f\ 



SPRING. 



tskes up when wholly comprefl'ed and clofed, or when 

 drawn out. 



Strength or force of a fpring is \ifed for the leaft force or 

 weight which, when the fpring is wholly compreflcd or 

 clofed, will reftrain it from unbending itlelf. Hence alfo 

 the force of a fpring bent, or partly clofed, is ufcd for the 

 leaft force or weight which, when the fpring is bent 

 through any ipace lefs than its whole length, will confine 

 it to the Hate it is then in, witliout fuffering it to unbend 

 any farther. 



We have before Itatcd, that the principle which this au- 

 thor aflumes is, that if a fpring be either compreired or 

 ftretched by any external force, its refiltance is proportional 

 to the fpace by which it is removed from its original and 

 natural fituation. Thus, fuppofe, for inllance, a fpring 

 (P/rt/^ XXXVIII. Mechanics, fg. 6.) C L, refting with the 

 end, L, againll any moveablt- fupport, but otherwife lying in 

 its natural fituation, and at full liberty ; then, if this fpring 

 be preded inwards by any force, p, or from C towards L, 

 through the fpace of one inch, and can be there detained 

 by that force, p, the refinance of the fpring and the force, 

 p, exaftly counterbalancing one another ; then will the force 

 2 p bend tlie ipring through the fpace of two inches, ■^p 

 through three inches, ^p through four inches, &c. The 

 ipace, C /, {fig- 7-) through which the fpring is bent, or 

 by which its end, C, is removed from its natural fituation, 

 being always proportional to the force which will bend it 

 fo far, and will detain it fo bent. 



And if one end, L, be fadened to an immoveable fup- 

 port {Jig. 8.), and the other end, C, be drawn outwards 

 to /, and be there detained from returning back by any 

 force, p, the fpace, C /, through which it is fo drawn out- 

 v/ards, will be always proportional to the force, p, which i$ 

 able to detain it in that fituation. 



And the fame principle holds in all cafes, where the 

 fpring is of any form whatloever, and is in any manner 

 whatfoever forcibly removed from its natural fituation. 



It may be here obferved, that the fpring, or elailic force 

 of the air, is a power of a different nature, and governed 

 by different laws from that of a material fpring. For fup- 

 pofing the line L. C {Jg- 6.) to reprefent a cylindrical 

 volume of air, which by compreffion is reduced to L / 

 (Jig. 7-), or by dilatation is extended to L / {Jig. 8.), 

 its elailic force will be reciprocally as L / ; whereas the 

 force or refiftance of a fpring will be direftly as C /. 



This principle being premifed. Dr. Jurin gives us a ge- 

 neral theorem concerning the atlion of a body llriking on 

 one end of a fpring, while the other end is fuppofcd to red 

 againll an immoveable fupport. And left any ojijeftion 

 (hould be fcrmcd ngainft the poflibiiity of an immoveable 

 fupport, fince any body, how great foever, may be removed 

 out of its place by the leaft force, he obfervcs, tliat the ob- 

 jeflion may be eafily removed. Thus, if the fpring L M 

 (Jig. g.) be fuppofed continued to N, fo that L N ;= 

 L M, if a body M, with any velocity in the dirediion 

 M L, llrikes one end of the fpring, and a body N, at the 

 fame time, with an equal velocity, and a contrary direftion 

 NL, ttrikes the other end, N, of the cuutiiiucd fpring, 

 the point L, the end of the firft fuppofed fpring, will be 

 immoveable. 



Theorim. — If a fpring of the ftrength P, and tlie length 

 C L {fig- lo.), lying at full liberty on an horizontal plane, 

 rell with one end, L, againft an immoveablf fupport ; and 

 a body of the weight M, moving with ihe velocity V, 

 in the diredlion of the axis of the Ipring, Itrike direftly «in 

 the other end, C, aad thereby force the fpring inwardi. 



or bend it through any fpace C B ; and a mean propor- 



tional, C G, 



be taken between the line CL x -j^, and 



2 a, a being the height to which a heavy body would af- 

 cend in vacuo, with the velocity V ; and upon the radius 

 R = C G, be erefted the quadrant of a circle G F A : 

 then, 



1. When the fpring is bent through any right fine of 

 that quadrant, as C B, the velocity, v, of the body, M, 

 is to the original velocity, V, as the cofine to the radius; 



that is, 1. = V X -^ . 



2. The time, t, of bending the fpring through the fame 

 fine C B, is to T, the time of a heavy body's afcending 

 in vacuo with the velocity V, as the correfponding arc to 



2 a ; that is, / = T 



GF 



After demonftrating this propofition, the author pro- 

 ceeds to draw from it feveral curious corollaries, which, for 

 greater fimphcity, he divides into three dillinifl claffes, ac- 

 cording as they appertain to the following cafe? : vix. 



Cafe I — When the fpring is bent through its whole 

 length, or is entirely comprcffed or doled, before the mov- 

 ing force of the body is dellroyed and its motion ceafes. 



Cafe 2. — When the moving force of the body is con- 

 fumed, and its motion ceafes before the fpring \i ben{ 

 through its whole length, or wholly clofed. And 



Cafe 3. — When the moving force of the body is ex- 

 panded, and its motion ceafes at the inllant that the fpring 

 is bent through its whole length, and is entirely clofed. 



We (hall enumerate a few of the moft interefting corol- 

 laries in each of thefe claffcs, premifiiig firft, that P = 

 ftrength of the fpring, L = its length, V :=z the initial 

 velocity of the body clofing the fpring, M = its mafs, ; = 

 time fpent by the body in clofir.g the fpring, A = height 

 from which a heavy body will fall in vacuo in a fecond of 

 time, a = the height to which a body would afccnd in 

 vacuo with the velocity V, C = the velocity gained by the 

 fall, m =: the circumference of n circle, whofe diameter 

 is I. 



Case i. Cor. — When the fpring is Lent through any 

 right fine, C B, (Jg. 10.) the lofs of velocity is to the 

 onginal velocity, as the verfed fine to the radius ; or V — 



Gf . 



■D = V X -_ -, where v is the prefent, and V the original 



velocity. 



2. When the fpring is bent through any fine, C B, the 

 diminution of tlie fquare of the velocity is to the fquare of 

 the original velocity, as the Iquarcol the fine to the fquare 



CB' 



of the radius ; that is, V — v'' = V- X -wr- 



3. When the fpring is bent llirough any fpace, /, the ve. 



body is equal t 

 / 2M a - pi 



r , , , ■ . ■,, / 2MLa - P/' 

 locity, V, of the body is equal to V x .. / ^ . . 



or to V X * / 



2 Ma 



4. The time, /, of bending the fpring through any 

 fpace, /, is proportional to the aic, G F, divided by ^/ a ; 



I being 



