent to 2π radians. The equationfor v t would then have been v t = (2π/360)(ω degrees per second )(r),which would have been much messier. Simplicity, then, is the reasonfor using radians rather than degrees; by using radians we avoidinfecting all our equations with annoying conversion factors.Since the velocity of a point on the object is directly proportionalto the angular velocity, you might expect that its acceleration wouldbe directly proportional to the angular acceleration. This is not true,however. Even if the angular acceleration is zero, i.e., if the objectis rotating at constant angular velocity, every point on it will havean acceleration vector directed toward the axis, e. As derived onpage 209, the magnitude of this acceleration isa r = ω 2 r . [radial acceleration of a pointat a distance r from the axis]e / Even if the rotating objecthas zero angular acceleration,every point on it has an accelerationtowards the center.For the tangential component, any change in the angular velocitydω will lead to a change dω·r in the tangential velocity, so it is easilyshown thata t = αr . [tangential acceleration of a pointat a distance r from the axis]self-check CPositive and negative signs of ω represent rotation in opposite directions.Why does it therefore make sense physically that ω is raised tothe first power in the equation for v t and to the second power in the onefor a r ? ⊲ Answer, p. 923Section 4.2 Rigid-Body Rotation 267