# Scalar invariance across inertial and non-inertial frames

1. Sep 8, 2008

### e2m2a

Suppose a rotating object, denoted as a rotator, rotates around a vertical axis rigidly attached to a sliding object, denoted as the slider. The slider is constrained to move only in a straight line along a linear track along a y-axis with respect to a x-y coordinate system. We hold the slider stationary and set the rotator in motion in the counter-clockwise direction. We assume no friction is present, so by the rotator's own rotational inertia, it continues to rotate without stopping and without any need for additional impulses. At some point in time as the rotator passes the zero degrees position, we release the slider. The y-component of the centrifugal reactive force acting on the axis, causes the slider to accelerate in the positive y-direction. With respect to an observer in the frame of the slider, the centripetal force acting on the center of mass of the rotator is orthogonal to the tangential velocity of the center of mass of the rotator. Thus, the scalar dot product of the centripetal force and the tangential velocity is zero. Because the scalar product is zero, is this not an invariant? Would this still hold with respect to a laboratory frame? That is, would not the scalar dot product of the centripetal force and the tangential velocity with respect to the laboratory frame still be orthogonal? I have had discussions with one individual who stated because the slider is moving with respect to the laboratory frame, you must add this instanstaneous velocity component to the tangential velocity of the rotator, and this would cause the angle between the centripetal force and the tangential velocity not to be orthogonal. Is this true?