THE INERTIA OF THE CONNECTING ROD— MACGREGOR. 195 



'^crement of the kinetic energy of the rod. As the rod is rotating 

 about the point 0, instantaneously fixed, its kinetic energy is 

 ^ u 'mk-, where u is its angular velocity about 0, m its mass, 

 and k its radius of gyration about an axis through perpendicu- 

 lar to the plane of motion. Hence the equation of energy is : 



(P^-- S)dc = d(h co'mk'), 



s 

 or 



pH- S = ^-r* cJmk"). 

 s dc " 



In this equation P, 2? and s are known as pointed out above, 

 for all crank positions, i. e., for all values of c. The mass iJi is 

 known. The angular velocity u may easily be found ; for it is 

 equal to the linear velocity of the crank-pin divided by s, the 

 distance of the pin from ; and the angular velocity of the crank 

 being given, together with its length, the linear velocity of the 

 crank-pin may be obtained at once. The radius of gyration, k, 

 about is equal to the square root of the sum of the squares of 

 ±he radius of gyration, h, about a parallel axis through G, the 

 'Centre of mass of the rod, which is constant and may be calcu- 

 lated, the form and dimensions of the rod being given, and of the 

 'distance, d, of G from 0, which may be found for all crank 

 positions by measuring the length of the line GO in diagrams 

 similar to Fig. 1. All the variable quantities of the above equa- 

 ,tion except S may thus be expressed as functions of c. It is 

 'therefore sufficient for the determination of 8. 



Usually, however, the problem under consideration is presented 

 in this way : — By what amount is the component, normal to the 

 ^rank, of the effort on the crank-pin too great, when calculated on 

 the assumption that the connecting rod has no inertia ? Or in 

 other words, what pull normal to the crank must the crank-pin 

 exert on the rod, in order that the rod may move in the given 



way f 



The equation of energy modified so as to be a direct answer to 



this question takes a somewhat simpler form. For if S' lie the 



component normal to the crank of the eftbrt on the crank-pin, 



calculated on the assumption referred to, we have, putting ta = o, 



P(p/s)-S' = 0. 



