ON GEOMETRY |
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If we construct the triangle
DCE by drawing CD parallel and equal to AA', and CE parallel and equal to BB', we see that the angleis equalto the angle because the sides are parallel, and it is also equal to the
angle whose sides are perpendicular to
AA' and BB'. The trianglesABO and CDE are similar because they are isosceles and the angles between the equal sides are equal. Clearly the side DE =w, represents the supplementary velocity which transforms AA' into BB'. We know that in similar triangles the sides are proportional so we can write.By inspection of our figures we see that,
the radius of the circle. The chord
AB may be taken as the arc AB of the circle, providedthe 'time'-interval is taken sufficiently small. Let us write chord. We
have. If we divide both sides of our equation by / we have
But
w/t =A, the acceleration, and. In words,the centripetal acceleration is equal to the square of the velocity in the circle divided by the radius.
The above formula is of structural importance because it is the foundation for the empirical proof of Newton's law of gravitation. For our purpose it is important for other reasons, to be stated later.
There are two more diagrams which should be considered, in this connection. Fig. 17 represents the plane circular motion of a point
P whose orbit in the plane XY is the circle PAB. In three-dimensional space-time the plane circular orbit of motion would be represented by the static cylindrical helix (or screw-line) with axis parallel to the 'time' axis T. (Fig. 17.) We should note that the motion is dynamically circular in the X Y plane, yet a three-dimensional space-time representation gives us a stationary helix.Similarly for vibrational movements which could be represented in one dimension by to-and-f ro movements on the
X axis from A to B and from B to A. (Fig. 18.) If we introduce our space-time form of representation by introducing the Taxis, our vibrational world-line would be represented structurally by a wave-line along the T axis. In particular, if the vibrational motion is simply harmonic, a proper choice of the 'time' unit makes the wave-line a sine curve.^{9} |
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