DISASTROUS EFFECTS OF COLLISION ON RAILWAYS. 



309 



I will now give the mathematical details to which allusion has been 

 made. 



It will be as well to begin with the very simple case of a body 

 whose centre of gravity is G, sup- 

 ported on a perfectly smooth hori- 

 zontal plane PQ by two props BD, 

 CE, and urged by a horizontal 

 impulsive force F at the point A. 

 Draw GHKL vertical. 



Let GH = h, 

 DL = a, 

 GL = b, 



M the mass, 



k its radius of gyration about G, 



V the horizontal velocity communicated to D, 



a the angular velocity about D resulting from the impact. 



Since the velocities communicated to G are 

 aa vertically upwards, and 

 V ' — ba horizontally forwards; 

 we have by the usual principles of motion, 



M(F-ba) = F, 



Maa = R, 

 Mtfa = Fh - Ra, 

 R denoting the vertical reaction at D. 



Hence M(a 2 + k*)a = Fh, 



Fh 

 a M (a 2 + k 2 ) ' 

 ._ F . F a* + k 2 + bh 

 r= M + ba= M a~V¥ 



Consequently, if h is nothing, no angular motion will result from 

 the shock; this is the case when the horizontal impulse F is directed 

 through the centre of gravity. 



Vol. VII. Part III. L l 



