Elements of Analytical Mechanics

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Comparing this value of Fwith that given in the discussion of motion due to gravity alone (Art. 92), we see that F is the same function of z, but a different function of /.
CONSTRAINED MOTION.
155 These results can be more readily obtained by considering- the motion to be free, the body being acted upon by the result- ant of all the forces, including the normal reaction. This re- sultant is Mg sin a — F, and it acts in the direction of the path.
139. Let the circumference OAB (Fig. 52) be in th
...e plane xz, and tangent to the axis of x at O. Then we have for the point A, Eq. (372), or — 2 = ^/* sin' a = OA sin a = OB sin* a . . (377) /=/^. (378) which is the time of fall from rest down the diameter of the circle.
Hence the time required for the body to pass over any chord of a given circle, the plane of the circle being vertical and the body starting from the upper extremity of the vertical diameter, is independent of its length; that is, the circumference is the locus of simultaneous arrival down all right lines in a vertical plane, these lines having a common point at the origin of motion.


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