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0593_C04*_fm Page 97 Monday, May 6, 2002 2:06 PM

Kinematics of a Rigid Body 97

4.9 Relative Velocity and Relative Acceleration of Two Points
on a Rigid Body
Consider a body B moving in a reference frame R as in Figure 4.9.1. Let P and Q be points
fixed in B. Consider the velocity and acceleration of P and Q and their relative velocity
and acceleration in reference frame R. Let O be a fixed point (the origin) of R. Let vectors
p and q locate P and Q relative to O and let vector r locate P relative to Q. Then, from
Figure 4.9.1, these vectors are related by the expression:

=
+
pq r (4.9.1)
By differentiating we have:

V = V + ddt or V − V = d dt= V (4.9.2)
R P R Q R R P R Q R R P Q
r
r
Because P and Q are fixed in B, r is fixed in B. Hence, from Eq. (4.5.2), we have:
ddt= ωω × r (4.9.3)
R R B
r
Then, by substituting into Eq. (4.9.2), we have:

R P Q R B R P R Q R Q
V = ωω × r or V = V + ωω × r (4.9.4)
By differentiating in Eq. (4.9.4), we obtain the following relations for accelerations:

R
R P Q = αα B × r+ ωω B ×( ωω B × r)
R
R
a (4.9.5)
and
a + αα
R
R
R a = R Q R B × r+ ωω B ×( ωω B × r) (4.9.6)
P
B

Q P
r

q
p
FIGURE 4.9.1
A body B moving in a reference R
frame R with points P and Q fixed O
in B.