P and Q will balance in every position on C. And since the same is true of all such pairs of heavy bodies equidistant from C, the whole line will balance on C in every position, and therefore C is the center of gravity of the line. 46. PROP. XXI. gravity of a triangle. To find the center of Let ABC be a triangle formed of lines ranged, at equal distances, parallel to any one of the sides, the lines B b E G themselves being made up of Bisect AB in E and BC in F; join AF, CE by lines intersecting in G;-G is the center of gravity of the triangle. For Af fb AF : FB, (from the equiangular triangles, ABF, Abf), :: AF : FC, · BF = FC. :: Af: fc, (from the equiangular triangles, AFC, Afc). And since the first and third terms of this proportion are the same, fb = fc, and therefore the straight line be would balance in any position on f. Prop. xx. In the same manner all the other lines parallel to BC might be shewn to balance in any position on the points that AF cuts them in, therefore the whole triangle would balance on AF in any position, and therefore the center of gravity of the triangle is in AF. Similarly it may be shewn that the center of gravity of the triangle is in the line CE. Wherefore G, the intersection of AF and CE, is the center of gravity of the triangle ABC. 47. PROP. XXII. When a body is placed on a horizontal base, it will stand or fall, according as the vertical line drawn from its center of gravity falls within or without the base. Let GH be a vertical line drawn through the center of gravity of the body, meeting the horizontal plane on which the body is placed in H; and let ABH be a horizontal line Al B H A B H drawn through H and terminated at A and B, points in the boundary of the base of the body. (1) Let H fall between A and B. Then AB being considered as a horizontal lever acted on at H by the weight of the body, no motion can take place round A, because the tendency of the weight would be to draw the lever downwards round A, a motion which the resistance of the plane will prevent taking place. For the same reason no motion can take place round B, and therefore the body will remain at rest, if the vertical drawn through the center of gravity fall within the base. 47 YDROSTATICS. planatory articles and paragraphs that are ents form no part of the Course required to be idate for the degree of B.A. Those in smaller ive of the Definitions and Propositions which they low. The whole of the extra matter so introit a very small, and a very easy, portion of the who come to this subject for the first time will omit it. CHAPTER I. HYDROSTATICS investigates the conditions by the pressures produced by fluids when ate of rest. DEF. A FLUID is a material body which e divided in any direction, and the parts of which be moved among one another by any force howsmall. [Water, Air, Gas, Mercury, Steam, are all inances of fluids.] FLUIDS have been divided into Elastic, and (1) ELASTIC FLUIDS are those of which the dimensions are increased or diminished when the pressure pon them is increased or diminished. (2) If H fall without AB, as in the second figure, no motion of the lever ABH can take place round as fulcrum, because the effect of the weight acting at H would be to draw AH downwards, a motion which the plane prevents. But if B be considered as the fulcrum, the effect of the weight at H would be to make the line AB to move upwards from the plane round B, and as there is no force to prevent this motion taking place, the body will fall over at B. Wherefore, "When a body, &c." Q. E. D. 48. PROP. 23. When a body is suspended from a point [round which it can swing freely], it will rest with its center of gravity in the vertical line passing through the point of suspension. N Let AB be the body in any position, C the point from which it is suspended, G its center of gravity, NGK a vertical line drawn through G, CN a perpendicular from Con NGK. Then since the weight of the body acts in the vertical through G its center of gravity, it may be supposed to be B applied at N; and if CN be considered as a lever moveable round C as fulcrum, since there is no force to counteract W acting perpendicularly at N on CN, motion must ensue. But if NG pass through C, CN vanishes, the weight is sustained by the immoveable fulcrum C, and the body is at rest. Wherefore "When a body, &c." Q. E. D. |