1839.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



255 



MOMENTUM OF FALLING BODIES. 



Sir, — As the following remarks on the momentmn of falline; bodies 

 are I believe of iiuporlance, and connected with the communications 

 of C. E. C. and B. on the subject, you will oblige me by giving them 

 a place in your journal. 



The questions for consideration are, the effect of a moving power 

 on the resistance offered to a pile while driving, and the comparative 

 effects of a weight acting simply as such on a pile, and when striking 

 it with velocity. These ellects are best measured by the depths 

 penetrated mider the same circumstances. 



It is demonstrated by writers on dynamics, tliat the effect of a 

 moving power in overcoming an uniform resistance is as the sipiare of 

 the velocity, and Sineaton in his "Experimental examination of the 

 quantity and proportion of mechanic power," has shown " that tlie 

 quantities of mechanic power to be expended are as the squares of 

 the velocities to be generated, aud vice versa. If, therefore, the re- 

 sistance to be overcome was uniform, the force of a ram or monkey 

 would be as the square of the velocity into the weight, or as v'b. But 

 the resistance offered by a pile when driving, is made up of the resis- 

 tance to penetration and the resistance of friction ; the tirst of which, 

 in homogenous substances, is nearly uniform, and the second increases 

 as the pile is driven, and in a ratio nearly to the depth penetrated. 

 Put a = the velocity that would be destroyed in one second by the 

 resistance to penetration, x =: the depth penetrated, and ;«.r =: the 

 velocity that would be destroyed in one second by the friction at the 

 depth X: we then have (Sinq)son's Fluxions, art. 218, vol. 1,) {a-\-mx) 

 dx-:^vdv, and by integrating and solving the resulting quadratic, we 



get.t: 



\/ 1 — 5 — -for the depth penetrated with the velocity v, 





and therefore the resistance imder the assumed circumstances will be 

 as this quantity in which a and m are known quantities, to be deter- 

 mined by experiment from the substance penetrated and the nature of 

 the pile. If we sujipose a to be very small compared with lux, as 

 would be the case with a rough pile in a bog when it had penetrated 



to some depth, we get.rrz-v/ — or, the effect of a sirolie is proportional 



to the velocity what the resistance increases as the depth. 



Without entering farther into the nature of the resistances to be 

 overcome in pile-driving, it appears that the force of the stroke will 

 be as the velocity when the resistiuice increirses as the dei)th; as the 

 squai-e of the velocity when the resistance is uniform ; and lus some 

 other function of the velocity when the resistance does not follow 



these laws. It also appears from the equation .r : 



that tlie effect increases in a gi'eater ratio than the velocity, but as in 

 all practical cases, the velocity of the ram will be greater than that 

 the superstructure can ever be supposed to have, the advantage will 

 be in favour of practice, and therefore practically we may suppose the 

 force to be as the velocity. The nature and mass of tlic pile will 

 have considerable influence on the force of the stroke. If the pile is 

 of straight-grained fir, it will be driven farther by the stroke than one 

 of beech or oak, and if shod and capped with iron, the effect will be 

 considerably increased from a greater degree of elasticity, as the effect 

 would be decreased if shod anil capped with cork or any other yield- 

 ing substance. Also, if bv represents the quantity of motion in the 



bv 

 ram, and;; the mass of the pile, will be the joint velocity after 



the stroke, which, as I have before remarked, being less than r, \\ ill 

 make the effect less. 



With respect to the effects of a body in the state of rest upon a pile, 

 and when striking it, I am of opinion that in most cases there is a dis- 

 parity in the forces, for as the weight produces no effect, how are we to 

 compare them ? A weight is compared to a weiglit by its own stand- 

 ard ; and a moving force to a moving force by a different standard ; 

 but when we connnence comparing both, we find the consideration 

 attended with no small difficully. in the spring balance used by B, 

 (Journal, vol. II, no. Ui, p. iti,) a weight on the spring keeps it steadily 



a point that the same weight, moving with a velocity of jj foot per- 

 second, would attain when its motion was destroyed ; but in the case 

 of a weight pressing on a pile, if it is not such as to cjusc penetrafion, 



1 do not see bow it can be conqiared with a moving force that does, 

 and if it does penetrate, the forces are best measured by the depth; 

 but after a few^ strokes of a monkey, (U- after tlie first, the effect of the 

 weight resting on the pile bec(,ines nothing. 



The object in bridge liuilding is to give firmness to the super- 

 structure by strengthening the foundations. For this purpose it w ill 



be best to sfii'pose the iucuubent ^yeigUt »wv»'s witli v. veluvjt/ «i' 



one or two feet per second, and to compare this force with (he entire 

 force used in driving the piles. Example: — Suppose an abulinent 

 measuring 30x20x 1*', each foot weighing li cwt., supported on 150 

 piles, each pile being driven with a monkey weighing G cwt., falling 

 from a height of IG feet. Here the force of the piles = 32 x l-'iOX 0:= 

 28,800. The weight of the abutment = 30x20x lOx U='0OUO cwt 



„ ' ,„ =: 3- , which shews the foundations are able to bear the force 

 y,ooo 



of the abutment moving with a velocity of more than three feet pei- 



second. The equation m^^b-\-Obr, given by your correspondent B, 



appears to me not to hold good for falling bodies, and certainly cannot 



appl)' to the example taken from " Hufton's Course," for, evidently 



b iS not similar to 9bv, and therefore the equation b-\-ebv cannot obtain. 



The fact of the matter is this, that in falling bodies, after the force 



ebp is expemled, b then acts by its weight, and very little consideration 



will shew that both cannot be added together. The ingenious method 



invented by Sir Christopher Wren for determining the effects of col- 



lision by suspending the bodies with threads of ecjual lengths, show 



the truth of Hutton's formulae ~— - and ■ „ | , , and the exiieriments 



U-fb B-]-b '■ 



of Smeaton for finding the mechanical power lost after the strokei 



prove the same tiling. 



The experiments made by B prove the effects of a moving power 



on his spring balance to be ——nearly, but until the case is shewn to 



be analogous to pile-driving, it would be incorrect fo use the result, 

 t-ralileo and Merseunus found the squares of the weights started from 

 the ground in ;i balance to be as the heights fallen; and Gra\esaiide, 

 in his Natural Philosophy, liy altering the apparatus, found results as 

 the square of the velocity ; therefore, in applying experiments of this 

 kind, it is cjuite necessary that the resistance in both cases be equal 

 and similar. 



B changes the equation Hiz::i-| — — into m=::b-\—^■^/s^^ it should 



be m^b-\-^\/Qis, for t^riy'Gls. The results, therefore, of tlie 



GOO 



examples he has given should be 2004--— ^G4x 25= 12,200, not 



8,685, and 100,0004-^ — '^— ^G4)001=220,000,iiot 181,000. By dis- 

 carding the factor 6, the results w ould be 12,000 and 120,000, which 

 are nearer the truth. 



Liimric/c, 

 •lith May, 1830. 



I am, sir, 



Yoin''s obediently, 

 John Nevillk. 



"ASPECTS AND PROSPECTS." 



Sir — Instead of the epithet "judicious," I think that of " ingenious " 

 might very well have been applied to Repton's remarks gi\en in your 

 February number ; since they certainly are of that ingeniously per- 

 plexing kind wliicli tend to make people fancy objections and incon- 

 veniences where none were before thought of. I at least am so far 

 from agreeing with him that aspect is of more importance than almost 

 any tiling else, that I consider it to be a matter of comparatively little 

 or no moment. In fact every aspect has something to recommend it, 

 and also something that may be alleged against it as a defect. 



According to Repton's theory, "an aspect due north is apt to be 

 gloomy, because no sunshine ever clieers a room so placed." No^v as 

 regards the exterior, such an aspect is certainly not at all to be reconi- 

 inended for that of the principal architecturiil front, because it will 

 never catch the sun, except its evening rays in summer time, conse- 

 quently will not show itself to the same advantage as when set off by 

 strong light and shade. But so far from the rooms themsehes being 

 necessarily gloomy, they w ill in siunmer time be far more agreeable 

 than those more exposed to the snn. A room facing the north and 

 looking out upon a pleasure ground or landscape liglited up by a 

 brilliant sun, enjoys a most cheering and animated picture, so placed, 

 be it observed, as to be viewed with the greatest effect. '\\'liethcr 

 such room be in itself gloomy or not, will depend upon the iucbitect, 

 iqion its design, (it ting up, and furniture. It inavbe of more iliaii ordi- 

 narily cheerful character, while another fu-iiig the south, shall lie (piite 

 the reverse. I am of opinion therefore that liepton either has nut 

 sufficiently explained himself, or else does not liimself uildcj.sti'U'.l whrtt 



