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 . finding x we may putd V = volume of any lamina parallel to YZ, F being the baseof such a lamina, each point of the lamina having the same , (equations (2), § 23), but and = \fxdV, V=fdV=fFdx; F:F%::«?:h;., .:F=fix, h. q ~L 4 7 4 For a frustum, x — —. ~-% ~; while for a pyramid, hlt be- — 3ing = 0, x = -rh. Hence the centre of gravity of a pyramid is one fourth the altitude from the base. It also lies 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 . finding x we may putd V = volume of any lamina parallel to YZ, F being the baseof such a lamina, each point of the lamina having the same , (equations (2), § 23), but and = \fxdV, V=fdV=fFdx; F:F%::«?:h;., .:F=fix, h. q ~L 4 7 4 For a frustum, x — —. ~-% ~; while for a pyramid, hlt be- — 3ing = 0, x = -rh. Hence the centre of gravity of a pyramid is one fourth the altitude from the base. It also lies in the line - .ft™ --^K joining the vertex to the centre of gravity _/[ d 1 of the base. yv-J^T Problem 8.—If the heaviness of the ma- ^>>^\ x ~a Serial °f tne above cone or pyramid varied ~/y directly as x, y^ being its heaviness at the fig. 22. Dase F^ we would use equations (1), § 23, putting y = j3 x; and finally, for the frustum, _ 4 h;-h* x~ 5 • h: - v 4 and for a complete cone x = —hr 27. The Centrobaric Method.—If an elementary area dF berevolved about an axis in its plane, through an angle a < 2tt. PARALLEL FORCES AND THE CENTRE OF GRAVITY. 25. the distance from the axis being = x, the volume generated isd V = axdF, and the total volume generated by all the dFsof a finite plane figure whose plane con-tains the axis and which lies entirely on oneside of the axis, will be V = fdV =afxdF. But from §23, afxdF^ aFx;ax being the length of path described bythe centre of gravity of the plane figure,we may write : The volume of a solid of revolution generatedby a plane figure, lying on one side of the axis, equals thearea of the figure multiplied by the length of curve describedby the centre of gravity of the figure. A corresponding statement may be made for the surfacegenerated by the revolution of a line. The arc a must be ex-pressed in n measure in numerical work. 27a. Centre
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectenginee, bookyear1888
