In the photograph above is a desktop model of a Cavendish balance. A mirror attached to the shaft of the pendulum bob reflects a laser beam to a scale to indicate the position of the pendulum bob. To perform an actual experiment with this device would take several hours, but it is useful as an aid when you wish to describe the experiment. If you would like to set the pendulum oscillating, you can partially turn in one of the arresting screws until the clamp just engages the cross bar of the pendulum, then turn it back out. (One screw is set underneath each ball of the pendulum bob. The one underneath the ball on the end facing the front of the table is clearly visible in the photograph.)

The Universal Law of Gravitation

From at least the time of the ancient Greeks, people sought to understand the motion of objects that fall to the earth’s surface after having been dropped from some height, and also the motion of the sun, moon and planets. Over time, great progress was made, most notably by the following people: Galileo Galilei determined that all objects, when dropped from the same height, fall to earth at the same rate (the acceleration is independent of mass, as long as drag is not significant). Nicolaus Copernicus pointed the way to a working model of planetary motion by devising a heliocentric model for the solar system. (The existing model, proposed by Ptolemy, was geocentric.) Tycho Brahe made meticulous measurements of the positions of the planets at various times, which resulted in a large set of accurate data. Johannes Kepler was able to use these data to deduce his three laws of planetary motion. It was not until the work of Isaac Newton, however, that anyone considered that the motion of falling objects on earth, and the motion of the planets, might be related.

In 1665, when Cambridge University closed because of the plague, Isaac Newton began to think about the possibility that the earth’s gravity acted not only near the earth’s surface, but extended much farther out. Guided by Kepler’s third law, he deduced that the force of gravity between two objects went in inverse proportion to the square of the distance between them. He then compared the acceleration of the moon in its orbit about earth to that of an object falling near the surface of the earth, and found that they were, indeed, in inverse proportion to the squares of their distances from (the center of) the earth. Long afterward, he wrote, as quoted in Westfall, Richard S., Isis 71(1), March, 1980, pp. 109-121:

And the same year I began to think of gravity extending to the orb of the Moon and (having found out how to estimate the force with which a globe revolving within a sphere presses the surface of the sphere) from Kepler’s rule. . . I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: And thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, and found them to answer pretty nearly.

“The same year” refers to 1665, which Newton mentions earlier in the section from which this is extracted. (Parts of the same passage are also quoted in Physics, Part One, by Resnick and Halliday, cited below.) Newton found that he could derive Kepler’s laws from his laws of motion and his law of gravitation. The law of universal gravitation, which Newton did not publish until roughly twenty years after he had formulated it, may be stated, as in College Physics by Sears and Zemansky (cited below):

Every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.

Mathematically, this is:

F ∝ mm′/r²

which we can also write:

F = Gmm′/r²

in which G is a constant called the gravitational constant.

The Cavendish Experiment

In the late eighteenth century, scientists argued over what the mean density of the earth was. By then, people had found that a plumb bob would be measurably deflected from vertical by a nearby mountain, and that by measuring this deflection and thus determining the attraction between the mountain and the pendulum, if they knew the density of the mountain, then by comparing the attraction between it and the plumb bob to that between the earth and the plumb bob, they could find the density of the earth. The densities of the mountains in question were not well known, however, so the results of these experiments were not accurate. The first person to make an accurate determination of the mean density of the earth was Henry Cavendish.

Some time in the mid-1780s, John Michell constructed a torsion balance with which he intended to determine the mean density of the earth. He was also hard at work trying to build a large telescope, but illness made this difficult. Cavendish and Michell were friends, and Cavendish knew of these projects. In a note to Michell regarding his struggles with them, he referred to the torsion balance experiment as “weighing the world.” Michell died before he could complete either apparatus, and the torsion balance went to Francis Wollaston. Since Wollaston did not have appropriate facilities for setting up the instrument, he gave it to Cavendish. Cavendish took the apparatus, set it up, made certain modifications, and proceeded to perform meticulous experiments, whose results he published in a paper titled, “Experiments to determine the Density of the Earth”.

The heart of the torsion balance was a pendulum that consisted of two lead balls, each about two inches in diameter and weighing 11262 grains (1.61 lb; 0.732 kg), hung about six feet apart on a (horizontal) rod, which was suspended from the middle by a thin wire 40 inches long. This torsion pendulum was enclosed in a wooden case with large glass windows. Suspended from a swing arm above the case, was a pair of two large spherical weights, each about 1 foot in diameter and weighing 2439000 grains (348 lb; 158 kg). These were suspended on rigid rods, and by rotating the swing arm, Cavendish could switch their positions with respect to the pendulum. To minimize temperature gradients, which would cause air currents, Cavendish isolated the apparatus by putting it in its own closed room, and arranged sytems by which he could operate it from an adjoining room. He read the scales that indicated the position of the pendulum by means of a telescope.

To perform the experiment, Cavendish set the weights in either the positive position (they pull the pendulum to higher scale readings) or the negative position (they pull the pendulum to lower scale readings), and observed a series of oscillations. He timed the periods, and he noted the locations of the extrema and midpoint of each oscillation. He then switched the position of the weights and repeated the experiment. He performed multiple experiments with the weights in both positions, and at least one experiment with them in the midway position.

To analyze the data, he found the force necessary to displace the pendulum by one division on the scale, which is inversely proportional to the square of the period, and expressed it as a ratio to the weight of the ball. Next, he calculated that each large weight was the equivalent of 10.64 spherical feet of water, and noted that its attraction on a particle at the center of the ball was to that of a spherical foot of water on an equal partical placed at its surface as 10.64 times the square of the inverse ratio of the distances (6/8.85)² to 1. (One spherical foot is the volume of a sphere whose diameter is one foot. 8.85 inches is the distance between centers of the balls and weights.) Since the geometry of the apparatus was not exactly right, Cavendish calculated a correction factor for the force (0.9779), by which he multiplied this result. Taking the mean density of the earth to the density of water as D to 1, with the mean diameter of the earth and the expression for the force required to displace the pendulum by one division, he derived an expression for the mean density of the earth in terms of the period and the number of scale divisions by which the pendulum is displaced (D = N²/(10683 B); B is the number of divisions).

Since the attraction of the weights on the balls varies with distance, their locations can effect the period of oscillation of the pendulum. He found that when they were moved from the positive position to the negative position or vice versa, no correction was necessary, but that when they were moved from the midway position to either the positive or the negative position, the correction was significant, and he included it in his data table. He also calculated several other corrections, but found them small enough that they were not necessary.

From the data that he collected, Cavendish calculated that the mean density of the earth was 5.48 times the density of water. This value is well within one percent of the currently accepted value of 5.51 times the density of water. (See, for example, Earth Fact Sheet (NASA).) Since the average density of the earth’s crust is considerably less than this, the core of the earth must be significantly denser than the crust. Cavendish compared his value to that obtained by Nevil Maskelyne, in his experiment near the Scottish mountain Schiehallion, of 4-1/2 times the density of water. He did not, however, calculate the value of the gravitational constant, G, or the mass of the earth. (See Henry Cavendish and the density of the earth .)

With Cavendish’s value for the mean density of the earth, and his stated radius of the earth, we can calculate the mass of the earth and then G. Cavendish gave the mean diameter of the earth as 41,800,000 feet, which gives a volume of 3.82 × 10²² ft³. The density of water is 62.4 lb/ft³, which multiplied by 5.48 gives the density of the earth as 342 lb/ft³. This multiplied by the volume gives the mass of the earth as 1.31 × 10²⁵ lb. Though we often use it this way, the pound is not a unit of mass. The unit of mass is the slug, and one pound is the force required to accelerate a mass of 1 slug at a rate of 1 ft/s². So the mass of the earth is 1.31 × 10²⁵ lb/32.2 ft/s² = 4.06 × 10²³ slugs.

The force of gravity on a mass m at the surface of the earth is F = mg, which also equals GMm/r², where M is the mass of the earth. Thus, g = GM/r², and G = gr²/M. So G = (32.2 ft/s²)(20,900,000 ft)²/4.06 × 10²³ slugs = 3.46 × 10^-8 ft³/slug·s². Since 1 lb = 1 slug·ft/s², this also equals 3.46 × 10^-8 lb·ft²/slug². This is close to the accepted value of 3.44 × 10^-8 lb·ft²/slug².

In SI units, the radius of 20,900,000 feet converts to 6,370,000 m, which gives the volume of the earth as 1.08 × 10²¹ m³. 5.48 g/cm³ equals 5,480 kg/m³, and together, these give 5.92 × 10²⁴ kg. G = (9.81 m/s²)(6,370,000 m)²/5.92 × 10²⁴ kg = 6.72 × 10^-11 m³/kg·s². Since 1 N = 1 kg·m/s², this also equals 6.72 × 10^-11 N·m²/kg². This is close to the accepted value of 6.67 × 10^-11 N·m²/kg². Both this value and the one above in English units, are well within one percent of the accepted value. (NIST gives 6.67330 × 10^-11 m³/kg·s², which equals 3.43928 × 10^-8 ft³/slug·s².)

If we now calculate the force between one ball and its weight (which is half the force on the pendulum), we have F = (3.44 × 10^-8 lb·ft²/slug²)(1.61 lb/32.2 (lb/slug))(348 lb/32.2 (lb/slug))/[8.85 in/(12 in/ft)]² = (3.44 × 10^-8 lb·ft²/slug²)(0.0500 slug)(10.8 slug)/(0.738 ft)² = 3.42 × 10^-8 lb! In SI units, it is F = (6.67 × 10^-11 N·m²/kg²)(0.732 kg)(158 kg)/[(8.85 in)(2.54 cm/in)(1 m/100 cm)]² = (6.67 × 10^-11 N·m²/kg²)(0.732 kg)(158 kg)/(0.225 m)² = 1.52 × 10^-7 N! This is indeed a tiny force, which shows just how delicate an experiment this is.

References:

1) Sears, Francis Weston and Zemansky, Mark W. College Physics, Third Edition (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1960), pp. 82-84.
2) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 81-82, 334-341.