698 MECHANICS the whole constrained to move in one plane, it is easy to see that the complete path of any point R of PQ is a species of figure of eight. A por tion of that curve on each side of a point of inflexion (where the cur vature vanishes) <^ w / ~~ 5 Q> was found to be sufficiently straight for prac tical purposes. Peaucel- But the rigor- lier cell, ous solution of this problem has only been arrived at in recent times ; and the beautiful device of Peaucellier, which we will briefly explain, has led to a host of remark able investigations and discoveries in a field regarded till lately as perfectly hopeless. A simple mode of arriving at Peaucellier s result is as follows. Let PQ, PK (fig. 32) be equal links, and PO a link of a different length, all jointed together at P. Suppose O to be fixed, and Q and R constrained to move in a fixed straight line OQR, what is the relation between OQ and OR 1 We have, if PS be perpendicular to QR, 31 - OP 2 =OS 2 + SP 2 , whence OP 2 - QP 2 = OS 2 - QS 2 = OQ. OR . OR is constant ; so 33 - Defini tion of physical particle. Thus the rectangle under OQ and that, if R were to describe a straight line, Q would describe a circle having O on its circumference. In practical application, to keep O, Q, R in one line, Q - the parts of the link- work 00 are doubled symmetrically about that line, so that it takes the form of a jointed rhombus PQP R (fig. 33) with two equal links, PO, OP attached at the extremities of a diagonal. As a very curious result of this ar rangement, if OQ have its length changed by any very small amount, the corre sponding change of length of OR is directly as OR 2 or inversely as OQ 2 . Hence, as will be seen later, a con stant force (towards or from 0) acting at Q will be balanced by a force (from or towards O) acting at R and varying inversely as the square of OR. DYNAMICS. Definitions and General Considerations, 96. We commence with a few necessary definitions. A "physical particle " is a purely abstract conception, embody-
- n => to g etner ^ e ideas of inertia and of a geometrical point.
It is, so to speak, a mathematical fiction, embracing only those properties which are required for our temporary purpose. Any mass, however large, can be treated as a particle, provided the forces to which it is subject are exerted in lines passing through its "centre of inertia" or " centre of mass " (this term will presently be defined), so as to be incapable of setting the mass into rotation. This is, to a first approximation, true of planetary motions, but when we look more closely into that question, so as for instance to take account of the oblate forms of the planets, we have to deal with forces which produce rotatory effects, such as "precession" and "nutation." 97. The "quantity of matter" in a body, or the Mass 1 " mass," is proportional to the " volume" and the " density" dens:l! conjointly. The "density" may therefore be defined as the quantity of matter in unit volume. If M be the mass, p the density, and V the volume of a homogeneous body, we have at once M-V ft provided we so take our units that unit of mass is the mass of unit volume of a body of unit density. Hence the dimensions of p are [ML" 3 ]. As will be presently explained, the most convenient unit mass is an imperial poimd, or a gramme, of matter. 98. The " quantity of motion," or the " momentum," of a moving body is proportional to its mass and velocity conjointly. As already stated this is, like velocity, a directed quantity, or " vector." Its dimensions are, of course, [MLT 1 ]. 99. " Change of quantity of motion," or " change of momentum," is proportional to the mass moving and the change of its velocity conjointly. Change of velocity is to be understood in the general sense of 32. Thus, with the notation of that section, if a velocity represented by OA be changed to another represented by OB, the change of velocity is represented in magnitude and direction by AB. 100. " Rate of change of momentum," or " acceleration of momentum," is proportional to the mass moving and the acceleration of its velocity conjointly. Thus ( 36) the acceleration of momentum of a particle moving in a curve is MA : along the tangent, and Mw 2 /p in the radius of absolute curvature. The dimensions of this quantity are [MLT" 2 ]. 101. The "vis viva," or "kinetic energy," of a moving body is proportional to the mass and the square of the speed conjointly. If we adopt the same units of mass and velocity as above, there is particular advantage in defining kinetic energy as half the product of the mass into the square of its speed. Its dimensions are [ML 2 T~ 2 ]. 102. " Rate of change of kinetic energy," thus defined, is the product of the speed into the component of accelera tion of momentum in the direction of motion. For Mom turn. Chan mo mi turn. Rate cliang mom( turn. Kiuet energ Rate chnng of it. The dimensions are [ML 2 T~ 3 ]. 103. The " space-rate of change of kinetic energy" is Space
2
ds of it. and its dimensions are [M.LT 2 ], the same as those of "force" ( 104). 10-i. "Force," as we have already seen, is any cattse Force which alters a body s natural state of rest, or of uniform motion in a straight line. The three elements specifying a force, or the three elements which must be known before a clear notion of the force under consideration can be formed, are its place of application, its direction, and its magnitude. The place of application may be a surface, as when one body presses on another ; or it may be throughout the whole mass of a body, as in the case of the earth s attraction for it. The " measure of a force " is the rate at which it produces momentum, or the momentum which it produces in unit of time, which is the same as what we have already called " rite of change of momentum." According to this method of measurement the standard or unit force is that force
which, acting on the unit of matter during the unit of