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It is often, however, convenient to use instead of the absolute unit ($ 188), the gravitation unit-which is simply the weight of unit mass. It must, of course, be specified in what latitude the observation is made. Thus, let W be the mass of a body in pounds; g the velocity it would acquire in falling for a second under the influence of its weight, or the earth's attraction diminished by centrifugal force; and P its weight measured in kinetic or absolute units. We have
186. According to the common system followed in modern mathematical treatises on dynamics, the unit of mass is g times the mass of the standard or unit weight. This definition, giving a varying and a very unnatural unit of mass, is exceedingly inconvenient: and its clumsiness is in great contrast to the clear and simple accuracy of the absolute method as stated above, to which we shall uniformly adhere, except when we wish, in describing results, to state forces in terms of the gravitation unit, as the vernacular of engineers in any locality. In reality, standards of weight are masses, not forces. It is better, though less usual, to call them standard masses than standard weights; as weight properly means force, and ambiguity is the worst fault of language. They are employed primarily in commerce for the purpose of measuring out a definite quantity of matter; not an amount of matter which shall be attracted by the earth with a given force.
Whereas a merchant, with a balance and a set of standard masses, would give his customers the same quantity of matter however the earth's attraction might vary, depending as he does upon masses for his measurement; another, using a spring balance, would defraud his customers in high latitudes, and himself in low, if his instrument (which depends on forces and not on masses) were correctly adjusted in London.
It is a secondary application of our standards of mass employ them for the measurement of forces, such as steam pressures, muscular power, etc. In all cases where great accuracy is required, the results obtained by such a method have to be reduced to what they would have been if the measurements of force had been made by means of a perfect spring-balance, graduated so as to indicate the forces of gravity on the standard masses in some conventional locality.
It is therefore very much simpler and better to take the imperial pound, or other national or international standard mass, as, for instance, the gramme (see Chapter IV.), as the unit of mass, and to derive from it, according to Newton's definition above, the unit of force.
187. The formula, deduced by Clairault from observation, and a certain theory regarding the figure and density of the earth, may be employed to calculate the most probable value of the apparent force of gravity, being the resultant of true gravitation and centrifugal force,
in any locality where no pendulum observation of sufficient accuracy has been made. This formula, with the two co-efficients which it involves, corrected according to modern pendulum observations, is as follows:
Let be the apparent force of gravity on a unit mass at the equator, and
g that in any latitude 1; then
g= G (1 + .00513 sino 1). The value of G, in terms of the absolute unit, to be explained immediately, is
32:088. According to this formula, therefore, polar gravity will be
= 32.088 x 1.00513 = 32.252. 188. As gravity does not furnish a definite standard, independent of locality, recourse must be had to something else. The principle of measurement indicated as above by Newton, but first introduced practically by Gauss in connection with national standard masses, furnishes us with what we want. According to this principle, the standard or unit force is that force which, acting on a national standard unit of matter during the unit of time, generates the unit of velocity.
This is known as Gauss' absolute unit; absolute, because it furnishes a standard force independent of the differing amounts of gravity at different localities.
189. The absolute unit depends on the unit of matter, the unit of time, and the unit of velocity; and as the unit of velocity depends on the unit of space and the unit of time, there is, in the definition, a single reference to mass and space, but a double reference to time; and this is a point that must be particularly attended to.
190. The unit of mass may be the British imperial pound or, better, the gramme; the unit of space the British standard foot or, better, the centimetre; and the unit of time the mean solar second.
We accordingly define the British absolute unit force as the force which, acting on one pound of matter for one second, generates a velocity of one foot per second.'
191. To render this standard intelligible, all that has to be done is to find how many absolute units will produce, in any particular locality, the same effect as the force of gravity on a given mass. The way to do this is to measure the effect of gravity in producing acceleration on a body unresisted in any way. The most accurate method is indirect, by means of the pendulum. The result of pendulum experiments made at Leith Fort, by Captain Kater, is, that the velocity acquired by a body falling unresisted for one second is at that place 32-207 feet per second. The preceding formula gives exactly 32.2, for the latitude 55° 35', which is approximately that of Edinburgh. The variation in the force of gravity for one degree of difference of latitude about the latitude of Edinburgh is only .0000832 of its own amount. It is nearly the same, though somewhat more, for every degree of latitude southwards, as far as the southern limits of the British Isles. On the
other hand, the variation per degree would be sensibly less, as far north as the Orkney and Shetland Isles. Hence the augmentation of gravity per degree from south to north throughout the British Isles is at most about 12000 of its whole amount in any locality. The average for the whole of Great Britain and Ireland differs certainly but little from 32-2. Our present application is, that the force of gravity at Edinburgh is 32.2 times the force which, acting on a pound for a second, would generate a velocity of one foot per second; in other words, 32.2 is the number of absolute units which measures the weight of a pound in this latitude. Thus, speaking very roughly, the British absolute unit of force is equal to the weight of about half an ounce.
192. Forces (since they involve only direction and magnitude) may be represented, as velocities are, by straight lines in their directions, and of lengths proportional to their magnitudes, respectively.
Also the laws of composition and resolution of any number of forces acting at the same point, are, as we shall show later (221), the same as those which we have already proved to hold for velocities; so that with the substitution of force for velocity, $$ 30, 31 are still true.
193. The Component of a force in any direction, sometimes called the Effective Component in that direction, is therefore found by multiplying the magnitude of the force by the cosine of the angle between the directions of the force and the component. The remaining component in this case is perpendicular to the other.
It is very generally convenient to resolve forces into components parallel to three lines at right angles to each other; each such resolution being effected by multiplying by the cosine of the angle concerned.
194. [If any number of points be placed in any positions in space, another can be found, such that its distance from any plane whatever is the mean of their distances from that plane; and if one or more of the given points be in motion, the velocity of the mean point perpendicular to the plane is the mean of the velocities of the others in the same direction.
If we take two points Ay, Ay, the middle point, P2, of the line joining them is obviously distant from any plane whatever by a quantity equal to the mean (in this case the half sum or difference as they are on the same or on opposite sides) of their distances from that plane. Hence twice the distance of P., from any plane is equal to the (algebraic) sum of the distances of A1, A, from it. Introducing a third point Ag, if we joir. AzP, and divide it in Pg so that A,Pz=2P, P2, three times the distance of P, from any plane is equal to the sum of the distance of Az.and twice that of P, from the same plane: i. e. to the sum of the distances of A1, A2, and Ag from it; or its distance is the mean of theirs. And so on for any number of points. The proof is exceedingly simple. Thus suppose P.n to be the mean of the first n points A2, A2,....A,; and An+1 any.
n+1 : RA
other point. Divide AntPn in P, so that Anti Pn+1=nPn+1P, Then from P., Pn+1, Ant1, draw perpendiculars to any plane, meeting it in S, T, V.
Pn+z Draw PnQR parallel to STV. Then
:: PnPn+1: PnAn+1::1:n+1. Hence n +1QP+=RAn+r Add to these n+1QT and its equal nP,S+RV, and we get S T
n+1(QPn+1+QT)=nPnS+RV + R An+19
i.e. n+1 Pm+T=nP, S+An+1V. In words, n+1 times the distance of P, from any plane is equal to that of A with n times that of P, i.e. equal to the sum of the distances of A4, A2,....An+1 from the plane. Thus if the proposition be true for any number of points, it is true for one more e-and so on
—but it is obviously true for two, hence for three, and therefore generally. And it is obvious that the order in which the points are taken is immaterial.
As the distance of this point from any plane is the mean of the distances of the given ones, the rate of increase of that distance, i.e. the velocity perpendicular to the plane, must be the mean of the rates of increase of their distances-i.e. the mean of their velocities perpendicular to the plane.]
195. The Centre of Inertia or Mass of a system of equal material points (whether connected with one another or not) is the point whose distance is equal to their average distance from any plane whatever ( 194).
A group of material points of unequal masses may always be imagined as composed of a greater number of equal material points, because we may imagine the given material points divided into different numbers of
In any case in which the magnitudes of the given masses are incommensurable, we may approach as near as we please to a rigorous fulfilment of the preceding statement, by making the parts into which we divide them sufficiently small.
On this understanding the preceding definition may be applied to define the centre of inertia of a system of material points, whether given equal or not. The result is equivalent to this :
The centre of inertia of any system of material points whatever (whether rigidly connected with one another, or connected in any way, or quite detached), is a point whose distance from any plane is equal to the sum of the products of each mass into its distance from the same plane divided by the sum of the masses.
We also see, from the proposition stated above, that a point whose distance from three rectangular planes fulfils this condition, must fulfil this condition also for every other plane.
· y =
The co-ordinates of the centre of inertia, of masses W2, W2, etc., at points (,, Yu, 2), (x, y, z), etc., are given by the following formulae :W.X + W.X2 +etc. Σuυ Ewy
wz Wi+w, +etc. Σ
Ew These formulae are perfectly general, and can easily be put into the particular shape required for any given case.
The Centre of Inertia or Mass is thus a perfectly definite point in every body, or group of bodies. The term Centre of Gravity is often very inconveniently used for it. The theory of the resultant action of gravity, which will be given under Abstract Dynamics, shows that, except in a definite class of distributions of matter, there is no fixed point which can properly be called the Centre of Gravity of a rigid body. In ordinary cases of terrestrial gravitation, however, an approximate solution is available, according to which, in common parlance, the term Centre of Gravity may be used as equivalent to Centre of Inertia ; but it must be carefully remembered that the fundamental ideas involved in the two definitions are essentially different.
The second proposition in § 194 may now evidently be stated thus :—The sum of the momenta of the parts of the system in any direction is equal to the momentum in the same direction of a mass equal to the sum of the masses moving with a velocity equal to the velocity of the centre of inertia. 196. The mean of the squares of the distances of the centre of
inertia, I, from each of the points of a system is less than the mean of the squares of the distance of any other point, O, from them by the square of 01. Hence the centre of inertia is the point the sum of the squares of whose
distances from any given points is a minimum. For OP=01? +1P? +20110, P being any one of the points and PQ perpendicular to 01. But IQ is the distance of P from a plane through I perpendicular to 0. Hence the mean of all distances, IQ, is zero. Hence
(mean of IP2)=(mean of OP)-01?, which is the proposition.
197. Again, the mean of the squares of the distances of the points of the system from any line, exceeds the corresponding quantity for a parallel line through the centre of inertia, by the square of the distance between these lines.
For in the above figure, let the plane of the paper represent a plane through I perpendicular to these lines, O the point in which the first line meets it, P the point in which it is met by a parallel line through any one of the points of the system. Draw, as before, PQ perpendicular to 01. Then PI is the perpendicular distance, from the axis through I, of the point of the system considered, PO is its distance from the first axis, of the distance between the two