been just sufficient yet further to elongate the bar by the same distance 7*, which whole elongation of 21 could not have remained; because the strain upon the bar is only that necessary to keep it elongated by l. The extremity of the bar would therefore, under these circumstances, have oscillated on either side of that point which corresponds to the elongation 7.

349. Eliminating 7 between equations (485) and (486), we obtain

whence it appears that the work expended upon the elongation of a bar under any strain varies directly as the square of the strain and the length of the bar, and inversely as the area of its section.†

The mechanical principle involved in this result has numerous applications; one of these is to the effect of a sudden variation of the pressure on a mercurial column. The pressure of such a column varying directly with its elevation or depression, follows the same law as the elasticity of a bar; whence it follows that if any pressure be thrown at once or instantaneously upon the surface of the mercury, the variation of the height of the column will be twice that which it would receive from an equal pressure gradually accumulated. Some singular errors appear to have resulted from a neglect of this principle in the discussion of experiments upon the pressure of steam, made with the mercurial column. No such pressure can of course be made to operate in the mathematical sense of the term instantaneously; and the term gradually has a relative meaning. All that is meant is, that a certain relation must obtain between the rate of the increase of the pressure and the amplitude of the motion, so that when the pressure no longer increases the motion may cease.

From this formula may be determined the amount of work expended prejudicially upon the elasticity of rods used for transmitting work in machinery, under a reciprocating motion-pump rods, for instance. A sudden effort of the pressure transmitted in the nature of an impact may make the loss of work double that represented by the formula; the one limit being the minimum, and the other the maximum, of the possible loss.

THE MODULI OF RESILIENCE AND FRAGILITY.

2

350. Since U=¦E(†) ̊KL (equation 486), it is evident

that the different amounts of work which must be done upon different bars of the same material to elongate them by equal

fractional parts (†),

are to one another as the product KL.

Let now two such bars be supposed to have sustained that fractional elongation which corresponds to their elastic limit; let U, represent the work which must have been done upon the one to bring it to this elongation, and M, that upon the other: and let the section of the latter bar be one square inch and its length one foot; then evidently

U=M ̧KL. . . . .
.. (488).

M, is in this case called the modulus of longitudinal resilience.*

It is evidently a measure of that resistance which the material of the bar opposes to a strain in the nature of an impact, tending to elongate it beyond its elastic limits.

If M, be taken to represent the work which must be similarly done upon a bar one foot long and one square inch in section to produce fracture, it will be a measure of that resistance which the bar opposes to fracture under the like circumstances, and which resistance is opposed to its fragility; it may therefore be distinguished from the last mentioned as the modulus of fragility. If U, represent the work which must be done upon a bar whose section is K square inches and its length L feet to produce fracture; then, as before,

U‚=M,KL . . . . . (489).

If P. and P, represent respectively the strains which would elongate a bar, whose length is L feet and section K inches,

The term "modulus of resilience" appears first to have been used by Mr. Tredgold in his work on "the Strength of Cast Iron," Art. 304.

to its elastic limits and to rupture; then, equation (487)

U =M. KL="1⁄2 KE
P2L;

...M.

P2 = K2E

P2

Similarly M=E

(490).

These equations serve to determine the values of the moduli M, and M, by experiment."

e

351. The elongation of a bar suspended vertically, and sustaining a given strain in the direction of its length, the influence of its own weight being taken into the account.

Let a represent any length of the bar before its elongation, Ax an element of that length, L the whole length of the bar before elongation, w the weight of each foot of its length, and K its section. Also let the length x have become x, when the bar is elongated, under the strain P and its own weight. The length of the bar, below the point whose distance from the point of suspension was x before the elongation, having then been L-x, and the weight of that portion of the bar remaining unchanged by its elongation, it is still represented by (L-x) w. Now this weight, increased by P, constitutes the strain upon the element Ar; its elongation under this strain is therefore represented (equation 485) by P+(L-x)w Ax, and the length A, of the element when

KE

The experiments required to this determination, in respect to the principal materials of construction, have been made, and are to be found in the published papers of Mr. Hodgkinson and Mr. Barlow. A table of the moduli of resilience and fragility, collected from these valuable data, is a desideratum in practical science.

Integrating between the limits 0 and L, and representing by L, the length of the elongated rod,

If the strain be converted into a thrust, P must be made to assume the negative sign; and if this thrust equal one half the weight of the bar, there will be no elongation at all.

352. THE VERTICAL OSCILLATIONS OF AN ELASTIC ROD OR CORD SUSTAINING A GIVEN WEIGHT SUSPENDED FROM ITS

EXTREMITY.

Let A represent the point of suspension of the rod (fig. 1. on the next page), L its length AB before its elongation, and the elongation produced in it by a given weight W suspended from its extremity, and C the corresponding position of the extremity of the rod.

Let the rod be conceived to be elongated through an additional distance CD=c by the application of any other given strain, and then allowed to oscillate freely, carrying with it the weight W; and let P be any position of its extremity during any one of the oscillations which it will thus be made to perform. If, then, CP be represented by x, the corresponding elongation BP of the rod will be represented by +, and the strain which would retain it permanently

KE

L

at this elongation (equation 485) by(+); the unbalanced pressure or moving force (Art. 92.) upon the weight W, at the period of this elongation, will therefore be repreKE KE

sented by(+)-W, or by; since W, being the

L

strain which would retain the rod at the elongation, is KE

represented by (equation 485).

* Whewell's Analytical Statics, p. 113.

The unbalanced pressure, or moving force, upon the mass W varies, therefore, as the distance x of the point P from the given point C; whence it follows by the general principle established in Art. 97., that the oscillations of the point P extend to equal distances on either side of the point C, as a centre, and are performed isochronously, the time T of each oscillation being represented by the formula

The distance from A of the centre C, about which the oscillations of the point P take place, is represented by L+; so that, representing this distance by L,, and substituting its value for 1, we have

353. Let us now suppose that when in making its first oscillation about C (fig 2.) the weight W has attained its highest position d1, and is therefore, for an instant, at rest in that position, a second weight w is added to it; a second series of oscillations will then be commenced about a new centre C1, whose distance L, from A is evidently represented by the formula

wL

So that the distance CC, of the two centres is

ΚΕ

; and the

greatest distance C,D,, beneath the centre C1, attained in the second oscillation, equal to the distance C,d, at which the oscillation commenced above that point. Now C,D,=C1d1=Cd1

wL
;

+CC1 = CD+CC1 = c + KE the amplitude d,D, of the

second oscillation is therefore 2(c+ KE).

K K