Relative motion of points on a rigid body
Relative velocity
of points on a rigid body: Consider points A and B on a rigid body. The relative position
of B with respect to A is given by r_{B/A}.
The
relative velocity of B with respect
to A is given by _{}. Since the distance between points A and B does not change
in a rigid body, r_{B/A} is constant and one can
calculate the derivative of r_{B/A} using the lemma to get
_{}
where
_{} is the angular
velocity of the rigid body. Since v_{B/A} is the cross product of _{} and r_{B/A}, the relative velocity of B with respect to A is
perpendicular to both _{} and r_{B/A}.
Relative acceleration
of points on a rigid body: The relative acceleration of point B
with respect to A is given by the
derivative of v_{B/A}. Using the above
expression for relative velocity, one gets
_{}
2D motion: In 2D, the angular
velocity and angular acceleration can be written as
_{}
The
following are graphical representations of the relative velocity and relative
acceleration of points on a rigid body.

The
left figure shows that the motion of
point B relative to A describes a circle since the distance
between the points does not change. As a result, the velocity of B relative to A is tangent to this circle. The right figure shows the components
of acceleration. Since the relative motion is on a circle, the acceleration of B relative to A has a component tangent to the circle (_{}= tangential component) and a component towards the center of
rotation (_{}= centripetal component).
Since
the relative motion of two points on a rigid body always is described by motion
on a circular path, one can also use polar coordinates with a constant radial
coordinate r (i.e., _{}), _{}, and _{}. This results in
expressions for relative velocity and acceleration given as
_{}
Another
method of describing relative motion is by using normal and tangential
coordinates. This results in
_{}
ã Mehrdad Negahban and the
University of Nebraska, 19962002.
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Department
of Engineering Mechanics, University of Nebraska, Lincoln, NE 685880526