B 



THE EFFECTS OP MACHINE'S WHEN IN MOTION. 147 



Let W express the weight C 



or mass of theannuhis; and zv 

 that of the cross beam a b, 

 which beam is of equal thick- 

 ness. Then the p' of the beam A 

 at ^ is 3 -a: by Prob. 6, Cor. 3 ; 

 which reduced to B, is -^, by 

 Cor. to Prob. 7, and the p' of 

 the ring is ^fl^^xlW^ by the last Prob. Henc^ 



/ of the whole is -—^ x\ +"^- 



Cor. If W express the weight of the tvvo ends 

 Aa, B^, being parts of the 

 annulus, whose center is 



S, and if the weight of * I ^ ._.g ~t^ .\ g 



the beam ah, whose center iJT' LJ 



is also S, be expressed by 



w, as before; then the jf>' 



of the whole beam, and both ends together, will be 



1^' 



ri — ^ a ' 



Prob. 1 1. When the two circular ends are braced 

 to the beam ab by the braces cd, cd, on both sides 

 of the beam : it is required to determine the // of 

 the whole, acting at B, when moving on the center 

 S. 



d--....- 



Let W and w repre- 

 sent the weight of the two 

 circular ends, and the 

 beam ab, respectively, as 

 in the last Prob. ; and let 

 the length c d, be ^', and the weight of the two braces 

 at one end, be zv\ Now if s be supposed the center 

 of rotation, then the case in Prob. 9 would apply. 

 And because sc varies so little from se or S«, in 2^ 

 beam of considerable length, that any deviation 

 from the truth which might arise from considering 

 sc cis r, would be so trifling as to render any further 



K 2 i^ivestigation 



