152 



THE POPULAR EDUCATOR. 



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 resistsd 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 o, 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, o being under 



Fig. 18. 



Fig. 19. 



O. And so, also, if o wore exactly p.^ove o, as in o c D, in the 

 vertical lino produced upwards, the weight would press down- 

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

 position, O E F, where a will not bo cither above or below o, 

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

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

 equilibrium. There are thus two positions in which the body 

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

 the lowest position it can attain and the other in the highest. 

 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 

 placed. 



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 

 or 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 



Fig. 20. 



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 door 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 roj 

 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 itt* 

 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- 

 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. 22 

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

 bases 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, G! and G, p, b-; the 

 corresponding centre and perpendicular of the body to the left. 

 Now, since the table, by its resistance distributed equally over 

 the base X Y z of the first body, prevents its moving down- 

 wards, and this resistance at every point is perpendicular to- 

 the floor, these resistances, taken together, are a system of 

 parallel forces, and have a parallel centre somewhere in that 

 base. Let this centre be o. Join now o p ; and, as the same 



reasoning holds good of 

 the body to the left, let 

 O, P, be the corresponding 

 line in it. Moreover, let 

 x, Xj, be the points in 

 which the lines O p, o, P, 

 cut the circumference, or 

 boundary of tho bases- 

 x Y z, x, Y, z,. The body 

 to the right is thus acted 

 on by two forces ; the re- 

 sistance at o upwards 



supporting it, and the weight at G pulling downwards. But, 

 as the point p falls, in this case, outside tho base x Y z, there 

 is nothing to prevent the body obeying it by turning over on 

 its edge at x. 



But, in the other case, where PJ is within tho base, the weight 

 at G, tends to make the body fall inwards, turning on its edge 



Fig. 21. 



