to the ellipse which touches the sides of the triangle at their middle oints.
p The graphical methods of determining the moment of inertia of a plane system of particles with respect to any line in its plane may be briefly noticed. It appears from § 5 (fig. 31) that the linear moment of each particle about the line may be found by means of a funicular polygon. If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line. For if the line in question be the axis of y, the first process gives us the values of mx, and the second the value of E(mx.x) or E(mx'). The construction of a second funicular may be dis ensed with by the employment of a planimeter, as follows. In fig. 59 p is the line with respect to which moments are to be taken, and the masses of the respective particles are indicated by the
corresponding segments of a
line in the force-diagram,
drawn parallel to p. The
funicular ZABCD . . . corresponding
to any pole O is
constructed for a system of
forces acting parallel to p
through the positions of the
particles and proportional to
the respective masses; and its
successive sides are produced
K to meet p in the oints H, K,
L, M, . . As expizained in § 5,
the moment of the first particle
is represented on a cer-H/
tain scale by HK, that of the
second by KL, and so on.
The quadratic moment of the
first particle will then be
represented by twice the area
AHK, that 0 the second by
twice the area BKL, and so
on. The quadratic moment of the whole system is therefore represented by twice the area AHEDCBA. Since a quadratic moment is essentially positive, the various areas are to taken positive in all cases. If k be the radius of gyration about p we find k' =2 X area AHEDCBA X ON -E-ad,
where aB is the line in the force-diagram which represents the sum of the masses, and ON is the distance of the pole O from this line. If some of the particles lie on one side of p and some on the other, the quadratic moment of each set may be found, and the results added. This is illustrated in fig. 60, where the total quad;-atig I5 ' 2 <1
1.—""' " /
moment is represented by the sum of the shaded areas. It is seen that for a given direction of p this moment is least when p passes through the intersection X of the first and last sides of the funicular; i.¢. when p goes through the mass-centre of the given system; cf. equation (15).-PART
§ 12. Rectilinear M ation.-Let x denote the distance OP of a moving point P at time t from a fixed origin O on the line of motion, this distance being reckoned positive or negative according as it lies to one side or the other of O. At time t+6t let the point be at Q, and let OQ = x+5x. The mean velocity of the point in the interval 6t is 6x/61. The limiting value of this when 6t is infinitely small, viz. dx/dt, is adopted as the definition of the velocity at the instant t. Again, let n be the velocity at time t, u+5u that at time t-l-5t. The mean rate of increase of velocity, or the mean acceleration, in the interval Bt is then 5u/6t. The limiting value of this when 6t is infinitely small, viz., du/dt, is adopted as the definition of the acceleration at the instant t. Since 14 = dx/dt, the acceleration is also denoted by dix/dt2. It is often convenient to use the “ fiuxional ” notation for differential coefficients with respect to the time; thus the velocity may be represented by at and the acceleration by ii or There is another formula for the acceleration, in which u is regarded as a function du du dx du
of the position; thus; iii, = NEC. The relation between x and I in any particular case may be illustrated by meansof a curve constructed with t as abscissa and x as ordinate. This is called the curve of positions or space-time curve; its gradient represents the velocity. Such curves are often traced mechanically in acoustical and other experiments. A curve with t as abscissa and n as ordinate is called the curve of velocities or velocity-time curve. Its gradient represents the acceleration, and the area gudt) included between any two ordinates represents the space described in the interval between the corresponding instants (see fig. 62).
So far nothing has been said about the measurement of time. From the purely kinematic point of view, the t of our formulae may be any continuous independent variable, suggested (it may be) by some physical process. But from the dynamical standpoint it is obvious that equations which represent the facts correctly on one system of time-measurement might become seriously defective on another. It is found that for almost all purposes a system of measurement based ultimately on the earth's rotation is perfectly adequate. It is only when we come to consider such delicate questions as the influence of tidal friction that other standards become necessary. The most important conception in kinetics is that of “ inertia.” It is a matter of ordinary observation that different bodies acted on by the same force, or what is judged to be the same force, undergo different changes of velocity in equal times. In our ideal representation of natural phenomena this is allowed for by endowing each material particle with a suitable mass or inertiacoejicient m. The product mu of the mass into the velocity is called the momentum or (in Newton's phrase) the quantity of motion. On the Newtonian system the motion of a particle entirely uninfluenced by other bodies, when referred to a suitable base, would be rectilinear, with constant velocity. If the velocity changes, this is attributed to the action of force; and if we agree to measure the force (X) by the rate of change of momentum which it produces, we have the equation gl (mu) = X. (1)
From this point of view the equation is a mere truism, its real importance resting on the fact that by attributing suitable values to the masses m, and by making simple assumptions as to the value of X in each case, we are able to frame adequate representations of whole classes of phenomena as they actually occur. The question remains, of course, as to how far the measurement of force here implied is practically consistent with the gravitational method usually adopted in statics; this will be referred to presently.
The practical unit or standard of mass must, from the nature of the case, be the mass of some particular body, e.g. the imperial pound, or the kilogramme. In the “ C.G.S.” system a subdivision of the latter, viz. the gramme, is adopted, and is associated with the centimetre as the unit of length, and the mean solar second as the unit of time. The unit of force implied in (1) is that which produces unit momentum in unit time. On the C.G.S. system it is that force which acting on one gramme for one second produces a velocity of one centimetre per second; this unit is known as the dyne. Units of this kind are called absolute on account of their fundamental and invariable character as contrasted with gravitational units, which (as we shall see presently) vary somewhat with the locality at which the measurements are supposed to be made.
If we integrate the equation (1) with respect to t between the limits t, t' we- obtain
m“'- mu = If X dl. (2)
The time-integral on the right hand is called the impulse' of the force on the interval t'-t. The statement that the increase of