a body at its surface with 3600 times the force that it would if the body were 60 times as far from its center, or at the distance of the moon. Consequently the acceleration toward the earth of a body at the distance of the moon should be of the acceleration of gravity at the earth's surfaces. But the acceleration of the moon toward the earth may be computed from the formula where R is the radius of its orbit (240000 miles) in feet and P is its period of orbital revolution (27.322 days) in seconds. Substituting, we have where it is not The two results while the acceleration of gravity at the pole, affected by the earth's rotation is 32.26 ft. sec. therefore agree as exactly as could be expected with the data used. We conclude, then, that the motion of the moon and the fall m FIG. 78. of an apple or a stone are both according to the same law of gravitation. 153. Determination of the Gravitation Constant.-To determine the constant of gravitation the force of attraction between two known masses must actually be measured. The extreme minuteness of this attraction between small masses makes the exact determination of its value very difficult. It was first accomplished by Cavendish in 1798, using a form of apparatus indicated in figure 78. Two small spherical balls m and m' were mounted on the ends of a light crossbar which was suspended by a fine silver wire at its center. Two large spherical balls of lead M and M' weighing 158 kilograms apiece were suspended one near m and the other near m' but on opposite sides so that their attractions tended to turn the bar in the same direction. To protect the suspended bar from being disturbed by air currents it was entirely enclosed in a narrow box, its deflections being observed by a telescope through a glass window. Having observed the deflection of the bar when the large masses were in the positions shown, the masses were moved into the dotted positions where their attractions produced a deflection of the bar in the opposite direction. From these observations, combined with a measurement of the force required to turn the suspended bar through a given angle, the force of attraction between the masses was determined. Quite recently (1889) Boys, who had discovered the remarkable elastic properties of fine quartz fibers, devised an apparatus similar in principle to that of Cavendish, but much more compact, in which the small suspended masses were hung by a quartz fiber so fine that larger deflections and greater accuracy of measurement were attained. According to Boys' determination, C=6.6576 X 10-8 in C. G. S. units. That is, the attraction between two masses of one gram each concentrated at two points a centimeter apart, or of two spherical masses of one gram each with a distance of one centimeter between centers is .000,000,066,6 dyne. Two kilogram masses 10 centimeters between centers attract with a force of .000666 dyne, or about seven ten-millionths of a gram weight. The constant of gravitation has also been reckoned by estimating the mass contained in an isolated mountain and then measuring its deflecting effect on a plumb-line near its base. 154. Mass of the Earth.-When the gravitation constant is known the mass of the earth itself may readily be determined. For consider the earth as attracting a gram mass at its surface. The force of attraction is g dynes or approximately 980, and from the law of gravitation Take M = mass of the earth, F=980, m=1, r = radius of earth in centimeters, and C-6.66 × 10-8. All of these quantities are known except M, which may be calculated. In this way the mean density of the earth is found. to be 5.527, a result which is especially interesting as the average density of the surface materials of the earth is only about 2.5. 155. Mass of a Planet. So also the mass may be found of any planet having a satellite whose distance and period of orbital revolution about the planet can be observed. For the attraction between the m M 7.2 planet and satellite is expressed by C, while the centripetal force in case of a satellite of mass m and period P and moving in a circle of 4π2m r, and since it is the attraction which holds the satellite radius r, is P2 in its orbit we have In the equation the mass of the satellite m cancels, and as all the other quantities except M are known, the mass of the planet may be computed. 156. Significance of Kepler's Third Law.-Let M represent the mass of the sun, E the mass of the earth, r the mean distance between them, and P the period of the earth's revolution about the sun. Then, as in the last paragraph So also if J is the mass of some other planet, such as Jupiter, and if r1 and P1 represent its distance from the sun and period of revolution in its orbit, respectively, we have 1 MJC 42 Jr1 or MC r3 (2) If the constant of gravitation C has the same value in case of the sun and earth as it has in case of the sun and Jupiter, then which is precisely what Kepler's third law asserts to be true throughout the solar system. It is concluded, therefore, that the same gravitation constant holds everywhere throughout the solar system and probably throughout the material universe. 157. Variation of Gravity on Earth.-The force of gravity is not the same everywhere on the earth's surface. There are three circumstances which determine this variation, namely, the fact that the earth is not a sphere, its rotation, and the height above sea level of the given station. The earth is approximately an oblate spheroid having its polar radius less than its equatorial by 13.2 miles or 21.2 kilometers, and in consequence of this the value of g at the poles is greater than at the equator by 1.6 cm. sec., due to this cause alone. But there is another circumstance which still further reduces the value of g at the equator. The rotation of the earth affects both the direction and amount of the acceleration g. For the resultant attraction F of the earth on a gram of matter situated at A (Fig. 79) is directed toward the center O, but this resultant attraction serves both to supply the centripetal force f, which holds the mass on the earth as it rotates, and also the force which we call its weight which gives it acceleration g when dropped. The centripetal acceleration f is directed 4 π2 perpendicular to the polar axis and is equal to P2 r, where P is the period of rotation of the earth and r is the distance AB. The distance AB=R cos l where R is the radius of the earth and I the latitude of A. Evidently, then, f is a maximum at the equator and has zero value at the poles. Since F is the resultant of ƒ and g and is Pole directed toward the center of the earth, At latitude 45° the plumb-line points away from the center of the earth about 6.9 miles. At the equator the centrifugal force of a mass of 1 gram is 3.36 dynes. Hence the acceleration of gravity is less at the equator than at the poles by 3.36 cm. sec. on this score alone. The height of a place above sea level also affects the value of g, as it must diminish with the increase in distance from the center of the earth. If h represents the height in centimeters or in feet, the corresponding change in g is (.000003)h. Though on account of the irregular shape and distribution of the earth's mass the exact value of g at any place can be determined only by pendulum experiments, an approximate |