Page 153 - Dynamics of Mechanical Systems

0593_C05_fm Page 134 Monday, May 6, 2002 2:15 PM

134 Dynamics of Mechanical Systems

This means that during the time that Q is a center of zero velocity, P moves in a circle
about Q. Indeed, during the time that Q is a center of zero velocity, B may be considered
to be in pure rotation about Q. Finally, observe in Eq. (5.4.2) that ω is normal to the plane
of motion. Hence, we have:

v = ωω r or v = r ω (5.4.3)
P
P
P
where the notation is defined by inspection.
If a body B is at rest, then every point of B is a center of zero velocity. If B is not at rest
but has planar motion, then at most one point of B, in a given plane of motion, is a center
of zero velocity.
To prove this last assertion, suppose B has two particles, say O and Q, with zero velocity.
Then, from Eq. (5.3.10), we have:

v = v + ωω × r (5.4.4)
Q
O
where r locates O relative to Q. If both O and Q have zero velocity then,

ωω× = 0 (5.4.5)
r
If O and Q are distinct particles, then r is not zero. Because ωω ωω is perpendicular to r, Eq.
(5.4.5) is then satisfied only if ωω ωω is zero. The body is then in translation and all points have
the same velocity. Therefore, because both O and Q are to have zero velocity, all points
have zero velocity, and the body is at rest — a contradiction to the assumption of a moving
body. That is, the only way that a body can have more than one center of zero velocity,
in a plane of motion, is when the body is at rest.
We can demonstrate these concepts by graphical construction. That is, we can show that
if a body has planar motion, then there exists a particle of the body (or of the mathematical
extension of the body) with zero velocity. We can also develop a graphical procedure for
finding the point.
To this end, consider the body B in general plane motion as depicted in Figure 5.4.1.
Let P and Q be two typical distinct particles of B. Then, if B has a center O of zero velocity,
P and Q may be considered to be moving in a circle about O. Suppose the velocities of P
and Q are represented by vectors, or line segments, as in Figure 5.4.2. Then, if P and Q
rotate about a center O of zero velocity, the velocity vectors of P and Q will be perpen-
dicular to lines through O and P and Q, as in Figure 5.4.3.
Observe that unless the velocities of P and Q are parallel, the lines through P and Q
perpendicular to these velocities will always intersect. Hence, with non-parallel velocities,
the center O of zero velocity always exists.

P B
B v
P
P Q
Q
v Q

FIGURE 5.4.1 FIGURE 5.4.2
A body B in general plane motion. Vector representations of the velocities of particles
P and Q.