THE EFFECTS OF MACHINES V/HElSr IN MOTION. 155 



Then f W is the p' of the wheel at B, to the 

 center S, (for we shall suppose the part of the axle 

 which passes through the wheel to be of the same 

 density with the wheel;) and \ zv the/ of the axle 

 at C, and which, reduced to B, will be -^. Hence 

 ^^^^^^— is the / of the wheel and ai^le together, at 

 B. Then will P+^.2I±±:''-h^ exnress the mass after 

 beino- in motion : and P— ^ as in the former case, 



being the moving power, by putting ^=:;:P -I ;f~~~i 



and proceeding as in the former case, we shall have ^ 

 —■^\/f-\-tV— ^ ■ t ; or by restoring the value of f, 



2^ _^ — k/ • '— 1 „ . • 



a ^ lab 2aft 



Scholium. These problems comprehend all the 

 cases that can be of general use in combining the 

 lever with the wheel and axle ; or in their separate 

 application, v/hcn the power is acted on by gravity, 

 and M'hose motion is uniformly accelerated, the same 

 as that of bodies falling freely through any given 

 space. And since, in the preceding Problems, gra- 

 vity, or the space which a body falls freely through 

 in the first second of time, is considered as unity, 

 it follows that the accelerative force of .r in all these 

 cases being multiplied by 15-j'^ feet, (or Avhat may 

 be the measure in any particular latitude,) will give 

 the space in feet that .r would pass through in the 

 first second of time, and from which the space 

 which would be passed over in any other time may 

 t)e computed, since those spaces are as the squares 

 of the times in which they would be passed over 

 from the beginning. It is also easy to compute the 

 velocity of x after passing through a given space 

 in any given time, for that velocity will be in the 

 siibdupUcate ratio of the accelerative force : and 

 hence another maximum may be determined, viz. 

 the greatest possible effect of .r, after passing 

 throngh a g'rcm space. For if the square root of 

 the accelerative force be multiplied by.r, the product 

 will be m the momentum of x for any space passed 

 <2 over. 



