48 OF THE ECiUILIBniUM AND STREKGTH OF ELASTIC SUBSTANCES. 



begin to bend with the weight of a bar -of the same thick- 



1 



ness, equal in length to .8225X- 



-X 10,000,000 



laoxiio 



feet, or 571 feet; that is, with.a weight of about 120 pounds; 



neglecting the effect of the weight of the bar itself. In the 



same manner the strength of a bar of any other substance 



may be determined, either from direct experiments on it3 



m 

 flexure, or from the sounds that it produces. K /= — 



—rz. 822511, and -^v'f.saasnj^.oorv/n, whence, if we 



1)1' I 



know the force required to crush a bar or column, we may 

 calculate what must be the proportion of its length to its 

 depth, in order that it may begin to bend rather than 

 be crushed. The height of the modulus of elasticity for 

 iron or steel is about 9,000,000 feet, for wood, from 

 4,000,000 to 10,000,000, and for stone probably about 

 s.OOO.OOO : its weight for a square inch of iron 30,000,000 

 pounds, of wood from 1,500,000 to 4,000,000, and of stone 

 about 5,000,000: and the values of n are in the two first 

 cases from 200 to 250, and in the third about 2500, and 



»/n becomes 15 and 50, and- 12.3 and 41.1 respect- 



V 



ively ; so that a .column of iron or wood cannot support 

 without being crushed a longitudinal force sufficient to bend 

 it, unless its length be greaterthan 12 times its depth, nor a 

 column of stone, unkss its length be greater than 40 times 

 its depth. 



324. Theorf.m. When a longitudinal 

 force is applied to the extremities of a straight 

 prismatic beam, at the distance a from the 

 axis, the deflection of the middle of the beam- 

 will be fl.(sec.arc(^(^)-^y)- 1). 



If we suppose the length to be increased until f^ 



(—\ .—, or friicv'l ), the beam might be retained 

 e / 12 Vl2// 



by the force /in the form of a harmonic curve, of which 



a might be an ordinate, and the vertical ordinate would be 



as much greater than a as the radius i> greater than the sine 



of the arc corresponding to its distance from the origin of 



the curve, or as the secant of the nrc corresponding to its 



distance from the middle of the curve is greater than the 



radius, and the excess of this secant above the radius will 



express the deflection produced by the action of the force ; 



f / ^ \ 



but this arc is to the quadrant — as f to 1;c^ I — -, 1 , and 

 2 \V\I I 



/Sf\ ' 

 is therefore equal to ^/ f — V — 



Scholium. Hence it appears that when the other 

 quantities are constant, the deflection varies in the simple 

 ratio of «. The radius of curvature at the vertex is 



I'hm 



liaf. (sec. arc ^/ — /"T" )> ^™'" which the degree of ex- 

 tension and compression of the substance may be deter- 

 mined. 



39,5. Theorem. The form of an elastic 

 bar> fixed at one end, and bearing a weight 

 at the extremity, becomes ultimately a cubic 

 parabola, and tlie depression is ^ of the 

 versed si nfi of an equal arc, in the smallest 

 circle of curvature. 



The ordinate of the cubic parabola being ax' its fluxion, 

 is 2ax'.r, and its second fluxion 4axxi, which varies as x 

 the absciss. If the curvature had been constant, the second 

 fluxion would have been bii, the first fluxion bxi, and the 

 ordinate ^bxx ; but as ii is bix — xii, the first fluxion is 

 bxi — ^'x, and the fluent ^i'— ir', which, when b—x 

 becomes it', instead of t. 



3i26. Theorem. The weight of the modu- 

 lus of the elasticity of a bar is to a weight 

 acting at its extremity only, as four times the 

 cube of the length to the product of the 

 square of the depth and the depression. 



If the depression be d, the versed sine of an equal arc in 



the smallest circle of curvature will be jd, and the radius of 



cc 

 curvature — , e being the length ; but the radius of curva- 

 3d 



bbm 



ture is also expressed by , a bemg here equal to e, 



\1af 



therefore ——-^.lle^f—Zbbdm, andm=-— -./. If/ 

 3d I'lcf bbd 



be the weight of a portion of the beam of which the 



4e' 

 length is g, the height of the modulus will be —^.g. 



SciioLii'M. In an experiment on a bar of iron, men- 

 tioned by Mr. Banks, e was 18 inches, b and d each 1, / 

 490 pounds, and g about 150 feet : hence the height of the 

 modulus could not have been less than 3,500,000 feet. 

 But d was probably much less than this, as the depression 

 was only measured at the point of breaking, and m must 

 have been larger in the same proportion. 



327. Theorem. If an equable bar be 

 fixed horizontally at one end, and bent by 

 its own weight, the depression at the extrc- 



r 



