CENTRE OF GRAVITY. J 



MECHANICAL PHILOSOPHY. STATICS. 



713 



hemispherical heavy metallic body, to a cylinder of 

 elder-pith, cork, or any other li^ht substance. The 

 centre of gravity of the whole, will lie somewhere in the 

 heavy hemisphere ; and if the cylinder be removed to 

 any position, except it be laid almost absolutely on its 

 side, it will speedily recover its upright position, as 

 shown in the figure. Toys are constructed on this 

 principle, the cylinder being cut into the form of a 

 soldier ; a regiment of such mimic troops being pressed 

 nearly to the ground by passing a stick over them, seem 

 immediately to spring up and recover their position as 

 if by magic. (Fig. 67). The toys called tumblers, made 

 Fig. 67. f plaster of Paris, with a hemisphe- 



rical bottom loaded with iron or 

 lead to bring the 

 centre of gravity to 

 the lowest position 



possible, also owe their amusing properties to the same 

 principle. Screens have been invented which, by 

 contrivance, right themselves after being pressed 

 down. The preceding illustrations show how a pointed 

 stick may be easily balanced on the tip of the finger 

 by fixing two pen-knives in its side, thus convert- 

 ing an unstable into stable equilibrium ; and how 

 three pen-knives, placed in the position A, B, C, and D, 

 may be kept in equilibrium on the point of a needle 

 held in the hand. In both cases the centre of gravity 

 must fall below the point on which the bodies are balanced. 



A small figure A (Fi.^. 08) with its foot fixed to a 

 sphere B, through which passes Fig. 



a bent wire, having two leaden 

 balls C and I) attached to its 

 extremities, if placed loosely on 

 a stand E, will speedily recover 

 its upright position after being 

 moved from it ; the figure being 

 so constructed, that the centre 

 of gravity of the three bodies 

 A, C, and D, falls below the 

 point of support ; that is, where 

 the sphere B rests on the stand 

 E. 



The apparent paradox of a 

 double cone ascending an in- 

 clined plane by its own weight, 

 is produced by the construction 

 of the cone and plane being such 

 that the centre of gravity of 

 the cone really descends, and by 

 its descent causes the cone to 

 ascend the plane. 



Construct an inclined plane of two eqnal pieces of 

 straight wire A C and B C (Fig. 692), fixed at their 

 extremities to three upright pieces A E, B F, and C! D, 

 standing perpendicular to the horizontal stand DBF: 

 A E and B F being equal to each other, and CD less 

 than A E or B F. 



Let H be a double cone consisting of two right cones 

 united together by their 

 circular bases. 



If the distance A B or 

 E F be equal to or less 

 than that between the two 

 vertices of the cone, and 

 the difference between A E 

 and C D less than the 

 radius of the circular base of the double cone, upon 

 the cone near C with its circular base lirtwwn 

 the wires A C and B C, it will roll up the inclined piano 



VOL. I. 



till its extremities are stopped by the upright supports 

 at A and B. 



Fig. 69-(2). 



That the centre of gravity really descends in this case 

 will be readily seen by the diagram (3), which represents 

 an imaginary section of the inclined plane through C D, 

 perpendicular to E F, represented by the bines G' O, O D, 

 CD,G'C(2). 



Fig. C9-(3). 



Let the circles K L M, K' L' M' (3) represeut the two 

 positions of the circular base of the cone at the com- 

 mencement and end of its ascent up the inclined plane ; 

 G and G', the centres of these circles, will represent the 

 position of its centre of gravity at these periods, and the 

 line G G' the path of the centre of gravity during tho 

 ascent ; since the centre of gravity of the double cone 

 is the centre of its circular base. By construction, the 

 angle G' O D' is a right angle. Through C' and G' draw 

 C' N and G' B parallel to O D', and the radius G M of 

 the circle K L M perpendicular to C' N. Then if M' be 

 the point where the circle K' L' M' cuts G O, G R or 

 N M' will evidently represent the vertical descent of the 

 centre of gravity of the cone, while the cone itself is 

 ascending the plane. And since G' N (3) is equal to tho 

 difference between the two supports A E and C D (2) of 

 the inclined plane, in order that the cone may ascend by 

 the descent of its centre of gravity, this difference must 

 be less than the radius G' M' (3) of the base of the 

 cone. 



PROPOSITION XXI 



A body plated on a plane horizontal surface will xtand or 

 fall according as the vertical line drawn through its 

 centre of gravity passes within or ivitliout its base. 



Let ABC (Fig. 70 1 and 2) represent sections of two 

 bodies, by vertical planes passing through their centre of 

 gravity G, having their bases A B placed in contact 

 with a horizontal plane. 



Fig. 70. 



1. 2- 



In (1) the vertical G W passing through G falls within 

 A IS, and in (2) without A B. 



The whole eH'ect of the weight of the body is equiva- 

 lent in both cases to a single force equal to that weight 

 applied at G in the direction G VV. 



4 Y 



