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Invariance of scalar dot product across inertial and non-inertial frames

  1. Sep 6, 2008 #1
    I have a question concerning scalar invariance with respect to an accelerating and an inertial reference frame. Here is the problem. Suppose we have a rotating spherical object, which we denote as the rotator, attached to a near-massless wire. The other end of the wire slips loosely over a vertical axis, so that it can freely rotate around it. The vertical axis is rigidly attached to a second object, denoted as the slider. The slider is constrained to move in a straight line along a linear track with respect to the y-axis of a x-y coordinate system. In this discussion we assume there is no friction. Initially, we prevent the slider from moving, and the rotator is located 180 degrees with respect to the x-axis without any motion. We give the rotator an initial impulse in the negative y-direction. The rotator rotates after the impulse in the counter-clockwise direction. We release the slider so that it can move just as the rotator reaches the zero degrees position. The slider begins to accelerate at this point in the positive y-direction due to the y-component of the centrifugal reactive force acting on the vertical axis. This centrifugal reactive force arises as an equal and opposite force to the centripetal force acting on the center of mass of the rotating sphere. The rotator rotates until it reaches its 90 degrees position with respect to the x-axis. Here is the fundamental question and issue. With respect to the frame of the slider, an observer would see that the centripetal force is always orthogonal with respect to the tangential velocity of the rotator, regardless of the fact that he is in a non-inertial frame. Therefore, the scalar dot product of the centripetal force and the tangential velocity will always be zero. Since the scalar dot product is zero in this frame, should it not be zero in any other frame, such as the laboratory frame of an observer who is not on the slider? Another words, with respect to a laboratory frame, would not the centripetal force and the tangential velocity of the rotator always be perpendicular to each other, even though the slider is accelerating with respect to the laboratory frame?
  2. jcsd
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