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. 1906 Excerpt: ...centroid of a volume, area or line is the centroid of a mass of unit density occupying the volume, area or line. Other names in common use for the centroid are: centre of inertia, centre of gravity. The sum 1, mx is often called the first moment of the masses. To see how integrals replace sums, take a very simple case. ...
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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. 1906 Excerpt: ...centroid of a volume, area or line is the centroid of a mass of unit density occupying the volume, area or line. Other names in common use for the centroid are: centre of inertia, centre of gravity. The sum 1, mx is often called the first moment of the masses. To see how integrals replace sums, take a very simple case. Example 1. Find the centroid of a thin straight rod of uniform density and thickness. Let the length AB (Fig. 19) of the rod be a and the mass of unit length X; then the total mass M of the rod is Aa. A PQ B Fig. 19. Divide AB into n equal parts, of which PQ is one. Let AP=x PQ = Sx; then the mass of PQ is A&c. If we suppose the mass of PQ to be concentrated at P, the moment of the mass of PQ about A will be x. X.Sx; while if we suppose the mass of PQ to be concentrated at Q, the moment about A will be (x+Sx)kSx. We may assume the moment to be greater than x. kSx but less than (x + Sx)X.Sx. If x is the distance of the centroid from A the moment of the whole mass M about A is Mx. Hence Mx 2 xkSx, but Mx X(x + 8x) AS.r. x=0 a=0 But when the number n of the parts, such as PQ, into which the rod is divided, becomes infinite each of these two sums converges to the same limit ( 33), namely the integral from 0 to a of xkdx; therefore _ fa Mx= I xkdx=$X.a?=Ma, so that x=a. In this and similar cases it is sufficient to consider only one of the inequalities and to write Mx = 2x8x approx.; Mx = xXdx. Jo In practice the differential dx is frequently used instead of the increment Sx; we shall however adhere meantime to the notation of increments. CENTROIDS. THEOREMS OF PAPPUS. 91 In the same way, or from symmetry, we see that x=4ajZir. The student may show that the integral is the same if we suppose the elementary mass to be o-JVQ. MJy, concentra...
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