13 



MECHANICS. 



sufficient to consider a single force 

 equal to the sum of all these separate 

 attractions, drawing the centre of gravity 

 in a line perpendicular to an horizontal 

 plane, which line is called the line of 

 direction. In this line the centre of 

 gravity will always either move, or en- 

 deavour to move, and it tvill always 

 assume the lowest position which the 

 circumstances under which the body is 

 Situated will admit of. 



(44.) If by the application of external 

 force a body be so adjusted, that the cen- 

 tre of gravity be placed in the highest 

 position which the circumstances under 

 which the body is situated permit, the 

 body will remain at rest so long as it be 

 perfectly undisturbed ; but, as in the 

 case of suspension already mentioned, 

 the slightest disturbance will cause the 

 centre of gravity to descend to the low- 

 est position. Of these two positions in 

 which it is possible for the body to rest, 

 the former is called instable t and the 

 latter stable, equilibrium. 



(45.) If a body be placed upon a 

 plane, its stability will be determined by 

 the position of the line of direction with 

 respect to the base. 



Let A B C D (fig. 14.) be a body rest- 

 ing on the level plane L M. Let O be 



fig' 14. 



its centre of gravity, and O N the line 

 of direction falling within the base A D. 

 Since the whole force which gravity 

 exerts upon the body may be conceived 

 to be applied at O, in the direction 

 O N, that force will be supported or 

 resisted by the plane L M, and the body 

 will stand firm. 



But if the line of direction O N (fig. 1 5 .) 

 fall without the base A D, the case will 



fig. 



be different. The force of gravity will 

 now act upon the body, so as to make it 

 turn over the edge D, and fall upon the 



side D C. " For draw O D, and from N 

 draw N m perpendicular to D O, and 

 complete the parallelogram N m O n. 

 The diagonal N O being taken to repre- 

 sent the whole force of gravity, it may be 

 resolved (9.) into two forces, represented 

 by O m and O n. The force O m is re- 

 sisted by the plane L M at D, and the 

 force O n evidently tends to turn the 

 body round the point D, and to make it 

 fall upon the plane towards the point M. 

 If the line of direction fall upon the 

 edge D of the base, the body is in a 

 state of instable equilibrium. For let 

 O D (fig. 16.) be perpendicular to the 

 plane L M, and with D as centre, and 



the distance or radius D O, describe a 

 quadrant of a circle. It is evident that 

 if the point O be moved towards M, it 

 will move through a part of the circular 

 arc O E, and every part of this arc is 

 nearer to D M than O, and, therefore, 

 the point O must descend. The slightest 

 disturbance in this case will make the 

 body fall towards M. 



(4 6 .) In general, the higher the centre 

 of gravity of a body is, compared with the 

 extent of its base, the more easily will 

 it be overturned. This will be easily 

 explained. Let A B C D, (fig. 17.) be 



fig- 17. 



a body resting on the horizontal plane 

 L M, and on the base A D. Let O be 

 the centre of gravity, and O N the line 

 of direction. Draw O D, and with D as 

 centre, and the radius DO, let the cir- 

 cular arch O F be described. In order 

 that the body A B C D should turn over 

 the edge D, it will be necessary that the 

 edge A be lifted off the plane L M to 

 such an height, that the point O shall 

 be raised through the arch O E, beyond 

 the point E. 



Now let us suppose that, by placing a 

 load G K, (fig. 18.) over A B C D, the 

 centre of gravity be raised from O to O', 

 it will be only necessary to raise the 



