Mechanics of engineeringComprising statics and dynamics of solids: and the mechanics of the materials of constructions, or strength and elasticity of beams, columns, arches, shafts, etc . *dpdp — j?J_a [_COS(pJ0 Pdpj dcp. 22 MECHANICS OF ENGINEERING. (Note on double integration,—The quantitycos cp J p* dp \d<p, is that portion of the summation / / cos cppdpdcp which belongs to a single elementary sector (triangle), since all its-elements (rectangles), from centre to circumference, have thesame cp and dcp.)That is, 1 r3 p + a r* r+a 2 r sin a ^ ? 3 «/- - C0B ^ = Wa L™ * = 3 • ~^T - 1 r x a i
Mechanics of engineeringComprising statics and dynamics of solids: and the mechanics of the materials of constructions, or strength and elasticity of beams, columns, arches, shafts, etc . *dpdp — j?J_a [_COS(pJ0 Pdpj dcp. 22 MECHANICS OF ENGINEERING. (Note on double integration,—The quantitycos cp J p* dp \d<p, is that portion of the summation / / cos cppdpdcp which belongs to a single elementary sector (triangle), since all its-elements (rectangles), from centre to circumference, have thesame cp and dcp.)That is, 1 r3 p + a r* r+a 2 r sin a ^ ? 3 «/- - C0B ^ = Wa L™ * = 3 • ~^T - 1 r x a i i i. ~ ^ r Sln i P or, putting p = 2a = total angle oi sector, a? = „? 3—? 6 p — 4^ For a semicircular plate this reduces to x = o-. [iVtffe.—In numerical substitution the arcs or and /? usedabove (unless sin or cos is prefixed) are understood to be ex-pressed in circular measure (7r-measure); , for a quad-rant, fi = | = + ; for 30°,/? = |; or, in general, if p . -, 180° 1 *~\ in degrees = , then p in ^--measure = -. \ Problem 4. Sector of a flat ring; thin c \Ja plate, etc.—Treatment similar to that of Problem 3, the difference being that the. Fio. 19. J limits of the interior integrations areinstead of . Result, l—n — 4 r* — r* sin \fiX = 3 r? - r; ~~~W~% Parallel forces and the centre of gravity. 23Pboblem 5.—Segment of a circle; thin plate, etc.—Fig. 20. Since each rectangular element of any ver-tical strip has the same x, we may take thestrip as dF in finding x, and use y as thehalf-height of the strip. dF = 2ydx, andfrom similar triangles x : y :: (— dy) : dx,, xdx = — ydy. Hence from eq. (4),§23, - fxdF x--p- fxZydx~~~F~ VM F 2_ZF }&.x
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectenginee, bookyear1888
