Simple Nature - Light and Matter

e/Even if the rotating object

has zero angular acceleration,

every point on it has an ac-

celeration towards the center.

axis. We adopt a coordinate system, d, with an inward (radial) axis

and a tangential axis. The length of the infinitesimal circular arc

dstraveled by the point in a time interval dtis related to dθby the

definition of radian measure, dθ= ds/r, where positive and negative

values of dsrepresent the two possible directions of motion along

the tangential axis. We then havevt= ds/dt=rdθ/dt=ωr, or

vt=ωr. [tangential velocity of a point at a

distancerfrom the axis of rotation]

The radial component is zero, since the point is not moving inward
or outward,

vr= 0. [radial velocity of a point at a

distancerfrom the axis of rotation]

Note that we had to use the definition of radian measure in this
derivation. Suppose instead we had used units of degrees for our an-
gles and degrees per second for angular velocities. The relationship
between dθdegreesand dsis dθdegrees= (360/ 2 π)s/r, where the extra
conversion factor of (360/ 2 π) comes from that fact that there are 360
degrees in a full circle, which is equivalent to 2πradians. The equa-
tion forvtwould then have beenvt= (2π/360)(ωdegrees per second)(r),
which would have been much messier. Simplicity, then, is the rea-
son for using radians rather than degrees; by using radians we avoid
infecting all our equations with annoying conversion factors.
Since the velocity of a point on the object is directly proportional
to the angular velocity, you might expect that its acceleration would
be directly proportional to the angular acceleration. This is not true,
however. Even if the angular acceleration is zero, i.e., if the object
is rotating at constant angular velocity, every point on it will have
an acceleration vector directed toward the axis, e. As derived on
page 213, the magnitude of this acceleration is

ar=ω^2 r. [radial acceleration of a point

at a distancerfrom the axis]

For the tangential component, any change in the angular velocity

dωwill lead to a change dω·rin the tangential velocity, so it is easily

shown that

at=αr. [tangential acceleration of a point

at a distancerfrom the axis]

self-check C

Positive and negative signs ofωrepresent rotation in opposite direc-

tions. Why does it therefore make sense physically thatωis raised to

the first power in the equation forvtand to the second power in the one

forar? .Answer, p. 1056

Section 4.2 Rigid-body rotation 273