ELASTICITY. 



ELASTICITY. 



J which ny U iistimMsd with greater accuracy by a hand q a 

 to the pulley. Aaectsaaian *rfi ADI will thus be pro- 

 I. M well a* MI iaorani at tendon, which may then be compared 

 br the common Uw of aUtios ; and the experiment* show that as 



: i proportional to 

 by Hooke : " tt 



lone w the added weight* are small, " thk uOnaiiNi ia proportional to 

 the inasiii of maim." Thi* law i* that expraeMd by 









ponding to a portion : 



When a m&atm ehatie string i* suspended vertically it will be 

 The tension varies from point to point, 

 oal to the portion of the atring of which it 

 "If * be a portion of the atretched atring corre- 

 i * of the same unstretdhed, and y - A 9, * + A *, 

 r pair of portion* greater than the former, and a 

 the whole length of the string in it* natural state, the extension Ay 

 &x at the element A* i proportional to the weight of the remaining 

 ortion- <x + A-r>of thestring; henoe if g denote the weight of a 

 nit of the string, and the index of elasticity peculiar to the sub- 



tenoe, we have ultimately [DIFFERENTIAL CALCULUS] ? l=ge(a 



fix 



r), and therefore by the rules of the Integral Caletdut y x=ge 

 (f f _ ), to which no arbitrary constant need be added, because y 

 commence* at the same point with x : if we now make x=a,ve find 

 that g-.a? expresses the extension of the entire string. 



Similar principles may be easily applied to determine the form of 

 an elastic string suspended from two points, and stretched by its own 

 weight ; but in this case the curve (which differs from the common 

 catenary) cannot be considered as accurately determined without taking 

 into account the elasticity of inflexion as well as that of extension. 

 The mere mathematical problem may be seen in most mechanical 

 (Price's ' Calculus,' voL iii. chap. 5 ; Puisson, ' Mccanique ; ' 



consult alao Lagrange, ' Moc. Aualytique,' for the method of intro- 

 ducing the condition of elasticity in a system at rest.) 



An important practical branch of this subject, on the strength of 

 beams, which has been much advanced by Mr. Peter Barlow, and the 

 more recent experiments of Mr. Eaton Hodgkinson, of Manchester, we 

 reserve for a future article. [STRENGTH OF BEAMS.] 



When a uniform elastic string, fixed at one extremity and stretched 

 by a force applied at the other extremity, is abandoned to itself, it 

 will return to it* original form after a series of contractions and ex- 

 pansions, the force which solicits each point being proportional to its 

 distance from it* original place, though the successive oscillations go 

 on rapidly diminishing in extent in consequence of the resistances 

 encountered. The same law applies to the displacements of the mole- 

 cule* of elastic fluids and gases. 



For the laws of the mutual impact of elastic bodies, see the article 

 COLLISION, Ac. If a body is attached to an elastic string, which at the 

 other extremity is fixed, and be projected in any direction, the resolved 

 part of the centrifugal fore* which act* in the direction of the length 

 of the string tends to stretch it, and the centripetal force will be pro- 

 portional to r c, r being the length of the stretched and c of the 

 unstretched string : this force is attractive when r is greater than c, 

 and repulsive when less. Hence if we conceive a circle, of which the 

 centre is the fixed point, and the radius equal to c, the portions of the 

 orbit described externally to the circle ore concave, and those internally 

 are convex relatively to the centre of the circle, and there are as many 

 point* of contrary flexion [CTRVE] as there are intersections of the 

 trajectory and circle. Neither the law of the periodic times nor the 

 form of the orbit is similar to those belonging to the earth and planets : 

 the supposition, therefore, that attraction between the great masses 

 which compose the solar system is conducted through the medium of 

 interposed and invisible elastic strings is unfounded. 



When an elastic string, fixed at one end, is bent by a weight or other 

 force applied at a given point, the elasticity of inflexion acts normally 

 at each point of the curve, and ia some function of the curvature at 

 that point It is usual to suppose it proportional to the simple cur- 

 vature. On this supposition the figure of an elastic lamina in a vertical 

 position, fixed at its lower point, and bent by a small weight applied at 

 the top, may be determined. This problem has been treated by Killer, 

 Lagrange, and Poisson. The English reader may find the varieties of 

 the elastic curve discussed in the appendix to Whewell's Mechanics. 



The elastic force of a twisted string follows a law precisely similar to 

 that of one which is only stretched : the latter is proportional to the 

 extension, the former to the torsion. Thus, if a cylindrical elastic 

 thread, fixed at one extremity, be twisted by a force applied perpen- 

 dicularly to it* length, any straight line taken along the surface of the 

 cylinder will be converted into a helix ; and with a double torsion the 

 circular arc through which each point has been removed from ita 

 original place is doubled. And since this circular arc may be sub- 

 divided into any number of equal arcs, the successive resistances of the 

 elasticity to the additional torsions ore equal, supposing each pre- 

 ceding rasutaooe to bs sustained. Therefore the accumulated force of 

 !i M proportional t.. the an^lc through which an index woukl 

 if nxed at any |int |*T|H-i><liculrly to the length of the cylinder, 

 nr in the HntapMoa of its radius : but this law has limit* as well as 

 that for the elasticity of extension ; for the torsion may be continued 



until a strain is prodnced, when there will of 

 diminution of elastic force. 



Let AB represent an elastic string, 



bean i 



vertically from the 



point A, and attached at B to a cylindrical body D, of which the axis 

 BO is in the direction of the string produced, the string being primi- 

 tively in an untwisted state. Let the cylinder be turned round its 

 axis through an angle r BO, or a, which measure* the torsion generated 

 in A B, and also the elastic force tending to bring the system back to 

 its original state. Let the restraining force be now removed, and the 

 cylinder, abandoned to itself, will return to it* original place after a 

 series of isochronous oscillations, which are gradually diminish.-.! by 

 the resistance of the air and by the internal resistances of the molecules 

 of A B during the processes of being twisted and untwisted. 



Let 8 m stand for an element of the cylinder, situated at a distance r 

 from its axis, and 8 the angle of torsion, at any time after the com- 



mencement of this motion ; then is the angular velocity; and 



d t 



therefore the linear velocity of Sm is r ; the accelerating force or 



d t 



ratio of the increment* of velocity and time is r__ ; the force of 



d fl 



torsion, being proportional to the angle 9, may be represented by n 6 

 apjiluil at a distance = unity from the axis of the cylinder, perpi 

 l.irly to the radius, the constant n being the force of torsion corre- 

 sponding to an angle = unity. Now, by DVAJembert's priiniji! 

 impressed force, taken in a reversed direction, would make equilibrium 

 with all the effective forces : that is, the force - 9, at a distance - unity, 



would produce an equilibrium with the forces such as r t m acting 



d C 

 on 8 m at a distance r ; hence the corresponding moment*, which are 



n 8 x 1, and the sum of all, such as r 2 8 m - must be equal, but 



<l f 



of contrary signs ; and since - U the common accelerating force on all 

 d t* 



the particles 8 nt at a unit distance, we need only take the sum of the 

 product* r* 8 m, which is easily found in this case by the rules of the 

 integral calculus, and is called the moment of inertia of the cylinder. 

 Representing it by M K ', where x is the mass of the cylinder and K 

 its radius of gyration, we have the equation 



Put (for the sake of abridgment) e> = . ; then, by the methods for 



integrating differential equations, we find 0= A sin. (c t + n), where A 



d 9 

 and B are arbitrary constant* ; and for the velocity -TT = A oos. (c t + B). 



Now, we can determine the constant* by the circumstances of the 

 origin of the motiou; for when t=o, ,we have supposed the initial 



(/ Q 



torsion was a, or r B o, and ^ waa then nothing. Hence we have 

 = A sin B ; O = COB B ; therefore, squaring and adding, we have A J = O*, 

 and A=. Similarly, by division, tan B=oc, .'. B = j. The value of 9 



is therefore expressed at any time by o sin f c t + 5), or cos (c t}. 



\V ln-ri the cylinder makes half an oscillation the elastic thread is then 

 perfectly free from torsion ; and if T be the time of an entire oscil- 



lation, since 9 then vanishes, we find o=a cos. ( ) ! therefore CT = , 



and T = - , which shows that the successive oscillations are of the same 



c 



duration, and that the square of the time of one oscillation varies 

 directly as the moment of inertia, and inversely as the force of torsion, 

 estimated at a given distance from the string. 



The suspended body may be any other as well as the cylinder we 

 have supposed, with manifestly the same result*. For instance, in 



