CHAPTER III. CENTRE OF GRAVITY AND STABILITY. Weight. 50. That force by virtue of which a body, when abaudoned to itself, falls towards the earth, is called the force of gravity. The force of gravity acts upon every particle of a body, and, if resisted, gives rise to a pressure; this pressure is called the weight of the particle. The resultant weight of all the particles of a body is called the weight of the body. The weights of the particles are sensibly directed towards the centre of the earth; but this point being nearly 4,000 miles from the surface, we may, for all practical purposes, regard these weights as parallel forces; hence, the weight of a body acts in the same direction as the weights of its elementary particles, and is equal to their sum. Centre of Gravity. 51. The centre of gravity of a body is the point of application of its weight. The weight being the resultant of a system of parallel forces, the centre of gravity is a centre of parallel forces, and so long as the relative position of the particles remains unchanged, this point will retain a fixed position in the body, and this independently of any parti cular position of the body (Art. 43). The position of the centre of gravity is entirely independent of the value of the force of gravity, provided that we regard this force as constant throughout the dimensions of the body, which we may do in all practical cases. Hence, the centre of gravity is the same for the same body, wherever it may be situated. The determination of the centre of gravity is, then, reduced to the determination of the centre of a system of parallel forces. Equations (13) are, therefore, immediately appli plicable. Preliminary discussion. 52. Let there be any number of weights applied at points of a straight line. We may take the axis of X to coincide with this line, and because the points of application of the weights are on this line, we shall have, y = 0, y' = 0, &c.; z = 0, z' = 0, &c.; substituting these in the second and third of Equations (13), we have, Уг = 0, and 21 = 0. Hence, the point of application of the resultant is on the given line. In the case of a material straight line, that is, of a line made up of material points, the weight of each point will be applied at that point, and from what has just been shown, the point of application of the resultant weight will also be on the line; but this point is the centre of gravity of the line. Hence, the centre of gravity of a material straight line is situated somewhere on the line. Let weights be applied at points of a given plane. We may take the plane XY to coincide with this plane, and in this case we shall have, 2 = 0, 2′ = 0, &c.; these in the third of Equations (13) will give, hence, the point of application of the resultant weights is in the plane. It may be shown, as before, that the centre of gravity of a material plane curve, or of a material plane area, is in the plane of the curve, or area. If the bodies considered are homogeneous in structure, the weights of any elementary portions are proportional to their volumes, and the problem for finding the centre of gravity is reduced to that for finding the centre of figure. In what follows, lines and surfaces will be considered as made up of material points, and all the volumes considered will be regarded as homogeneous unless the contrary is stated. Centre of Gravity of a straight line. 53. Let there be two material points M and N, equal in weight, and firmly connected by an inflexible line MN. The resultant of these weights will bisect the line MN in S (Art. 40); hence S is the centre of gravity of the two points M and N. M N Fig. 31. Let MN be a material straight line, and S its middle point. We may regard it as composed of heavy material points A, A'; B, B', &c., equal in weight, and so disposed that for each point on one side of S, there is another point on the other side of it and equally distant from it. From what precedes, the AB S B'A N Fig. 32. centre of gravity of each pair of points is at S, and consequently the centre of gravity of the whole line is at S. That is, the centre of gravity of a straight line is at its middle point. P B Centre of Gravity of symmetrical lines and areas. 54. Let APBQ be a plane curve, and AB a diameter, that is, a line which bisects a system of parallel chords; let PQ be one of the chords bisected. The centre of gravity of the chord PQ will be upon AB, and in like manner, the centre of gravity of any pair of points lying at the extremity of one of the parallel chords will be found upon the diam Fig. 33. eter; hence, the centre of gravity of the entire curve is upon the diameter (Art. 52). The entire area of the curve is made up of the system of parallel chords bisected, and since the centre of gravity of each chord is upon the diameter, it follows that the centre of gravity of the area is upon the diameter. Hence, if any curve, or area, has a diameter, the centre of gravity of the curve, or area, lies upon that diameter. If a curve or area has two diameters, the centre of gravity will be found at their point of intersection. Hence, in the circle and ellipse the centre of gravity is at the centre of the curve. If a surface has a diametral plane, that is, a plane which bisects a system of parallel chords terminating in the surface, then will the centre of gravity of the extremities of each chord lie in the diametral plane, and consequently, the centre of gravity of the surface will be in that plane. The centre of gravity of the volume bounded by such a surface, for like reason, lies in the diametral plane. Hence, if a surface, or volume, has a diametral plane, the centre of gravity of the surface, or volume, lies in that plane. If a surface, or volume, has three diametral planes intersecting each other in a point, that point is the centre of gravity. Hence, the centre of gravity of the sphere and the ellipsoid lie at their centres. We see, also, that the centre of gravity of a surface, or volume, of revolution lies in the axis of revolution. Centre of Gravity of a Triangle. Ꭺ 55. Let ABC be any plane triangle. Join the vertex A with the middle point D of the opposite side BC; then will AD bisect all of the lines drawn in the triangle parallel to the base BC; hence, the centre of gravity of the triangle lies upon AD (Art. 54); for a like reason, the centre of gravity of the triangle lies upon the line BE, drawn from D Fig. 34. the vertex B to the middle point of the opposite side AC; it is, therefore, at G, their point of intersection. Draw ED; then, since ED bisects AC and BC, it is parallel to AB, and the triangles EGD and AGB are similar. The from D to A. D A Fig. 34. Hence, the centre of gravity of a plane triangle is on a line drawn from the vertex to the middle point of the base, and at one-third of the distance from the base to the vertex. Centre of Gravity of a Parallelogram. 0 D E C 56. Let AC be any parallelogram. Draw EF bisecting the sides AB and CD; it will also bisect all lines of the parallelogram parallel to these sides; hence, the centre of gravity lies on it; draw also the line OH bisecting the sides AD and BC; for a similar reason, the centre of gravity lies on it: it is, therefore, at G their point of intersection. A F Fig. 35. Hence, the centre of gravity of a parallelogram lies at the point of intersection of two straight lines joining the middle points of the opposite sides. It is to be remarked, that this point coincides with the point of intersection of the diagonals of the parallelogram. Centre of Gravity of a Trapezoid. Λ B 57. Let AC be a trapezoid. Join the middle points, O and P, of the parallel sides, by a straight line; this line will bisect all lines parallel to AB and DC; hence, it must contain the centre of gravity. Draw the diagonal BD, dividing the trapezoid into two triangles. Draw also the lines DO and BP; take P Fig. 86. |