SPRING. 



2MLa , . 

 / being the right fine of the arc, and R = — = — being 



the radius. 



Case 2. Cor. i. — If the motion of the body ceafe 

 when the fpring is bent through any fpace, /, the initial 

 velocity is 



2. If the motion of the body ceafe when the fpring is 

 bent through any fpace /, the time, t, of bending it is 



/ = I" X 



\/2PA' 



or < = i" X 



x/: 



M/ 



2 \/ 2rA 2 V 2/ A 



where m = 3.1416, viz. the circumference of a circle to 

 diameter i. 



3. In the fame cafe, the time of bending the fpring is 



. , /ML /M/ J .. L ^ 

 proportional to i / —5—, or to * / ; and it — be 



giyen, t will be as y M ; and if both — and M be given, 



t will always be the fame, whatever be the original velocity, 

 or through whatever fpace the fpring is bent. 



4. If the motion of the body ceafe when the fpring is 

 bent through any fpace, /, the produft of the initial 

 velocity, and the time of bending the fpring, or V /, = 



mC I 

 1" X — V-, and is proportional to /, the fpace through 

 4A 



which the fpring is bent. 



Hence, any two of the three quantities, V, /, and /, be- 

 ing given, the other is readily determined. 



5. In the fame cafe the initial quantity of motion M V 



Case 3. Cor. i. — If the motion of the body ftriking 

 the fpring ceafe, when it is wholly clofed, the initial vclo- 



city V = C./-?A. 



' V 2MA 



/PL 



2 . The initial velocity V is proportional to -i / -^=-. 



y M 



3. If PL be given, either in the fame or in different 

 fprings, the initial velocity V is reciprocally as ^ M. 



4. The produft of the initial velocity, and the time fpent 



f^ T 



in clofing the fpring, or V t, — i" x ^;andispro- 



4A 

 portional to L, the length of the fpring. 



5. The initial quantity of motion, or M V, = 

 P /PLM 



6. M V is proportional to ^/ P L M, or to P /. And 

 if P L be given, either in the fame or different fprings, 

 M V is as v' M. 



P 

 7- If jj be given, either in the fame or in different 



fprings, the initial quantity of motion is as the length of 

 the fpring into the time of bending it. 



8. If a quantity of motion, M V, bend a fpring 

 through its whole length, and be confumed by it, no 



other quantity of motion equal to the former, as n M 



V 

 X — , will clofe the fame fpring, and be wholly confumed 

 n 



by it. 



9. But a quantity of motion, lefs or greater than M V, 

 in any given ratio, may clofe the fame fpring, and be wholly 

 confumed in clofing it ; and the time fpent in clofing the 

 fpring will be refpeftively lefs or gjreater, in the fame given 

 ratio. 



C PL 



I o. The initial vis viva, or M V', = — ; and 2 a M 



2 A 



= PL. 



11. The initial vis viva is as the reftangle under the 

 ftrength and length of the fpring : that is, M V is as 

 PL. 



P 



12. If -p be given, the initial vis viva is as P', or as L'. 



13. If the vis viva M V' bend a fpring through its whole 

 length, and be confumed in clofing it ; any other vis viva, 



equal to the former, as n' M X — , will clofe the fame 



n 



fpring and be confumed by it. 



14. But the time of clofing the fpring by this vis viva 



V . 

 «■■ M X — » will be to the time of clofing it by the vis 

 n~ 



viva M V", as « to I. 



15. If the vis viva M V" be wholly confumed in clofing 

 a fpring of the ftrength P, and length L ; then the vit 

 viva n'MV will be fuffictent to clofe, l, either a fpring 

 of the ftrength n' P, and length L ; 2, or a fpring of the 

 ftrength n P, and length n L ; 3, or of the ftrength P, 

 and length n'' L : 4, or if n be a whole number, the num- 

 ber n'- of fprings, each of the ftrength P, and length L, 

 one after another. We may add, that it appears from hence, 

 that the number of fimilar and equal fprings a given body 

 in motion can wholly clofe, is always proportional to the 

 fquares of the velocity of that body. And it is from this 

 principle that the chief argument, to prove the force of a 

 body in motion to be as the fquare of its velocity, is de- 

 duced. See Force. 



The theorem above-mentioned, and its corollaries, will 

 hold equally, if the fpring be fuppofed to have been at firlt 

 bent through a certain fpace, and by unbending itfelf to 

 prefs upon a body at reft, and thereby to drive that body 

 before it, during the time of its expanfion : only V, in- 

 ftead of being the initial velocity with which the body ftruck 

 the fpring, will now be the final velocity with which the 

 body parts from the fpring, when totally expanded. 



It is alfo to be obferved, that the theorem, &c. will hold 

 equally good, if the fpring, inftead of being preffed in- 

 wards, be drawn outwards by the aftion of the body. The 

 like may be faid, if the fpring be fuppofed to have been 

 already drawn outwards to a certain length, and in reftor- 

 ing itfelf draw the body after it. And, laftly, the theo- 

 rem extends to a fpring of any form whatever, provided 

 L be the greateft length it can be extended to from its 

 natural fituation, and P the force which will confine it 

 to that length. See Phil. Tranf. N° 472. fed. 10. or 

 vol. xliii. art. 10. 



Spring is more particularly nfed, in the Mechanic ArtSt 

 for a piece of tempered ft.eel, put into feveral machines to 



give 



