A treatise on mechanics, applied to the arts; including statics and hydrostatics / By the Rev. H. Moseley ... Published under the direction of the committee of General Literature and Education, appointed by the Society for Promoting Christian Knowledge.

  • Moseley, Henry, 1801-1872.
Date:
1847
Catalogue details

Licence: Public Domain Mark

Credit: A treatise on mechanics, applied to the arts; including statics and hydrostatics / By the Rev. H. Moseley ... Published under the direction of the committee of General Literature and Education, appointed by the Society for Promoting Christian Knowledge. Source: Wellcome Collection.

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    Then by the last proposition, moment of id = mt. p -f mt. Pt* „ R — mt. r + nit. p2 „ R — mt. R2+mt. p „ R4 = mt. r3 -f mt. p4 -f CO. r=: -f- CO. „ Rn = mt. Rni+Pn Therefore adding these equations together and striking out: similar terms on the tivo sides we get moment Ra = mt. p + mt. iq + mt. iq-f-. . -f mt. iq. r„ is manifestly the resultant of all the forces of the system. Hence, therefore, it follows that the moment of the resultant force is equal to the sum of the moments of the components, in all cases. If the forces be in equilibrium their resultant equals nothing ; the sum of their moments about any point, therefore, equal's nothing. The demonstration of this proposition applies to every pos¬ sible case of forces in the same plane, and, therefore, to the case of parallel forces. But in this case the same line drawn from the point about which the moments are measured, is per¬ pendicular to all the forces of the system. Thus, in the fig. Art. 45, the Line m ??q is perpendicular to the directions of all the forces Pn p2, p3, p4. So that, in the case of parallel forces, we have only to draw a line from the point about which the moments are measured, perpendicular to any one of the forces of the system ; we shall then obtain the moment of each force by multiplying it by its distance from the point measured on this line. Also, if r be the resultant of all the forces, and it intersect the line m ml produced in a point which we call r, we have, by the proposition. R X m r = pf X m m 1 4- r, X m m2 + iq x m m3 — p4 X m — P5 X M m5 p. X M7)i +P, X mm, + P, X Mm- — P, X M m. — Pr X Mvi, M r — —1-------iL---^---2-? R. In which expression the moments of p4 and p5 are taken nega¬ tively, because they tend to turn the system in a contrary way from the rest. Also, in the case of parallel forces, the resultant r is equal to the sum of the components (Art. 4G) ; it being here also observed that those must be taken negatively, whose tendency * The moments of all' the forces which tend to turn the system in an * V opposite direction are here supposed to be taken negatively.