188 



THE POPULAR EDUCATOR. 



1. A body has one Centre of Gravity, and only one. 



2. The Centre of Gravity is not changed by the body being 

 turned round after any manner in any direction. 



It thus appears that the weights of all the separate atoms of 

 any mass of matter are equal to a single weight supposed to act 

 at some point within that mass, or, as sometimes happens (and 

 we shall see), even without, equal to their sum. There is great 



advantage in this simplification ; for, instead of having to con- 

 sider millions of diminutive forces acting at all its points, 

 we direct our attention to only one force, acting at only one 

 point. 



You can now understand how it is that a piece of card or thin 

 board may be supported on the point of a rod, wire, or needle. 

 All that is necessary is to bring the point under the centre of 

 gravity of the board ; then, the resultant of all the forces by 

 which its several parts are pulled downwards passing through 

 that centre, will be resisted by the rod, and there will be equili- 

 brium ; the card will be balanced. 



Another consequence follows. Let the body be of any shape, 

 regular or irregular ; and suppose that, having determined its 

 centre of gravity, we fix or support that point in some way so 

 that the body may freely turn round it, as on a pivot, in every 

 direction. Then, since, as I have shown, the centre of gravity 

 cannot change as the body turns round, whatever position I 

 place it in, the centre remains supported, and the resultant 

 weight, G P, passing through it, will be resisted by its supports, 

 and the body will be in equilibrium, as in Fig. 18, where G is the 

 supported centre of gravity. 



Now suppose that instead of this centre we make the body 

 pivot round some other one of its points, o (as in Fig. 19). Then, 

 if I place it so in the position o A B, that the centre of gra- 

 vity, G, may lie exactly under o, as a plumb-line would hang, 

 the weight acting along the line, o G, may be taken to have o 

 for its point of application, by which, as it is fixed, it will be 

 resisted. In such case there will be equilibrium, G being under 



E _ -KB; 



/ c--::MI 



\ 



Fig. 18. Fig. 19. 



O. And so, also, if G were exactly above o, as in o c D, in the 

 vertical line produced upwards, the weight would press down 

 wards on o, and be there resisted. But if I put it in any other 

 position, o E F, where G will not be either above or below o 

 the weight acting downwards, in the direction G P, will not be 

 opposite to the line o G of resistance of o, and there cannot be 

 equilibrium. There are thus two positions in which -the body 



Fig. 20. 



may be at rest, both on the vertical line through o ; but one in 

 ;he lowest position it can attain and the other in the higliest. 

 We thus learn that 



1. If a body be suspended by or supported at its centre of 

 gravity, it will be at rest, whatever be the position in which it is 

 jlaced. 



2. If the body be suspended by or supported at any other 

 point, it will be at rest when the 



centre of gravity is in its highest 

 lowest possible position on the 

 vertical line through the point of 

 suspension or support. 



If two points A, B (Fig. 20), are 

 fixed, all the points of the line A B are 

 fixed, but the body is free to turn 

 round that line ; and if in that case 

 the centre of gravity is somewhere 

 on A B, as G, it also is fixed, and the 

 weight there concentrated will be 



borne by the two points of support, A B, divided between 

 them in two portions inversely proportional to their distances, 

 A G, B G, from the centre of gravity. The body will, therefore, 

 be in equilibrium in every position into which it can be turned 

 round the line A B. But if, when two points are fixed, this 

 centre is not on the line A B, it is free to move round it. There 

 are, therefore, two positions, a lt G 2 , in a plane vertically passing 

 through this line one below, the other above, in which it may 

 rest, and the result is similar to that stated in the above pro- 

 positions. Familiar examples of this are furnished by all pieces 

 of machinery in which bodies move round fixed axles, such as 

 the fly-wheel of a steam-engine, or the smaller wheels round 

 which the bands pass, which set the printing presses at work in 

 the machine-room all the points along the line which runs down 

 the centre of the axle are at rest. A trap-door, which opens both 

 downwards and upwards, is another instance ; in that case the 

 centre of gravity is under or above the axle-line of the hinges 

 when the do'or hangs in equilibrium. 



But bodies may be kept in equilibrium in other ways than that 

 of fixing points within their substance. A horse poised in the 

 air, as it is about to be lifted into a transport ship, by a rope 

 which descends from the top of a crane and is attached to a belt 

 which goes round his body, is an instance. Here the centre 

 of gravity of the lifted animal is under the point of support 

 and on the line of direction of the rope which transmits its 

 weight to the crane above. But observe, in this case, there is 

 only one position of equilibrium namely, the lowest. The rope 

 not being rigid, you cannot wheel the horse half round, heels up 

 in the air (Fig. 21) until he reaches the highest position the 

 chain would allow him to reach, and make his weight thence 

 press downwards on the crane. To do this a rigid bar should 

 take the place of the rope. 



But bodies are most commonly kept at rest by being sup- 

 ported from below by the 

 earth, either on the 

 ground itself, or on some 

 floor, table, etc. What 

 conditions will secure 

 a steady equilibrium ? 

 First, there must be some 

 base or bottom to the 

 body on which it may 

 rest, such as the bottom 

 of a teapot or candle- 

 stick. Secondly, it must 

 be broad enough to keep 

 the body steady, to pre- 

 vent its upsetting or 

 rocking. A candlestick 

 resting on the socket 

 into which the candle is 

 put, would soon over- 



Fig. 21. 



turn, and the slightest touch would set an egg rocking. 



Now, in order to ascertain the equilibrium and stability of 

 bodies so placed, let us suppose two of the forms in Fig. 2! 

 to rest on a level table, touching it on the two perfectly fiat 

 basi* x Y z, x, Y, z,, there represented. Let G be the centre of 

 gravity of that to the right, and G P the perpendicular to the 

 table through that point. Let, moreover, GJ and G, ^ be the 



