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0593_C05_fm Page 135 Monday, May 6, 2002 2:15 PM

Planar Motion of Rigid Bodies — Methods of Analysis 135

B
P
Q

P v P Q v Q

O
FIGURE 5.4.3 FIGURE 5.4.4
Location of center O of zero velocity by the inter- A body in translation with equal velocity particles
section of lines perpendicular to velocity vectors. and center of zero velocity infinitely far away.
If the velocities of P and Q are parallel with equal magnitudes, and the same sense, then
they are equal. That is,

v = v Q and ωω × r = 0 (5.4.6)
P
where the last equality follows from Eq. (5.3.10) with r locating P relative to Q. If P and
Q are distinct, r is not zero; hence, ωω ωω is zero. B is then in translation. Lines through P and
P
Q
Q perpendicular to v and v will then be parallel to each other and thus not intersect
(except at infinity). That is, the center of zero velocity is infinitely far away (see Figure
5.4.4).
If the velocities of P and Q are parallel with non-equal magnitudes, then the center of
zero velocity will occur on the line connecting P and Q. To see this, first observe that the
relative velocities of P and Q will have zero projection along the line connecting P and Q:
That is, from Eq. (5.3.10), we have:

v = v + ωω × r or v P/Q = ωω × r (5.4.7)
Q
P
Hence, v P/Q must be perpendicular to r. (This simply means that P and Q cannot approach
or depart from each other; otherwise, the rigidity of B would be violated.) Next, observe
Q
that if v and v are parallel, their directions may be defined by a common unit vector n.
P
That is,
v = v n v = v Q n, and v P/Q = v P Q n (5.4.8)
P
Q
P
/
,
Q
where v , v , and v P/Q are appropriate scalars. By comparing Eqs. (5.4.7) and (5.4.8) we see
P
Q
P
that n must be perpendicular to r. Hence, when v and v are parallel but with non-equal
magnitudes, their directions must be perpendicular to the line connecting P and Q. There-
P
fore, lines through P and Q and perpendicular to v and v will coincide with each other
Q
and with the line connecting P and Q (see Figure 5.4.5).
Next, observe from Eqs. (5.3.10) and (5.4.3) that, if O is the center of zero velocity, then
P
the magnitude of v is proportional to the distance between O and P. Similarly, the
Q
magnitude of v is proportional to the distance between O and Q. These observations
enable us to locate O on the line connecting P and Q. Specifically, from Eq. (5.4.3), the
distance from P to O is simply v /ω.
P
From a graphical perspective, O can be located as in Figure 5.4.6. Similar triangles are
formed by O, P, Q and the “arrow ends” of v and v .
Q
P