RES
384
RES
which the fibres are supposed to stretch equally, and breakall at once, corresponds to the case of a rod moving parallelto itself, where the centre of extension or percussion does notappear, as being confounded with the centre of gravity.
The base of fraction being a surface, whose particularnature determines its centre of percussion, it is necessary thatthis should be first known, to find on what'point of the ver-tical axis of that base it is placed, and how far it is from theaxis of equilibrium. Indeed, we know in the general, tliatit always acts with so much the more advantage as it isfarther from it; because it acts by a longer arm of a lever ;and, of consequence, it is the unequal consistency of the fibresin M. Mariotte’s hypothesis which produces the centre ofpercussion; but this unequal resistance is greater or less,according as the centre of percussion is placed more or lesshigh on the vertical axis of the base, in the different surfacesof the base of the fracture.
To express this unequal resistance, accompanied with allthe variation it is capable of, regard must be had to the ratiobetween the distance of the centre of percussion from the axisof equilibrium, and the length of the vertical axis of the base;in which ratio, the first term, or the numerator, is alwaysless than the second, or the denominator ; so that the ratiois always a fraction less than unity ; and the unequal resist-ance of the fibres in M. Mariotte’s hypothesis is so much thegreater, or, which amounts to the same, approaches so muchnearer to the equal resistance in Galileo’s hypothesis, as thetwo terms of the ratio are nearer to an equality.
Hence it follows, that the resistance of bodies in M. Mari-otte’s system is to that in Galileo’s, as the least of the termsin the ratio is to the greatest. Hence, also, the resistancebeing less than what Galileo imagined, the relative weightmust also be less ; so that the proportion already mentioned,between the absolute and relative weight, cannot subsist inthe new system, without an augmentation of the relativeweight, or a diminution of the absolute weight; which dimi-nution is had by multiplying the weight by the ratio, whichis always less than unity. This done, we find that the abso-lute weight, multiplied by the ratio, is to the relative weightas the distance of the centre of gravity of the body from theaxis of equilibrium, is to thp distance of the centre of gravityof the base of the fracture from the same axis ; which isprecisely the same thing with the general formula given byM. Varignon for the system of M. Mariotte. In effect, afterconceiving the relative weight of a body, and its resistanceequal to its absolute weight, as two contrary powers appliedto the two arms of a lever, in the hypothesis of Galileo, thereneeds nothing to convert it into that of M. Mariotte, but toimagine that the resistance, or the absolute weight, is becomeless, everything else remaining the same. One of the mostcurious, and perhaps the most useful questions in thisresearch, is to find what figure a body must have, that itsresistance may be equal in all its parts, whether it be loadedwith an additional weight, or as only sustaining its ownweight.
To this end, it is necessary that some part of it should beconceived to be cut off by a plane parallel to the fracture, sothat the momentum of the part retrenched be to its resistancein the same ratio as the momentum of the whole is to its ■resistance. These four powers act by arms of levers peculiarto themselves, and are proportional in the whole, and in eachpart, of a solid of equal resistance. From this proportion,Varignon deduces two solids, which shall resist equally in alltheir parts, or be no more liable to break in one part than inanother. Galileo had previously found one of these, whichis that in which the sides are parabolical : the other, foundby Varignon , is in the form of a trumpet, which is to be fixed
into the wall by its greater end ; so that its tli, °!
its leno )9
the arm of the lever by which it acts, is incr e; b e >
in*
weight, is always diminished in proportion as i [s ^the arm of the lever by which it acts, is increase ^remarkable, that, however different the two systemthe solids of equal resistance are the same in both- .ports’!
tin 9
The following is a general synopsis of the most ^results which have been drawn by different writesubject, both practical and theoretical. fur e, bf 9
1. The resistance of a beam or bar, to a f ra ® n 0 f tb e
force acting laterally, is as the solid made by a. seC ' j; s ta ne0beam in the place where the force is applied, info t tb eof its centre of gravity from the point or linebreach will end. C °^ S
2. In square beams, the lateral strengths are as
of their breadths or depths. ^ are
3. In cylindric beams the resistances or streng
the cubes of the diameters. , a j-e c0lJ
4. In rectangular beams, the lateral strengthjointly as the breadths and squares of the depths- ^ gec (io a
5. The lateral resistances of any beams, wa ° u |j e 9 of
are similar figures and alike placed, are as the clike dimensions of those figures. „„ r ro"' er
tbe
:e dimensions of those figures. . r0 tve r ‘‘
6. The lateral strength of a beam, with its nm 0 p,var'wards, is to its strength with the broad ei ® of
of the broader face to the hrea
upwards,as the breadth
narrower. , t j, e
7. The lateral strengths of prismatic beams o ^. gtamaterials, are as the areas of the sections, and t ®their centre of gravity, directly, and as their
weights reciprocally. e prop eI ? a
8. When the beam is fixed at both ends, t ^ e / a fi0D sider 1
80“
must
lid
Box, yew, plum-tree, oak
Elm, ash .
Walnut, thorn .. tre e.
Red fir, holly, elder, plane crab-tree, app
Beech, cherry-tree, hazel.
Alder, asp, birch, white fir r willow-
iron .
Brass.
Bone.
Lead .
Fine free-stone .
cU 01 ,
A cylindric rod of good clean fir, of an 40^^' e ef e(t
drawn in length, will bear at its jjear a ^° ol {e>' etlC g(
a spear of fir, of two inches diameter, jj cif cl “'’ r op e !ton weight. A rod of good iron, of an ^ he£0P eawill bear nearly three ton weight. its e r 0 pe> 0
an inch circumference, will bear A rod of fir > ° r *Hence Emerson concludes, that if a r0
streP^
beam ^only thM C0Se ’ we
9. CylimiLl i ength ofthe former. ^
Proportional to S< ^ uare P risms have their lateral a ®j jS ,
direcS anl „ - th , e cubes of their diameters, or10 Similn 61 r ^ en £ths and weights inversely .inversely"!! ' ^ >Sma and Coders have their
The foil a ^- 01 iona ^ 1° their linear dimensions. e %pet ymen on 7£ S reSults are wholly drawn fro®writers hv erent dances, by Emerson ^° A ~the nrecp/o means °f which the propositions a ,
computations.^ ^ ° f this article may ^
The relative Resistances or Strengths of Wood a ndother Bodies. vrov arti0