Gravitation Constant And Mean Density Of The Earth - Encyclopedia

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GRAVITATION CONSTANT AND MEAN DENSITY OF THE EARTH The law of gravitation states that two masses M 1 and M2, distant d from each other, are pulled together each with a force G. M 1 M 2 /d 2, where G is a constant for all kinds of matter - the gravitation constant. The acceleration of M2 towards M 1 or the force exerted on it by M 1 per unit of its mass is therefore GM1/d2. Astronomical observations of the accelerations of different planets towards the sun, or of different satellites towards the same primary, give us the most accurate confirmation of the distance part of the law. By comparing accelerations towards different bodies we obtain the ratios of the masses of those different bodies and, in so far as the ratios are consistent, we obtain confirmation of the mass part. But we only obtain the ratios of the masses to the mass of some one member of the system, say the earth. We do not find the mass in terms of grammes or pounds. In fact, astronomy gives us the product GM, but neither G nor M. For example, the acceleration of the earth towards the sun is about o6 cm/sec. 2 at a distance from it about 15 X 10 12 cm. The acceleration of the moon towards the earth is about 0.27 cm/sec. 2 at a distance from it about 4 X Io 10 cm. If S is the mass of the sun and E the mass of the earth we have o6 = GS/ (15 X 1012)2 and 0.27 =GE/ (4X 1010)2 giving us GS and GE, and the ratio SÆ = 30o,000 roughly; but we do not obtain either S or E in grammes, and we do not find G.

The aim of the experiments to be described here may be regarded either as the determination of the mass of the earth in grammes, most conveniently expressed by its mass= its volume, that is by its "mean density" A, or the determination of the "gravitation constant" G. Corresponding to these two aspects of the problem there are two modes of attack. Suppose that a body of mass m is suspended at the earth's surface where it is pulled with a force w vertically downwards by the earth - its weight. At the same time let it be pulled with a force p by a measurable mass M which may be a mountain, or some measurable part of the earth's surface layers, or an artificially prepared mass brought near m, and let the pull of M be the same as if it were concentrated at a distance d. The earth pull may be regarded as the same as if the earth were all concentrated at its centre, distant R.

Then w = G.17rR 3 Am/R 2 = G. 3 ?rR Om,. .. (I) and p= GMm /d2.. . (2) 0= 3M2 w 471-Rd, ' p' If then we can arrange to observe w/p we obtain A, the mean density of the earth.

But the same observations give us G also. For, putting m=w/g in (2), we get In the second mode of attack the pull p between two artificially prepared measured masses M 1, M2 is determined when they are a distance d apart, and since p= G. M 1 M 2 /d 2 we get at once G= pd 2 /M 1 M 2. But we can also deduce A. For putting w = mg in (I) we get A ' G ' ,rR' Experiments of the first class in which the pull of a known mass is compared with the pull of the earth maybe termed experiments on the mean density of the earth, while experiments of the second class in which the pull between two known masses is directly measured may be termed experiments on the gravitation constant.

We shall, however, adopt a slightly different classification for the purpose of describing methods of experiment, viz: I. Comparison of the earth pull on a body with the pull of a natural mass as in the Schiehallion experiment.

2. Determination of the attraction between two artificial masses as in Cavendish's experiment.

3. Comparison of the earth pull on a body with the pull of an artificial mass as in experiments with the common balance. It is interesting to note that the possibility of gravitation experiments of this kind was first considered by Newton, and in both of the forms (I) and (2). In the System of the World (3 rd ed., 1737, p. 40) he calculates that the deviation by a hemispherical mountain, of the earth's density and with radius 3 m., on a plumb-line at its side will be less than 2 minutes. He also calculates (though with an error in his arithmetic) the acceleration towards each other of two spheres each a foot in diameter and of the earth's density, and comes to the conclusion that in either case the effect is too small for measurement. In the Principia, bk. iii., prop. x., he makes a celebrated estimate that the earth's mean density is five or six times that of water. Adopting this estimate, the deviation by an actual mountain or the attraction of two terrestrial spheres would be of the orders calculated, and regarded by Newton as immeasurably small.

Whatever method is adopted the force to be measured is very minute. This may be realized if we here anticipate the results of the experiments, which show that in round numbers O = 5.5 and G =1/15,000,000 when the masses are in grammes and the distances in centimetres.

Newton's mountain, which would probably have density about A/2 would deviate the plumb-line not much more than half a minute. Two spheres 30 cm. in diameter (about 1 ft.) and of density II (about that of lead) just not touching would pull each other with a force rather less than 2 dynes, and their acceleration would be such that they would move into contact if starting I cm. apart in rather over 400 seconds.

From these examples it will be realized that in gravitation experiments extraordinary precautions must be adopted to eliminate disturbing forces which may easily rise to be comparable with the forces to be measured. We shall not attempt to give an account of these precautions, but only seek to set forth the general principles of the different experiments which have been made.

I. Comparison of the Earth Pull with that of a Natural Mass. Bouguer's Experiments. - The earliest experiments were made by Pierre Bouguer about 1740, and they are recorded in his Figure de la terre (1749). They were of two kinds. In the first he determined the length of the seconds pendulum, and thence g at different levels. Thus at Quito, which may be regarded as on a table-land 1466 toises (a toise is about 6.4 ft.) above sea-level, the seconds pendulum was less by 1/1331 than on the Isle of Inca at sea-level. But if there were no matter above the sea-level, the inverse square law would make the pendulum less by 1/1118 at the higher level. The value of g then at the higher level was greater than could be accounted for by the attraction of an earth ending at sea-level by the difference I/1118-1/1331= 1/6983, and this was put down to the attraction of the plateau 1466 toises high; or the attraction of the whole earth was 6983 times the attraction of the plateau. Using the rule, now known as "Young's rule," for the attraction of the plateau, Bouguer found that the density of the earth was 4.7 times that of the plateau, a result certainly much too large.

In the second kind of experiment he attempted to measure the horizontal pull of Chimborazo, a mountain about 20,000 ft. high, by the deflection of a plumb-line at a station on its south side. Fig. r shows the principle of the method. Suppose that two stations are fixed, one on the side of the mountain due south of the summit, and the other on the same latitude but some distance westward, away from the influence of the mountain. Suppose that at the second station a star is observed to pass the meridian, for simplicity we will say directly overhead, then a xi'. 13 By division G plumb-line will hang down exactly parallel to the observing telescope. If the mountain were away it would also hang parallel to the telescope at the first station when directed to the same star. But the mountain pulls the plumb-line towards it and the star appears to the north of the zenith and evidently mountain pull/earth pull= tan gent of angle of displacement of zenith.

Bouguer observed the meridian altitude of several stars at the two stations. There was still some deflection at the second station, a deflection which he {? ` estimated as 1/14 that at the w est of z first station, and he found on Fars} Station allowing for this that his observations gave a deflection of 8 seconds at the first. station. From the form and size of the mountain he found that if its density were that of the earth the deflection should be 103 seconds, or the earth was nearly 13 times as dense as the mountain, a result several times too large. But the work was carried on under enormous difficulties owing to the severity of the weather, and no exactness could be expected. The importance of the experiment lay in its proof that the method was possible.

Maskelyne's Experiment. 1774 Nevil Maskelyne (Phil. Trans., 1 775, p. 495) made an experiment on the deflection of the plumb-line by Schiehallion, a mountain in Perthshire, which has a short ridge nearly east and west, and sides sloping steeply on the north and south. He selected two stations on the same meridian, one on the north, the other on the south slope, and by means of a zenith sector, a telescope provided with a plumb-bob, he determined at each station the meridian zenith distances of a number of stars. From a survey of the district made in the years 1774-1776 the geographical difference of latitude between the two stations was found to be 42.94 seconds, and this would have been the difference in the meridian zenith difference of the same star at the two stations had the mountain been away. But at the north station the plumb-bob was pulled south and the zenith was deflected northwards, while at the south station the effect was reversed. Hence the angle between the zeniths, or the angle between the zenith distances of the same star at the two stations was greater than the geographical 42.94 seconds. The mean of the observations gave a difference of 54 . 2 seconds, or the double deflection of - the plumb-line was 54.2-42.94, say 11.26 seconds.

The computation of the attraction of the mountain on the supposition that its density was that of the earth was made by Charles Hutton from the results of the survey (Phil. Trans., 1778, p. 689), a computation carried out by ingenious and important;' methods. He found that the deflection should have been greater in the ratio 17804: 9933 say 9: 5, whence the density of the earth comes out at 9/5 that of the mountain. Hutton took the density of the mountain at 2.5, giving the mean density of the earth 4.5. A revision of the density of the mountain from a careful survey of the rocks composing it was made by John Playfair many years later (Phil. Trans., 1811, p. 347), and the density of the earth was given as lying between 4.5588 and 4.867.

Other experiments have been made on the attraction of mountains by Francesco Carlini (Milano Effem. Ast., 1824, p. 28) on Mt. Blanc in 1821, using the pendulum method after the manner of Bouguer, by Colonel Sir Henry James and Captain A. R. Clarke (Phil. Trans., 1856, p. 591), using the plumb-line deflection at Arthur's Seat, by T. C. Mendenhall (Amer. Jour. of Sci. xxi. p. 99), using the pendulum method on Fujiyama in Japan, and by E. D. Preston (U.S. Coast and Geod. Survey Rep., 18 93, p. 513) in Hawaii, using both methods.

In 1854 Sir G. B. Airy (Phil. Trans., 1856, p. 297) carried out at Harton pit near South Shields an experiment which he had attempted many years before in conjunction with W. Whewell and R. Sheepshanks at Dolcoath. This consisted in comparing gravity at the top and at the bottom of a mine by the swings of the same pendulum, and thence finding the ratio of the pull of the intervening strata to the pull of the whole earth. The principle of the method may be understood. by assuming that the earth consists of concentric spherical shells each homogeneous, the last of thickness h equal to the depth of the mine. Let the radius of the earth to the bottom of the mine be R, and the mean density up to that point be A. This will not differ appreciably from the mean density of the whole. Let the density of the strata of depth h be 8. Denoting the values of gravity above and below by g a and g b we have gb= G3?R? = G. 3 7rRO, and ga =G3(R+h)2 -}-G. 47r hS (since the attraction of a shell h thick on a point just outside it is G.47r(R+h) 2 h6/(R+h) 2 = G.47rh3).

Therefore ga = G.3,rRo (I - R + R ?) nearly, whence ga = 1 - 2h3h S, gb R -: -- R o 'A = 3/1/(_ 1 +21/+ga) R R Stations were chosen in the same vertical, one near the pit bank, another 1250 ft. below in a disused working, and a "comparison" clock was fixed at each station. A third clock was placed at the upper station connected by an electric circuit to the lower station. It gave an electric signal every 15 seconds by which the rates of the two comparison clocks could be accurately compared. Two "invariable" seconds pendulums were swung, one in front of the upper and the other in front of the lower comparison clock after the manner of Kater, and these invariables were interchanged at intervals. From continuous observations extending over three weeks and after applying various corrections Airy obtained g b /g a = 1.0000-5185 . Making corrections for the irregularity of the neighbouring strata he found A/5=2.6266. W. H. Miller made a careful determination of 8 from specimens of the strata, finding it 2.5. The final result taking into account the ellipticity and rotation of the earth is A = 6.565.

(Mitth. des K.U.K. Mil. Geog. Inst. zu Wien, ii., 1882, p. 77; 188 3, p. 59; vi., 1886, p. 97). R. von Sterneck repeated the mine experiment in 1882-1883 at the Adalbert shaft at Pribram in Bohemia and in 1885 at the Abraham shaft near Freiberg. He used two invariable halfseconds pendulums, one swung at the surface, the other below at the same time. The two were at intervals interchanged. Von Sterneck introduced a most important improvement by comparing the swings of the two invariables with the same clock which by an electric circuit gave a signal at each station each second. This eliminated clock rates. His method, of which it is not necessary to give the details here, began a new era in the determinations of local variations of gravity. The values which von Sterneck obtained for A were not consistent, but increased with the depth of the second station. This was probably due to local irregularities in the strata which could not be directly detected.

All the experiments to determine A by the attraction of natural masses are open to the serious objection that we cannot determine the distribution of density in the neighbourhood with any approach to accuracy. The experiments with artificial masses next to be described give much more consistent results, and the experiments with natural masses are now only of use and I S ' Station Due.South of Summit on Slope FIG. i. - Bouguer's Plumbline Experiment on the attraction of Chimborazo.

in showing the existence of irregularities in the earth's superficial strata when they give results deviating largely from the accepted value.

II. Determination of the Attraction between two Artificial Masses. Cavendish's Experiment (Phil. Trans., 1798, p. 469). - This celebrated experiment was planned by the Rev. John Mitchell. He completed an apparatus for it but did not live to begin work with it. After Mitchell's death the apparatus came into the possession of Henry Cavendish, who largely reconstructed it, but still adhered to Mitchell's plan, and in 1797-1798 he carried out the experiment. The essential feature of it consisted in the determination of the attraction of a lead sphere 12 in. in diameter on another lead sphere 2 in. in diameter, the distance between the centres being about 9 in., by means of a torsion balance. Fig. 2 shows how the experiment was carried out. A torsion rod hh 6 ft. long, tied from its ends to a vertical piece mg, was on account of the excellence of his methods. His work was undoubtedly very accurate for a pioneer experiment and has only really been improved upon within the last generation. Making various corrections of which is not necessary to give a description, the result obtained (after correcting a mistake first pointed out by F. Baily) is A = 5.448. In seeking the origin of the disturbed motion of the torsion rod Cavendish made a very important observation. He found that when the masses were left in one position for a time the attracted balls crept now in one direction, now in another, as if the attraction were varying. Ultimately he found that this was due to convection currents in the case containing the torsion rod, currents produced by temperature inequalities. When a large sphere was heated the ball near it tended to approach and when it was cooled the ball tended to recede. Convection currents constitute the chief disturbance and the chief source of error in all attempts to measure small forces in air at ordinary pressure.

Reich's Experiments (Versuche iiber die mittlere Dichtigkeit der Erde mittelst der Drehwage, Freiberg, 1838; "Neue Versuche mit der Drehwage," Leipzig Abh. Math. Phys. i., 1852, p. 383). - In 1838 F. Reich published an account of a repetition of the Cavendish experiment carried out on the same general lines, though with somewhat smaller apparatus. The chief differences consisted in the methods of measuring the times of vibration and the deflection, and the changes were hardly improvements. His result after revision was A= 5.49. In 1852 he published an account of further work giving as result A = 5.58. It is noteworthy that in his second paper he gives an account of experiments suggested by J. D. Forbes in which the deflection was not observed directly, but was deduced from observations of the time of vibration when the attracting masses were in different positions.

Let T i be the time of vibration when the masses are in one of the usual attracting positions. Let d be the distance between the centres of attracting mass and attracted ball, and S the distance through which the ball is pulled. If a is the half length of the torsion rod and 0 the deflection, 5=a0. Now let the attracting masses be put one at each end of the torsion rod with their centres in the line through the centres of the balls and d from them, and let T2 be the time of vibration. Then it is easy to show that a/d = Old= (T 1 - T 2) /(Ti +T2).

This gives a value of 0 which may be used in the formula. The experiments by this method were not consistent, and the mean result was 0=6.25.

Baily's Experiment (Memoirs of the Royal Astron. Soc. xiv.).- In 1841-1842 Francis Baily made a long series of determinations by Cavendish's method and with apparatus nearly of the same dimensions. The attracting masses were 12-in. lead spheres and as attracted balls he used various masses, lead, zinc, glass, ivory, platinum, hollow brass, and finally the torsion rod alone without balls. The suspension was also varied, sometimes consisting of a single wire, sometimes being bifilar. There were systematic errors running through Baily's work, which it is impossible now wholly to explain. These made the resulting value of A show a variation with the nature of the attracted masses and a variation with the temperature. His final result A = 5.6747 is not of value compared with later results.

Cornu and Baille's Experiment (Comptes rendus, lxxvi., 18 73, p. 954; lxxxvi., 1878, pp. 571, 699, zoos; xcvi., 1883, p. 1 493). - In 1870 MM. A. Cornu and J. Baille commenced an experiment by the Cavendish method which was never definitely completed, though valuable studies of the behaviour of the torsion apparatus were made. They purposely departed from the dimensions previously used. The torsion balls were of copper about zoo gm. each, the rod was 50 cm. long, and the suspending wire was 4 metres long. On each side of each ball was a hollow iron sphere. Two of these were filled with mercury weighing 12 kgm., the two spheres of mercury constituting the attracting masses. When the position of a mass was to be changed the mercury was pumped from the sphere on one side to that on the other side of a ball. To avoid counting time FIG. 2. - Cavendish's Apparatus.

h h, torsion rod hung by wire 1 g,; x,x, attracted balls hung from its ends; WW, attracting masses.

hung by a wire lg. From its ends depended two lead balls xx each 2 in. in diameter. The position of the rod was determined by a scale fixed near the end of the arm, the arm itself carrying a vernier moving along the scale. This was lighted by a lamp and viewed by a telescope T from the outside of the room containing the apparatus. The torsion balance was enclosed in a case and outside this two lead spheres WW each 12 in. in diameter hung from an arm which could turn round an axis Pp in the line of gl. Suppose that first the spheres are placed so that one is just in front of the right-hand ball x and the other is just behind the left-hand ball x. The two will conspire to pull the balls so that the right end of the rod moves forward. Now let the big spheres be moved round so that one is in front of the left ball and the other behind the right ball. The pulls are reversed and the right end moves backward. The angle between its two positions is (if we neglect cross attractions of right sphere on left ball and left sphere on right ball) four times as great as the deflection of the rod due to approach of one sphere to one ball.