On Crank Motion. 



61 



I will offer one other demonstration, which I do not re- 

 member to have seen published, and for which I am indebt- 

 ed to Mr. Hogan, a practical mechanician, and lecturer on 

 mechanics in Paris. 



B Suppose a force whose in- 

 tensity is represented by the 

 % line A.B. moving in the direc- 

 tion A.E. ; the whole quantity 

 of force expended in passing 

 from A to E may be represent- 

 ed by the surface of the paral- 

 lelogram ABEF=the intensi- 

 ty of the force X the distance 

 through which it acts. Sup- 

 pose farther, that the force or 

 power be applied to the ex- 

 tremity of a crank whose length 

 is A.C. and whose center is in 

 C. and that the crank has arri- 

 ved at any point c of its revolu- 

 tion : the force tending to ro- 

 tation, on the tangential force 

 in this point, according to the 

 laws of the resolution of forces, 

 J* will be represented by a c per- 

 pendicular to the line of force. 

 When the extremity of the crank shall have passed through 

 an infinitely small space, and have arrived in d, the arc c.d> 

 may be considered as a right line, the force tending to ro- 

 tation, in this point, will be b.d. and the quantity of force 

 tending to rotation during the passage from c to d will 



be= — — — xc.d.=gfx.cd ; the quantity of force applied 



2 

 during that passage will be A.B. x«6.=parallelogram abhi, 

 — and because of similar triangles. 

 gf:ce'.:Cf:cd whence 

 ce X Cf=gfx cd. 

 But c/=A.B. and ce=ab — therefore AB Xab=gfXcd, or 

 in plain English, in whatever point of its progress the effect 

 of the crank be considered, the power rendered is equal to 

 the power applied — this being true as regards all the parts, 

 must necessarily be true as regards the whole of its revolu- 

 tion. 



