Centre of Gravity of a Triangle. "Centres of gravity of bodies of uniform density." 105 197. It is often useful to know the centre of figure, which is also the centre of gravity in uniform bodies of that figure; we therefore give a few of the more common cases which may be determined either by pure calculation or by trial. The centre of gravity (or figure) of a uniform straight rod is at its middle point; of a circular area or ring is at the centre of the circumference; of a rectangle or of any parallelogram is at the crossing of the two diagonals; of a ball or sphere, hollow or solid, is at its centre; of any parallelopiped (or box-shaped figure) is at the common crossing of its three diagonals; of any cylinder is at the middle point of its axis; of an oval or ellipse, at the crossing of its two axes, or greatest and least diameters. All these figures are evidently symmetrical about the centre of figure, that is, any straight line passing through the centre of the figure is divided into two equal parts there; and consequently, an equal number of molecules lying on each side of that centre in every direction round about, the whole must balance about that point. 198. Two masses, A and B, at the ends of a rod, will balance about a point midway between them if they are equal; and if в be greater than A, then the centre of gravity, or point about which they balance, will lie as much nearer B as B is greater than A. E 199. A triangular or three-cornered plate, such as A B C (fig. 20), will balance about a line, A D, passing through one corner and the middle point of the opposite side, because just half of the plate lies symmetrically on each side of that line. Thus the centre of gravity lies somewhere in A D; it also lies in B E for a like reason. But it can only lie in both lines if it be at o, their common point of crossing. A little geometry shows that this B point is situated so that O D is one-half of O A, or one-third of A D. D Fig. 20. It is to be noted that the centre of gravity of three equal masses placed at the corners of the triangle would be the same point. Hence the placing of three equal weights or masses at the corners of a triangular plate already balanced, will not alter its balance. It is to be noted, however, that the centre of gravity of three rods forming a triangle, differs from that of the triangular area which 106 The Pyramid-Cone-Sphere. they contain, unless in the particular instance where the rods are all equal. Calculation shows that they balance about the centre of the circle, which can be inscribed within the triangle formed by joining the middle points of the rods. 200. A pyramid is a solid bounded by triangular faces, meeting in a point, such as is represented by A B C D, or by M N O P Q R S, in the figure, and such as would be enclosed by sheets of paper cut like A B C D B' and M N O P Q R S N' in fig. 21, and folded along the lines drawn from A to the different corners. As each of these triangular faces has its centre of gravity at a distance from A, two-thirds of the vertical height of A (art. 199), it is clear that the centre of gravity of the whole, when folded together so as to form the surface of a pyramid, will be vertically distant from A, two-thirds of the vertical height of A above the common base. Since a cone is but a kind of pyramid with an infinite number of sides, just as a circle is a polygon with an infinite number of sides, the centre of gravity of a conical surface will, in like manner, lie in its axis at a distance from its summit or vertex two-thirds of its whole vertical height. 201. The centre of gravity of a solid pyramid or cone lies in its axis, or line drawn from the vertex to the centre of gravity of the base-area, and at a distance from the vertex three-fourths of the whole axial height. The Centre of Gravity falls as low as possible. 107 "The centre of gravity seeks the lowest position." 202. Since the conjoint weight of a body acts as if concentrated at its centre of gravity, and the tendency of each particle is downwards, it is clearly the same thing to say that the centre of gravity of a body tends always to seek the lowest position possible under the circumstances. In a body hanging freely from a point, this position is, of course, vertically below the point of support; and if the body be moved from this position, its centre of gravity is raised, and will tend to fall back to it again. The following cases, which appear at first to be exceptions to the law, are really interesting proofs of it. A wooden cylinder or roller, e d c (fig. 22), placed on a slope, a b, will roll down, because its centre of gravity is thereby approaching the earth; but if there be a heavy mass of lead, c, introduced at one side, and if the roller be placed on the slope with the lead in the high position d, the lead, in falling down to the position c, will move the roller towards ¿, the apparent rolling up-hill being in truth a falling of the centre of gravity of the cylinder. a Fig. 22. If a billiard-ball, c (fig. 23), be placed upon the smaller ends of two cues, a b and c d, laid on a table with their points, c and a, together, but with the larger ends, b and d, so far apart that a Fig. 23. server, to be rising when really its centre is descending in obedience to gravity. A double cone, such as f, would similarly roll from c to e, and with still more of the fallacious appearance of rolling upwards, because its sides would always be resting on the upper and rising surfaces of the cues. 203. The beam or rod, c d (fig. 24), resting on the edge of the table, a b, would naturally fall if left to itself, because more than half of it is beyond the edge of the table; but, strange to say, an additional weight, e, attaching to its projecting part, as at b, by the cord, 108 Effect of Position of Centre of Gravity. Fig. 24. be, instead of pulling it down faster, will fix or steady it on the table, provided the weight be pushed inwards a little by a rod, de, resting against it and against a niche in the rod at d. It is evident that the rod, cd, in falling, must turn round the edge of the table at b; but in so doing, after the arrangement here - supposed, it must lift the weight, e, along the path, ef-which rise, as the weight is heavier than the rod (that is to say, as the common centre of gravity of the connected objects is near e), gravity prevents, and therefore the rod and weight will both remain supported by the table. An umbrella or walking cane, hanging on the edge of a table by a crooked handle, is an instance of the same kind. And the common toy of a little man standing on tiptoe upon the top of a pillar, and carrying in his hands two balls at the ends of a wire, is simply a combination of parts which places the centre of gravity of the whole below the support, in fact, a kind of pendulum. 204. The following experiment will serve as a striking illustration of this principle. Plunge into a cork, at the end, a (fig. 25), the prongs of two short forks of equal weight, making an angle with the cork of about 45°. If the forks are properly adjusted, the extreme edge of the cork, b, will rest firmly and be supported in a horizontal position on the edge of the wine glass, c. The dotted line, de, shows that there is an equal quantity of matter on each side of it, and the centre of gravity is right over b. The cork may be equally supported horizontally on the point of a pin. If the forks be brought a little more forward, the cork will rise, but will still be supported; if carried backward, it will remain supported, but sloping downwards. This, of course, depends on the change made in the position of the centre of gravity. Fig. 25. Conditions that a Body may stand or fall. 205. Attention must be paid to the form or position of a body when it is desired that it should not be readily pushed over or upset. 109 If, from its form or situation, a body cannot be overturned without its centre of gravity being lifted, its state of equilibrium or rest is said to be stable, that is, not easily disturbed, because, on being left to itself, it will by its gravity return to its old position. The rise of the centre of gravity in overturning, will depend on the breadth of the base compared with the height of the centre of gravity above the base. This is shown in the annexed drawings (fig. 26), which exhibit a series of combinations of base and height. The dot, c, marks the centre of gravity, and the curved line beginning from the dot marks the path of that centre, when the body is being overturned. This path is of course a portion of a circle which has the end of the base for a centre. The further inwards, therefore, that the centre of gravity is, horizontally, from the end of the base, the further of course is it from the top of the circle which it has to describe in moving, the steeper will be its commencing path, and the greater, consequently, will be the disturbance required to overturn the body. In the body A (fig. 26), which has a broad base with the centre of gravity low, this centre must rise almost perpendicularly before it can fall over, and the resistance to overturning is therefore nearly the whole weight of the body. Hence the firmness of a pyramid. In B, C, and D, the path of the centre begins less and less steep as the base is narrower, and hence they are so much the less stable. E is in a tottering position; for, the centre of gravity being directly over a base which is a mere point, the least inclination to either side places it on a descending slope, and the body must fall. In F (fig. 27) the position is tottering on one side, and stable on |