Equilibrium between hiving and Dead Forces. 243 



Fig. 1. 



A© w 



The question now is narrowed down to the con- 

 sideration of two forces, producing the same velocity 

 in the same amount of matter, w^ the one force grav- 

 ity =2^,* acting through the height AB = a, the other 

 force, W, acting through the distance BC=a;. 



Let us assume W=aw. Taking seconds and feet 

 as units of time and space, and putting t for time, s 

 for space, and g for the velocity in feet produced in 

 one second by the force of gravity, the relations are — 



For the force of gravity w <x.g=Z2>, 



t. s. V. ^ ~h 



I" ^g ^ X 



2" 2g 



&c. &c. 



For the force W=aic ccag=22,a^ 



t. s. 



1" ^ag 



%" 2ag 



&C. &LC. 



Now the velocity generated by gravity on the body w, falling 



the height a, will be {2gaf -, and the velocity produced on the 

 same body w, by the force 'W=a2v, (equal the friction,) acting 



through the space a:, will be (2«^.r)'-'. These velocities being 

 equal, we have — 



{2agx)^=^{2ga)^- 

 Substituting for ag and g their proportionals W and w, we 

 have — 



g 

 2g 



&c. 



ag 



2ag 

 &c. 



W 



(2Wx)^={2wa)^ 



Hence, 



W= 



aw 



(A) 



That is, W is the pressure or dead weight which will hold in 

 equilibrio the percussion of the ram to, falling the height a, and 

 driving a pile of unappreciable weight the depth x. The height 

 a must in this case be taken after the blow is made ; because, 

 during the percussion the ram is not only acting by virtue of its 



* It will be here observed that the gravitating force acting upon the weight w, 

 is equal to the weight of the body itself, taken if we please in pounds. Hence 

 the propriety of representing that force by to, equal to the weight of the body. 



