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48 Rotation Rigid Body About Fixed Axis

When a body rotates about a fixed axis, any point P located in
the body travels along a circular path. To study this motion it is first
necessary to discuss the angular motion of the body about the axis.
Angular Motion. Since a point is without dimension, it cannot
have angular motion. Only lines or bodies undergo angular motion.
For example, consider the body shown in Fig. 2-18,a and the angular
motion of a radial line r located within the shaded plane.
Angular Position. At the instant shown, the angular position of
r is defined by the angle θ measured from a fixed reference line to r.
Angular Displacement. The change in the angular position,
which can be measured as a differential dθ, is called the angular
displacement. This vector has a magnitude dθ of measured in degrees,
radians, or revolutions, where 1 rev = 2π rad. Since motion is about a
fixed axis, the direction of dθ is always along this axis. Specifically,
the direction is determined by the right-hand rule; that is, the fingers
of the right hand are curled with the sense of rotation, so that in this
case the thumb, or dθ points upward, Fig. 2-18,a. In two dimensions,
as shown by the top view of the shaded plane, Fig. 2-18,b, both θ and
dθ are counterclockwise, and so the thumb points outward from the
page.
Angular Velocity. The time rate of change in the angular
position is called the angular velocity ω (omega). Since dθ occurs
during an instant of time dt, then,
dθ
ω = . 2-49
dt
/
This vector has a magnitude which is often measured in rad s.
It is expressed here in scalar form since its direction is also along the
axis of rotation, Fig. 2-18,a.When indicating the angular motion in the
shaded plane, Fig. 2-18,b, we can refer to the sense of rotation as
clockwise or counterclockwise. Here we have arbitrarily chosen
counterclockwise rotations as positive and indicated this by the curl
shown in parentheses next to Eq. 2-49. Realize, however, that the
directional sense of ω is actually outward from the page.

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