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 rB/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, |rB/A| is constant and one can calculate the derivative of rB/A using the lemma to get

 

 

where is the angular velocity of the rigid body. Since vB/A is the cross product of and rB/A, the relative velocity of B with respect to A is perpendicular to both and rB/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 vB/A. Using the above expression for relative velocity, one gets

 

 

2-D motion: In 2-D, the angular velocity and angular acceleration can be written as

 

 

Example 1:

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).

 

Example 2

Example 3

Example 4

 

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

 

 


 

 

 

 

 


Mehrdad Negahban and the University of Nebraska, 1996-2002.

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Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526