THE EFFECTS OF MACHINES WHEN IN MOTION. 149 



Since W expresses the weight of the beam, 3- W 

 will express the p of the whole beam acting at B. 

 And since both ends are of equal length, P — .r, will 

 be the moving power, and P+^-W+.r is the mass 

 to be moved, with respect to angular velocity. 



Plence — :^^, — is the accelerativ^e, and is as the ve- 



locitjawith which P will move after having over- 

 come the resistance. But since ACzzCB, this quan- 

 tity is also the accelerative force of .v, and therefore 



the momentum of .v is TT", — , whichbeino-amaxi- 

 p+^w+x ' _^ 



mum, its fluxion is equal nothing : hence Jt'x—^^.vx 



X i" + i VV 4- ^i- — >r X Fi — ^i ^' — 0, 1 io^^ which, when 



reduced, wc have .v = v/^ +9PU i p — w— 3? ^ 



Prob. 13. Let the anus of the beam AB be of 

 unequal lengths, and let the whole beam be to 

 the shorter end, both in length and weight, as }i to 

 unity. And let W express the weight of the whole 

 beam. Then if P as a power be suspended at B, it 

 is required to determine the weight .v; so that it 

 may ascend, when overcome by P, with the greatest 

 momentum possible. 



Then by Problem 6, Cor. 2, A r -C ''"^ 



the p of the whole beam is equal ^_^^ 



^=^XW=g\V, by putting @ 



g— " ~f "T^ and the weio-ht of 

 the shorter end will be — , that of the lon^'cr "~^-'^ 

 by the same Cor. Now the weight of the longer arm 

 being —"7^, its weight when reduced to B will be 

 "~l' '"^ , and by the same reasoning the weight of the 

 shorter end AC, reduced to A, will be—: and as 



n-\ (BC) : 1 (AC) : :^:-^^zz the M'eight of 

 AC reduced to B. Ag-ain ; as yz-l : 1 : : .v : — ^^ 



o ' n — I 



r: the weio-ht of .r reduced to B. Hence- 



O ' 27J.n — I n — I 



is that weight, Avhich if applied at B, would precisely 

 balance the end AC, together with the weight ^'. 



. K 3 Hence 



