"Many years ago," he wrote, "the late Rev. John Michell of this Society, contrived a method of determining the density of the earth, by rendering sensible the attraction of small quantities of matter; but, as he was engaged in other pursuits, he did not complete the apparatus till a short time before his death, and did not live to make any experiments with it. After his death the apparatus came to the Rev. Francis John Hyde Wollaston, Jacksonian Professor at Cambridge who, not having conveniences for making experiments with it in the manner he could wish, was so good as to give it to me.

"The apparatus is very simple; it consists of a wooden arm, six feet long, made so as to unite great strength with little weight. This arm is suspended in an horizontal position, by a slender wire 40 inches long, and to each extremity is hung a leaden ball about two inches in diameter; and the whole is inclosed in a narrow wooden case to defend it from the wind.

"As no more force is required to make this arm turn round on its centre than what is necessary to twist the suspending wire, it is plain that if the wire is sufficiently slender, the most minute force, such as the attraction of a leaden weight a few inches in diameter, will be sufficient to draw the arm sensibly aside. The weights which Mr. Michell intended to use were eight inches in diameter. One of these was to be placed on one side of the case, opposite to one of the balls, and as near it as could be conveniently done, and the other on the other side, opposite to the other ball, so that the attraction of both these weights would conspire in drawing the arm aside; and when its position as affected by those weights was ascertained, the weights were to be removed to the other side of the case, so as to draw the arm the contrary way and the position of the arm was to be again determined; and, consequent]y, half the difference of these positions would show how much the arm was drawn aside by the attraction of the weights.

"In order to determine from hence the density of the earth, it is necessary to ascertain what force is required to draw the arm aside through a given space. This Mr. Michell intended to do by putting the arm in motion, and observing the time of its vibrations, from which it may easily be computed."

Cavendish then explained how he modified the apparatus so that it could be operated from a remote position, since the mass of an observer's body at close range would attract the small lead balls enough to introduce significant error. He also recognized the necessity for adding a mechanism "to accurately remove the weights to the other side of the case so as to draw the arm the contrary way." He equipped the arm with a small mirror that reflected the image of an ivory scale to a telescope equipped with a cross hair at the observing position, a scheme that served both to keep the observer away from the apparatus and to amplify the apparent displacements of the balance arm.

In spite of these major modifications of Michell's apparatus, Cavendish reported a puzzling effect. Now and again, for no apparent reason, the small balls of lead spontaneously "wiggled around," a phenomenon that Cavendish finally attributed to air currents in the case. This may well have been the source of the disturbance. On the other hand, his weights may have been influenced by stray electrostatic charges because he did not shield them from the flame of an open candle with which he illuminated the scale. Nevertheless, Cavendish was able to calculate the density of the earth as being 5.448 times that of water, almost precisely in the middle of Newton's estimated range, and he assigned a value to G of 6.754 X 10^-8 (as expressed in centimeter-gram-second units).

An inexpensive version of the Cavendish apparatus, modified to include electrostatic shielding, has been constructed and used successfully for evaluating G within 1 per cent of the currently accepted value by Sam Epstein of Los Angeles.

Epstein writes: "Essentially the Cavendish apparatus is a sensitive torsion balance and a balance arm suspended at the middle by a slender steel wire. A small lead weight suspended from each end of the arm is attracted to large weights by mutual gravitation. The rotation of the arm varies with the intensity of attraction and is measured by the deflection of a light beam from a small mirror fixed to a pair of hooks that support the balance arm, as shown in the accompanying illustration below].

"The complete balance assembly including the case is supported from the ceiling by a length of 3/4-inch pipe that terminates in flanges. The structure is braced by a set of three guy wires. The torsion wire attaches at the upper end to a special fitting that rests in the top end of the pipe: a brass flange and cap nut drilled and threaded to receive a small rod with a hook at one end for attaching the wire. The flange is centered in the pipe by four adjusting screws and can be rotated for adjusting the orientation of the wire and balance arm. The lower end of the torsion wire attaches to an S-fitting that engages a pair of wire hooks to which a small mirror fixed.

"The remaining apparatus, including the large weights, is mounted on a wooden platform that in turn rests on legs in the form of inverted lag screws, which serve as leveling adjustments. The platform carries two metal pails filled with sand; these act as massive supports for a pair of turntables used to shift the position of the heavy weights. The turntables roll on casters of the nonswiveling type that ride on Masonite disks laminated to plywood atop the metal pails. A 3/8-inch pivot anchored in the center of each plywood disk engages a companion hole in the center of the turntables. The pivots keep the turntables centered when the position of the attracting weights is altered. The gravitational attraction of the mass represented by the sand in the metal pails acts vertically on the small weights and therefore exerts no turning force on the balance arm. The attractive force of the brackets from which the large weights are suspended, however, can influence the motion of the arm and must be minimized. This is accomplished by making the brackets in V-sections of thin aluminum sheeting and providing the necessary strength by means of wire braces.

"Lead is an ideal material for the weights because of its high density, cheapness and low melting point. (Any other substance can be used, however, because all matter exerts gravitational attraction.) Lead can be bought in most plumbing establishments, where arrangements might also be made for melting and casting the weights. The construction requires about 75 pounds of lead. Standard fruit-juice cans of 46-ounce capacity serve as forms for the large weights. Make the castings by embedding clean, dry cans almost to the top in moist sand or earth and filling them to the brim with molten lead. Lead shrinks on cooling, so that more than one pouring is necessary. The sand absorbs heat from the sheet metal and keeps their soldered joints from melting. If the cans are not completely dry, the molten lead may spatter and cause severe burns. A face shield, hat, gloves and a garment with long sleeves should be worn for safety. Do not melt more lead at one time than can be safely carried and poured. Let the castings cool for at least an hour before removing them from the sand or earth. Then drill and tap the lead to take an accurately centered screw to which suspension hooks are attached. Each completed unit should weigh 16.8 kilograms. Although their weight need not be precisely this amount, the weights of each must be the same and should be determined to within 100 grams. The weights are equalized by filing lead from the heavier of the pair. After the weights have been suspended from the aluminum brackets and mounted on the turntables, measure and record the distance between the suspension wires and the center of the turntable to the nearest millimeter.

"The small weights carried by the balance arm are much easier to make. Simply stand two 1/2-inch brass tubes with 1/32-inch walls on a smooth metal surface and pour in the lead. The tubes should be about four inches long to allow for shrinkage when the lead solidifies. When the lead is cool, saw an inch from the tops and square the ends with a file Each weight is then drilled and fitted with a small screw eye; these must be accurately centered because the suspension wires are presumed to pass through the center of mass of each weight and the distance between the centers of mass of the large and small weights must be subsequently determined by measuring the distance between the suspending wires. The small weights, of about 100 grams each, must be equal and measured to the nearest gram.

"The balance-arm assembly is enclosed by an airtight case that includes a pair of metal-lined tubes for shielding the small weights. These shields can be made of clear plastic sheeting rolled and cemented. The electrostatic shielding that lines the tubes can be made of copper fly screen and is electrically connected to the grounded pipe support and guy wires. One side of the case is hinged for a door and sealed with weather stripping.

"Any optical system that can project a reasonably sharp image of a cross hair can be used to indicate the position of the balance arm. I improvised a system that uses a clear Mazda lamp for the source and a common reading glass and spectacle lens for focusing the image of a fine wire on the screen, as illustrated in the accompanying drawing [Figure 8]. Doubtless a satisfactory substitute could be constructed around a 35 millimeter slide projector.

"When the balance has been assembled as illustrated, carefully center the balance arm on its suspension hooks and hang the small weights in their shields. Measure and record the distance between the small weights to the nearest millimeter. If the arm is properly centered, this will be twice the length of the lever arm through which the force of gravitational attraction is exerted on the torsion wire as a moment, or twisting force. Now turn the adjustment head at the top of the suspension pipe to position the balance arm longitudinally with respect to the case. If the torsion wire has been twisted during installation, several turns may be required to align the balance arm. The balance arm can be considered properly aligned when the. small weights remain centered in the shielding tubes following an adjustment. Because the balance is sensitive and oscillates at a frequency on the order of only about one cycle per 10 minutes the inexperienced operator may tend to overshoot in making this adjustment. Make the final adjustments in small increments and wait at least 10 minutes between adjustments for the arm to respond. After centering the arm, align the optical system and thereafter observe the movements of the balance by measuring the movements of the cross hair on the scale.

"In order to measure the force required to twist the torsion wire, first carefully mark on the arm the points at which the balance arm rests in the suspension hooks; then remove the balance arm and small weights and replace the assembly with a brass rod a 1/4-inch in diameter that equals the total weight of the balance-arm assembly to within a gram (The balance-arm assembly of my apparatus weighs 239 grams and the length of my 1/4 inch brass bar of equal weight is 35 inches.) Center the brass bar carefully on the suspension hooks and, after the image of the cross hair has come to rest on the scale, give the brass rod a slight impulse to set it in oscillation and then close the door of the case.

"The objective is to determine the period of oscillation of the balance when the torsion wire is loaded with the weight of the balance-arm assembly and from the period of oscillation to calculate the force, or moment, required to twist the wire. The brass rod is substituted for the balance-arm assembly because the subsequent calculation requires that the moment of inertia of the load be known, and it is much easier to compute the moment of inertia of a simple geometric form, such as a brass rod, than that of a complex shape such as the balance arm. The brass rod must oscillate smoothly in a horizontal plane without evidence of swinging, conical rotation or teeter-totter. The period of the system, or the time required for the cross hair to move from one extreme position to the other and return, must be measured to the nearest second. Time at least 10 oscillations and record the average. Remove the brass rod, reinstall the balance-arm assembly according to the guide marks previously placed on it and check to be sure that the small weights are properly centered and in precise axial alignment with the pivots of the turntables. If they are not so centered, align them by shifting the turntables. The centers of mass of the large and small weights must be at the same level, a position that is determined by the length of the suspension wires.

"In order to determine the force of gravitational attraction between large and small weights, first rotate the turntables until all four weights are in a straight line. This is the neutral position in which the large weights exert no force tending to rotate the balance arm. Record the position of the cross hair on the scale. (The cross hair may move erratically from time to time but excursions of less than a millimeter may be ignored.) Next, rotate the turntable 90 degrees so that the two heavy weights are on opposite sides of the case. Observe the cross hair. It will oscillate slowly for a time and come to rest about five millimeters from the previously recorded neutral position. Stay at least three meters from the balance assembly to minimize the influence of your own gravitational attraction on the small weights. Record the position of the cross hairs when the balance reaches equilibrium. Then rotate the turntables 180 degrees to shift the large weights to the other sides of the case and record the position at which the cross hair comes to rest at the side opposite the neutral position. Keep shifting the large weights 180 degrees until a good average value for the total deflection of the cross hair-the distance between the extreme points-has been recorded.

"Simple arithmetic then makes it possible to solve for the value of G and the 'weight' and density of the earth [see Figure 9].

"The attractive force exerted between the large and small cylinders of lead used in this apparatus does not equal so much as the weight of a small gnat's wing. (It amounts to about a hundred-billionth of an ounce.) Even a relatively crude torsion balance responds easily to forces of this order, however, as indicated by the closeness to the generally accepted values of the values derived by the experiment."

Bibliography

READINGS IN THE PHYSICAL SCIENCES edited by Harlow Shapley, Helen Wright and Samuel Rapport. Appleton-Century-Crofts, Inc., 1948.

A SOURCE BOOK IN PHYSICS. William Francis Magie. McGraw-Hill Book Co., Inc., 1935.

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