6 



MECHANICS, 



A' x Ca+B*xCft+ D' x C rf = 



E' x C^ + F' x C/ + G-' x Cg ; that 

 is, that the sum of the efforts to turn the 

 board round in one direction, is equal to 

 the sum of the efforts to turn it round 

 in the other direction. 



(11.) We have hitherto neglected to 

 consider the weight of the machine itself, 

 the axis being always supposed to pass 

 through the centre of gravity. If this 

 be not the case, we have only to con- 

 skier the weight of the machine itself as 

 one of the weights or forces which are 

 applied to it, and that this force is ap- 

 plied in a vertical direction at the centre 

 of gravity. Thus, for example, in the 

 last experiment, let us suppose the 

 weights A', B', D', E', G'tobe the forces 

 which act upon the board. Let F be 

 the centre of gravity of the board, and 

 let F' be its weight acting in the vertical 

 line F F' passing through F. The former 

 investigation will remain unchanged, the 

 only difference being that the weight F' 

 is now that of the board conceived to be 

 concentrated at its centre of gravity F. 



(12.) It is scarcely necessary to say, 

 that if the sum of the moments of the 

 forces which tend to turn the body 

 round in one direction, be greater than 

 the sum of the moments of the forces 

 which tend to turn it in the opposite 

 direction, the body will move round its 

 centre in the direction Of the former. 



CHAPTER.HI . Straight Levers three 

 kinds Lever bearing severalWeights 

 Beam bearing a Weight and resting 

 on two Props Pressure on the Ful- 

 crum of a Lever Load borne on Poles. 



(13.) A LEVER considered as a bar or 

 rod, supported on a prop or fulcrum, is 

 of three kinds, according to the position 

 of the power and weight with respect to 

 the prop. 



If the prop be in the middle, the 

 lever is said to be of the first kind; if 

 the weight be in the middle, it is of the 

 second hind ; and if the power be in the 

 middle, it is a lever of the third kind. 



(14.) A lever of the first kind is re- 

 presented in fig. 4. If we neglect in 



trical reasoning or algebraical notation. The stu- 

 dent, however, will find his progress most materially 

 facilitated by the acquisition even of a very small 

 portion of the first elements of 'n-o.netry ani Alge- 

 bra. For students who only seek this limited know- 

 ledge of these sciences, there are perhaps no treatises 

 WbuJbjBan 1)3 read with more a 1 vantage than DAK- 

 ometry and ^Algebra. 



the first instance, the weight of the bar 

 itself, or suppose the centre of gravity 

 to be placed immediately over the prop 

 G, the efforts of the power and weight 

 to turn the lever in opposite directions 

 are P x p and W x w, p and w being 

 C G and B G ; and in order that equili- 

 brium should subsist, these must be 

 equal (7.) that is P x p = W x w. 



A lever of the second kind is repre- 

 sented in fig. 5 ; and one of the third kind 



Fig. 5. 



C 



ili'' 1 . ,i!,.:t 



in fig. 6. What" we ' have just ob- 

 served respecting the power and weight 



Fig. 6. 







in levers of the first kind, also applies 

 to those of the second and third kinds. 



(15.) The condition of equilibrium in 

 the straight lever being that the product 

 P x p should be equal to W x iv, it 

 follows, that the power P may be di- 

 minished indefinitely by increasing its 

 distancejt? from the prop indefinitely, 

 for what the magnitude of the product 

 P x p loses by the diminution of P, it 

 will gain by the increase of p. 



There is another way in which the 

 power which supports a given weight 

 by means of a lever may be indefinitely 

 diminished, and yet its distance from 

 the prop may be preserved. Let the 

 distance w of the weight from the prop 

 be diminished until the product W x w 

 becomes equal to P x p. 



