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0593_C05_fm Page 149 Monday, May 6, 2002 2:15 PM

Planar Motion of Rigid Bodies — Methods of Analysis 149

Equation (5.7.12) shows that if we know the velocities of two points of B we can
determine the angular velocity of B. Then, from Eq. (5.7.9), the coordinates (x , y ) of the
C
C
center of zero velocity can be determined. That is,
 x − x   x − x 
x = x − ˙ y P Q and y = y + ˙ x P Q (5.7.13)
C p P   C P P  
˙
˙
 y − ˙ y Q   y − ˙ y Q 
P
P
We can verify these expressions using the geometric procedures of Section 5.4. Consider,
for example, a body B moving in the X–Y plane with center of zero velocity C as in Figure
5.7.2. Let P and Q be two points of B whose positions and velocities are known. Then, the
magnitude of their velocities designated by v and v are related to the angular speed ω
P
Q
of B as:
+
v = v P = aω and v = v Q = ( a b)ω (5.7.14)
P Q
where a and b are the distances shown in Figure 5.7.2. By comparing and combining these
expressions, we have:
P)
Q (
v = v + bω or ω = v − v / b (5.7.15)
Q P
From the geometry of Figure 5.7.2 we see that:

P)
θ
v = ˙ y cos ,θ v = ˙ y cos , b = ( x − x /cosθ (5.7.16)
P P Q Q Q
By substituting into Eq. (5.7.15) we obtain:

˙ y − ˙ y
ω= Q P (5.7.17)
x − x
Q P
This verifies the second equation of Eq. (5.7.12); the first expression of Eq. (5.7.12) can be
verified similarly.

Y
v Q
ω
v P
θ
θ b
Q
θ P
a
FIGURE 5.7.2 C B
A body B with zero velocity center C O
and typical points P and Q. X