A Treatise on the Integral Calculus Founded on the Method of Rates (Paperback)

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. 1907 edition. Excerpt: ...of the point Piy P2, etc. from the plane. Thus, referring to rectangular coordinate planes, its distance x from that of yz is given by the equation x. Jp = pxxx J-p2x2 +... = 2px. Multiplying both sides by the mass of one of the equal particles, the equation becomes x-2m--mlx1-j-m2x2-(-... = mxy.. (1) where mx, m2, etc. are the masses of unequal particles situated at Plf P2y etc., and 2m is the total mass. The point (x, y z) whose three coordinates are similarly defined is the centre of gravity of unequal particles. XVI. CENTRE OF GRA VITY OF A CONTINUOUS BODY. 235 The second member of equation (1), that is, the aggregate of the distances multiplied by the masses, is called in Mechanics the Statical Moment of the total mass 2m with respect to the plane ofyz. The Centre of Gravity of a Continuous Body. 205-The property of the centre of gravity given in the preceding articles obviously extends to continuous bodies. That is to say, the position of this point, which is sometimes called the Centroid, is determined by the mean distances of the particles from three given planes; and, when given, it determines the mean distance from any plane. If the body is homogeneous, that is to say, if the same quantity of matter is contained in equal elements of volume, the mean distances are the ordinary arithmetical means. On the other hand, if the body is not homogeneous, its density at a given point constitutes the weight to be attributed to the element at that point, supposing, as in Art. 135, the geometric elements to be all equal. In any case, the variable density is simply used as a factor of the element both in finding the total mass 'as in Art. 135) and in finding the statical moment. 206. The centre of gravity of a homogeneous solid is also called...