8 ROTATIONAL MOTION 8.2 Rigid body rotationA
rigid bodyP’ (^)
Q P
B (^) axis of rotation
Figure 67: Rigid body rotation.
The instantaneous angular velocity of the body ω(t) is defined
ω = lim
δφ dφ
δt (^0) δt =^ dt.^ (8.1)^
Note that if the body is indeed rotat
→
ing rigidly, then the calculated value of ω
should be the same for all possible points P lying within the body (except for
those points lying exactly on the axis of rotation, for which ω is ill-defined). The
rotation speed v of point P is related to the angular velocity ω of the body via
v = σ ω, (8.2)
where σ is the perpendicular distance from the axis of rotation to point P. Thus, in
a rigidly rotating body, the rotation speed increases linearly with (perpendicular)
distance from the axis of rotation.
It is helpful to introduce the angular acceleration α(t) of a rigidly rotating
body: this quantity is defined as the time derivative of the angular velocity. Thus,
dω
α = =
dt
d^2 φ
dt^2
, (8.3)
where φ is the angular coordinate of some arbitrarily chosen point reference
within the body, measured with respect to the rotation axis. Note that angular