Outlines of a New Theory of Rotatory Motion

9781235651090: Outlines of a New Theory of Rotatory Motion

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1834. Excerpt: ... SECTION I. ON THE ACTUAL MOTION OF BODIES, AND ON THE FORCES CAPABLE OF PRODUCING ANY GIVEN MOTION. Signification of the terms, Simple Rotatory Motion, and Angular Velocity. The only rotatory motion of which we have a clear idea, is that of a body which turns on an immoveable axis. For we see plainly all the circumferences of the circles which the different points of the body describe about this axis, and which they can really describe at the same time, without changing in any respect their relative position, or what we may denominate the form of the body. We have an equally clear idea of the quantity or measure of this rotatory motion; for since all the points in it describe similar arcs in the same time, the ratio of the velocity of a point to the radius of the circle which it describes is the same for all points, and it is this constant ratio which forms the measure, or, as it is called, the angular velocity of rotation, (1) (1) Let OPp (fig. 1) be the axis of rotation, R, r, any two points describing the circles RR', rr', which must lie in planes perpendicular to Op, and therefore parallel to each other. Let RR', rr', be arcs described uniformly in the same time (t). Draw PS parallel to pr, and when pr comes into the position pr, let PS' be the corresponding position of PS. Then i SPS' = l rpr'. (Euc. xi. 10.) But since the body is rigid, SR = S'R', and therefore, / RPR' = /. SPS' = Lrpr'. Hence every point in the body describes round Op in the time (t) an angle = RPR'. Let (to) be the angle described in 1" by every point, then £RPR' = tw, RR' and the actual velocity (v) of R = w t = = i it. W, t V.... and = to, which is the same for every point, is PR called the Angular Velocity of the body, or the Velocity of Rotation. Composition of Rotatory ...

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