746 MECHANICS of a uniform dilatation, whose linear measure in every direction is na L JL , combined with a shear in each transverse section, whose !. 2 Q7^> 1 , O Q1 measure is !---. 91. The shorter axis of this shear is radial, and the magnitude of the shear is obviously greater for smaller values of r. The inner layer of the walls is thus the most distorted. The amount of the distortion is directly as the pressure, and inversely as the area of the section of the walls. When the walls are very thin the shear is practically the same throughout their thickness. "When they are very thick, the shear near the inner surface is nearly > 2n however fine be the bore. That near the outer surface is nearly which vanishes when the bore is very fine. Thus it appears that, if a stout tube bursts by the shear produced by internal pressure, little is gained either by making it of extremely great thickness or by making it of very small bore. The diagrams A and B in fig. 66 show, necessarily on a greatly exaggerated scale, the nature of the distortion produced at different A B CD Fig. 66. parts of the wall of the tube. They represent transverse sections of small, originally spherical, elements made by planes at right angles to the axis. The radial diameters are horizontal. A is an element close to the external surface, B an element near the inner surface. The increase per unit volume of the interior of the tube is Hal / 1 [ J_ al - ajj Jc a ^ n so that, if the tube be very thick in comparison with it:? bore, the increase is nearly U/n. In flint glass this is approximately about Trmr* when IT is a ton-weight per square inch. Cylinder 282. The reader who has followed the above investigation will under find no difficulty in obtaining the corresponding results for a cylin- external drical tube, closed at both ends and exposed to external pressure n, pressure, in the form _p Ua /_!_ ^J_ James500 (talk)f^! I 3& r* 2 _ dr The only comments we need make are (1) that the signs of these distortions are now negative ; (2) that they are (so far as change of volume is concerned) greater than for the internal pressure, as f has taken the place of a% as a factor in each term involving k ; (3) that the terms involving the rigidity are, except as regards sign, unchanged. The change of volume of every part of the walls is nr 1 is about for and the change of volume of the interior is 1 _ k n The numerical value of the factor n flint glass and about -f^^ for steel, when n is a ton-weight per square inch. There is, however, a peculiarity which (when the walls are thick enough) distinguishes this from the preceding case. For k is usually considerably greater than n, so that a fortiori 3k is greater than 2n. Hence, in the value of dp/dr, the term in n is always greater than that in k so long as the pressure is internal. Thus the radial elfect is compression at all parts of the walls. But, when the pressure is external, we may (if the walls be thick enough) find a value of r for which 1 3k a"- 1 In glass, this occurs when r=l 6a nearly. At this distance from the axis there is no radial change of length ; at greater distances there is radial compression, and at smaller radial extension. This is indicated in the diagrams C and D in fig. 66, which like the former are greatly exaggerated. They represent the distortion of small spherical elements of a thick tube, the first at the inner wall, the second at the outer surface. As before, these are sections made by a plane perpendicular to the axis of the cylinder. 283. In a spherical shell of internal and external radii n and Splin^ i, the equations become a little more simple on account of the slid] more complete symmetry. undu Using the same notation, so far as it is now applicable, we have oxter:. 7 press il The statical equation is With these we obtain, for external pressure n, the result -= n from which the other equation may be derived by differentiation. 284. The propagation of plane waves in an elastic solid has been discussed in ELASTICITY, and the mechanics of fluids is discussed under HYDROMECHANICS. GENERAL CONSIDERATIONS. 285. The preceding view of the subject of Abstract Genei Dynamics has been based entirely upon Newton s Laws of consi- Motion, which were adopted without discussion, as a com- c cva " plete and perfectly definite foundation ; and the terms employed, as well as the mode of treatment in general, have somewhat closely followed Newton s system. The only considerable apparent departure from that system is connected with the development of the idea of energy, and its application to the simplification of many of the methods and results. This also was, as we have seen, really introduced by Newton ; but it has been immensely extended since his time both by mathematical and by experimental processes. It is time that we should now return to the laws of motion, and examine more closely, in the light of what we have learned, one or two of the more prominent ideas which they embody. To do so fairly we must go back to Newton s own definitions of the terms which he employs. About many of these, which have already been quoted in 97-1 13, there is no difference of opinion But it is otherwise when we come to the definition of "force" ( 5, 104). There can be no doubt that the proper use of the term Force " force " in modern science is that which is implied in the statement of the first law of motion, as we rendered it in 1 from Newton s Latin. It is thus seen to be the English equivalent of the term vis impressa. Newton uses the word vis in other connexions, and with a certain vague ness inevitable at a time when the terminology of science was still only shaping itself ; but his idea of " force " was perfectly definite, and when this is in his mind the vague word vis is (when necessary) always qualified and rendered precise, either by the addition of impressa or in some equally unambiguous way. To render vis by " force," wher ever it stands without the impressa or its equivalents, is to introduce a quite gratuitous confusion for which Newton is not responsible. We have only to think of the multitude of terms, such as vis insita (inertia), vis acceleratrix (accel eration), vis viva (kinetic energy), &c., A:c., to see that all such complex expressions must be regarded as wholes, and that vis does not mean " force " in any one of them. 286. Thus in Newton s view force is whatever changes (but not " or tends to change ") a body s state of rest or of uniform motion in a straight line. He mentions, as instances, percussion, pressure, and central force. 1 Under the last of these heads he expressly includes magnetic as well as gravitational force. Thus o o 1 Vis centripeta. It has been already explained that such words as centripeta include impressa, so that the above rendering of Newton s
phrase is the obvious one.