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L8 Relative Motion using Rotating Axes

L8 Relative Motion using Rotating Axes - S Widnall J...

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S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L8 - Relative Motion using Rotating Axes In the previous lecture, we related the motion experienced by two observers in relative translational motion with respect to each other. In this lecture, we will extend this relation to another type of observer. That is, an observer who rotates relative to a stationary observer. Later in the term, we will consider the more general case of a translating (includes accelerating) and rotating observer as well as multiple observers who may be translating and rotating with respect to one another. Although governed by the same equations, it is useful to distinguish two cases: first is the rotating observer that observes a phenomena that a stationary observer would categorize as non-accelerating motion such as a fixed object or one moving at constant velocity (and direction). Such an observer might be a child riding on a merry-go-round or play-ground turntable. Such a rotating observer would observe a fixed object or one with constant velocity motion as tracing a curved trajectory with changing velocity and therefore accelerating motion. Our transformation from the rotating into the fixed system, must show that the observed motion was non-accelerating and thus force-free by Newton’s law. We should also be able to describe the motion seen by the rotating observer. The second case is where a physical phenomena takes place on a platform that is rotating, such as a mass- spring system fixed to a turntable. In this case, the motion, the forces and the frequency of oscillation will be a ff ected by the rotation and will di ff er from the system behavior in an inertial coordinate system. Our approach must demonstrate this di ff erence and provide the tools for analyzing the system dynamics in the rotating frame. As a matter of illustration, let us consider a very simple situation, in which a point a at rest with respect to the fixed observer A located at the origin of the coordinates x, y, z , point O , is also observed by a rotating observer, B , who is also located at point O . The coordinate system used by B , x , y , z is instantaneously aligned with x, y, z but rotating with angular velocity . In a), the coordinate system x, y, z and x , y , z are shown slightly o ff set for emphasis. 1
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Consider first case shown in a) , in which the point a located at position r does not move in inertial space. Observer A will observe v a = 0 for the point r . Now consider the same situation observed by B . In the rotating coordinate system, B will observe that the point r , which is fixed in inertial space appears to move backwards due to the rotation of B s coordinate system. v a/B = × r . (1) To reconcile these two observations, it is clear that to obtain the velocity in inertial coordinates, A must correct for the ”artificial” velocity seen by B due to the rotation of B s coordinate system by adding × r to the velocity reported by B .
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L8 Relative Motion using Rotating Axes - S Widnall J...

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