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. 1897 edition. Excerpt: ... at the pivot, it can have no motive effect whatever. 119. Now, the fundamental dynamical idea in rotation is the moment of a force (read sect. 57 again); and the following general statements are true, with their converses. (1) The moment of the resultant must equal the sum of the moments of the components ...
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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. 1897 edition. Excerpt: ... at the pivot, it can have no motive effect whatever. 119. Now, the fundamental dynamical idea in rotation is the moment of a force (read sect. 57 again); and the following general statements are true, with their converses. (1) The moment of the resultant must equal the sum of the moments of the components about any point in every possible case, otherwise the resultant would not be truly the resultant, because unable to replace the components in rotating power. That this condition is fulfilled by the diagonal of a parallelogram whose sides represent the component forces may be proved among other ways as follows: To show that the resultant given by the parallelogram of forces is equivalent to its components in rotating as well as in translating power--that is, that its moment about any point in the plane is equal to the sum of the moments of the two components. The moment of the force AB about a point O (fig. 34) is (see sect. 53) geometrically representable by twice the area of the triangle OAB; the moment of AD is similarly proportional to twice the triangle OAD, and that of AC is twice OAC: hence what we have to prove is the following equality between the areas, OAB ] OAC = OAD; the point O being in the plane of the parallelogram. Now OAC = OBD + ADB, because the bases pig. 34. are equal, and the height of the single triangle is equal to the sum of the heights of the others (this is an easy extension of Euc. I. 38--analytically obvious, thus and by inspection, OAD = OAB + OBD + ADB; therefore O AD = OAB + 0 AC; which was to be proved. (2) The algebraic sum of the moments of any number of forces about a point on their resultant is zero; in other words, the sum of the positive moments equals the sum of the negative. (The moments of two forces...
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