(to) Integrating over the base, to obtain one-third of the kinetic energy T, 3T = 2 pf '3 4R2(3x4-h4)dx/h 3 = pR2h4 / 1 35 V 3 (II) so that the effective k 2 of the liquid filling the trianglc is given by k 2 = T/Z p R 2 A = 2h2/45 = (radius of the inscribed circle) 2, (12) or two-fifths of the k 2 for the solid triangle.
(21) The comparison of this formula with experiment provides a striking confirmation of the truth of the kinetic theory but at the same time discloses the most formidable difficulty which the theory has so far had to encounter.
These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.
The determination of the series of configurations developing out of given initial conditions is not, however, the problem of the kinetic theory: the object of this theory is to explain the general properties of all gases in terms only of their molecular structure.
With liquid of density p, this gives rise to a kinetic reaction to acceleration dU/dt, given by 7rp b 2 a 2 b l b d J = a 2 +b2 M' dU, if M' denotes the mass of liquid displaced by unit length of the cylinder r =b.
ZI /t = - (a - s) M'Q 2 sine cos ° - EQ sin() =[ - (a - (3)M'U+E]V (8) Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid - aM' (Ç_vR), - (3M' (-- E -FUR), - EC' dR, (9) so that its equations of motion are M (Ç - vR) _ - aM' (_vR) - (a - $) M'VR, (io) M (Ç+uR) = - OM' (dV+U R) - (a - ()M'UR - R, '(II) C dR = dR + (a - Q)M'UV+0V; (12) and putting as before M+aM'=ci, M+13M' = c2, C+EC'=C3, ci dU - c2VR=o, dV +(c1U+E)R=o, c 3 dR - (c 1 U+ - c 2 U)V =o; showing the modification of the equations of plane motion, due to the component E of the circulation.
In point of fact it is found that the properties which are most easily explained are those connected with the gaseous state; the explanation of these properties in terms of the molecular structure of matter is the aim of the " Kinetic Theory of Gases."
Thus the estimation of kinetic energy is intimately affected by the choice of our base of measurement.
The energy is less than that of an ideal gas by the term npc. If we imagine that the defect of volume c is due to the formation of molecular aggregates consisting of two or more single molecules, and if the kinetic energy of translation of any one of these aggregates is equal to that of one of the single molecules, it is clear that some energy must be lost in co-aggregating, but that the proportion lost will be different for different types of molecules and also for different types of co-aggregation.
In the following table are given the values of the diameters of the molecules of six substances with which it is easy to experiment in the gaseous state, these values being calculated in different ways from formulae supplied by the kinetic theory.
B2' and this, by § 36, is also the ratio of the kinetic energy in the annular 4,1 interspace between the two cylinders to the kinetic energy of the liquid moving bodily inside r = b.
Originally impinged on that at rest is now represented by the energy, kinetic and potential, of the small motions of the individual molecules.
There is probably but little transformation of one form of kinetic energy into another in the plant.