THE ETflCTS OF MACHINES WHEM IN MOTtOK. 14! 



AS*=:i;^+^^4-2fi^; and therefore 2V^'i'-\-id^'v-\-2ad v 

 will be the fluxio n of all the/id^ in th e wh ole beam ABD- 



,_ ^\i.2'V''i; + 2d'-'v+2a^v ZZ >/ v* + 3'?^ + 6<t^ 



Hence w = >/ ^ 1 



which, when a vaniihes, and the be am coincides with 



the line AB, becomes equal a/ '^ ^ -; — and if J vanifli 



w=v v^T, for then D will coincide with S and A D B 

 will become two beams revolving on their extremities. 



Prob. 4. Let ABC re- a 



prefent a circular fuperfices, 

 or folid W'heel of uniform 

 thicknefs, fo that its weight 

 may be as its area ; and Jet 

 it revolve round its center S; 

 it is required to determine 

 the diftance zv of its center 

 of gyration from S. 



Put Ar= the area of the -^ 

 circle whofe diameter is unity, 

 of ABC. Then 4 Ar* is the area of ABC, whofe 

 fluxion is 8 A r' /• ; and therefore 8 A r' r is 

 the fluxion of all the pd^ in ABC. Hence w^: 



■^zzrvh which exprefTion applies to every 



and rzn radius 



flu. 8 



^flu. 8 A r r 



folid wheel of uniform thicknefs whofe radius is r. 



Prob. 5. Let ABC and 

 ebcht two concentric circles 

 whofe refpedive radii are 

 R,r ; — if the plane or folid 

 wheel whofe area is a b c 

 be taken away, and the re- 

 maining plane or folid Aa 

 B^C^, uniformly thick, be 

 conceived to revolve round 

 the center S ; it is required to determine the diftan.ce of 

 its center of gyration from S. 



Put a— the area of the circle whofe radius is unity, 

 then 4 A R= will be the area of the greater circle, and 



then 



