Equilibrium between Living and Dead Forces. 



243 



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 =w* acting through the height AB=a, the other 



Fig. 1 . 



A 



w 



W 



x. 



W 



aw. Taking seconds and feet 



as units of time and space, and putting i 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 ocg-=32, 





t. 



s. 



1 



2 



// 



i 





// 



&C. 



2<r 

 &C. 



V. 



g 



2? 



B 



&C. 



X 



For the force W=aw oc ag=32a, 



cQ 



w 



t. 



s. 



V. 



// 



1 



2" 

 &c. 



lag 



lag 

 &c. 



ag 



2ag 



&c. 



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



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

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



e space x, will be (2agx)~. These velocities being 



through th 

 e 1"al, We have 



{2agxy 



(2M 



i 



Substituting for ag and g their proportionals W and w, 



we 



Hence 



(2W>) 

 W 



I 



(2wa) 



I 



aw 



x 



(A) 



ram w, falling the height a, and 



That is, "W j s the pressure or dead weight which wilt 

 equilibrio the percussion of the 



nvin g a pile of unappreciable weight the depth x. The height 

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



Urin g the percussion the ram is not only acting by virtue of its 



is e ' T' 11 be llere obse «" v ed that the gravitating force acting upon the weight w, 



the l ° tb ° vve 'S nt °f the body itself, taken if we please in pounds. Hence 



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



