A 



15Q0n the maximum of mechanic powers, and 



Prob. 15. Let ABC be a solid wheel of" uniform 

 thickness and density, revolv- 

 ing on its center S : and let its 

 weight be W, and if P be a 

 Av^eight applied as a power, sus- 

 pended to a line passing freely 

 over the wheel, and to which 

 lin,e is fixed the weight .v at 

 the opposite end. The value 

 of ,v is required, in case of a 

 maximum. 



B 



Q © 



Since the weight and power are equally distant 

 from the center of motion, P— .r will be the moving 

 power : and by Problem 8, i W is the // of the wheel. 

 Hence P-l-| W + ^ is the mass to be moved, as to 



angular velocity. Then will 



P + gW+x 



be the accelera- 



- the motive force of .r, whose 



tive force and -^. , 



fluxion being equal to nothing, we have V.v+^FWx 



—QFWx—Wdx-^v'-x^io and .r — y/ VV " -h 6' P VV -j- ^1^ 



_2P--W. 



Prob. l6. LetA^, B b, be two circular ends, 

 iixed to the beam a b, these 

 ends being of equal thickness 

 as well as the beam. Let the 

 weight of both the former to- 

 gether be \V', and that of the 

 latter iv: and let the beam 

 move on its center S. — Then if 

 P be a given weight, acting as 

 a power at B, it is required to determine the weight 

 .V suspended at the other end under the circum- 

 stances of a maximum. 



& 



Now if S B 1= II and sbzz r, then by Problem 10, 

 thep' of the beam and heads, reduced to B, will be 

 ^^, where h — — -^-^-. And since the beam 



and 



i w' 



