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0593_C05_fm Page 133 Monday, May 6, 2002 2:15 PM

Planar Motion of Rigid Bodies — Methods of Analysis 133

0 = rΩcosθ − φ ˙ cosφ (5.3.23)
l

θ
φ
˙˙ x =− rΩ 2 cos − φl ˙˙ sin − φl ˙ 2 cosφ (5.3.24)
and

0 =−rΩ sinθ − φ ˙˙ cosφ + φ 2 ˙ sinφ (5.3.25)
2
l
l
Observe further from Figure 5.3.4 that x may be expressed as:
x = cosθ + cosφ (5.3.26)
l
r
and that from the law of sines we have:

l = r
sinθ sinφ (5.3.27)

By differentiating in Eqs. (5.3.26) and (5.3.27), we immediately obtain Eqs. (5.3.22) and
(5.3.23). Finally, observe that by differentiating in Eqs. (5.3.22) and (5.3.23) we obtain Eqs.
(5.3.24) and (5.3.25).

5.4 Instant Center, Points of Zero Velocity

If a point O of a body B with planar motion has zero velocity, then O is called a center of
zero velocity. If O has zero velocity throughout the motion of B, it is called a permanent
center of zero velocity. If O has zero velocity during only a part of the motion of B, or even
for only an instant during the motion of B, then O is called an instant center of zero velocity.
For example, if a body B undergoes pure rotation, then points on the axis of rotation
are permanent centers of zero velocity. If, however, B is in translation, then there are no
points of B with zero velocity. If B has general plane motion, there may or may not be
points of B with zero velocity. However, as we will see, if B has no centers of zero velocity
within itself, it is always possible to mathematically extend B to include such points. In
this latter context, bodies in translation are seen to have centers of zero velocity at infinity
(that is, infinitely far away).
The advantage, or utility, of knowing the location of a center of zero velocity can be
seen from Eq. (5.3.10):

v = v + ωω × r (5.4.1)
Q
P
where P and Q are points of a body B and where r locates P relative to Q. If Q is a center
Q
P
of zero velocity, then v is zero and v is simply:
v = ωω × r (5.4.2)
P