Measurement Of Gravity

Type of physical science: Classical physics

Field of study: Mechanics

Gravity, the most dominant universal force, attractive in nature, affects all forms of matter and even energy in spite of its extreme weakness. Traditional Newtonian gravitational theory, adequate for navigation and general astronomical purposes, requires modification when great precision in measurement is necessary.

Overview

Gravity is the natural tendency of objects to move downward toward the Earth, and such objects are said to have weight. Gravity has been traditionally described as a field with every particle of matter as a source of a gravitational field. The intensity of this field is affected by the distance from and position on the Earth's surface and the local mass distribution in relation to the total mass of the Earth.

The gravitational field produces an attractive force between bodies, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This statement summarizes the law of universal gravitation, as formulated by Sir Isaac Newton in 1687. This law assumes that the masses are distributed symmetrically about a sphere of constant radius and uniform density. Actual gravity surveys, however, demonstrate that no mathematical formula has been found that describes exactly the gravitational field of the earth, which is complicated by irregularities in the topography and mass distribution, combined with a pronounced flattening of the earth at its poles, caused by rotation.

The force of gravity varies with position on the earth's surface. The acceleration of free-falling bodies caused by the force of gravity is determined experimentally as greatest at the poles and smallest at the equator. The value for the acceleration of gravity, g, for example, is only 9.782 meters per second per second (m/s²) in the Canal Zone of Panama, but is 9.825 meters per second per second in Greenland, which is closer to the North Pole. This value of g near the equator is lessened by a factor, which is the square of the velocity of a point on the Earth's surface divided by the radius of the earth (V²/R). Since points closer to the equator move with a greater velocity, the value of g will be smaller at the equator.

Gravitational acceleration diminishes with altitude, an object at the earth's surface and near the equator that would have a value of g equal to 9.83 meters per second per second would have that value drop to 8.70 meters per second per second at an altitude of 400 kilometers.

This decrease is observed because the effect of the gravitational force on acceleration follows an inverse square law; that is, the farther the object is from the center of earth, the less the acceleration. At twice the distance, for example, the force and resulting acceleration would be only one-fourth of the original amount.

The equation for the law of universal gravitation has a constant G, the gravitational constant that was not known at the time Newton formulated the law. The constant G is assumed equal for all conditions and locations on the earth and in the universe. The weight of an object W is equivalent to its mass m multiplied by the acceleration g (W = mg). If the weight is equated to the pull of gravity from the law of universal gravitation, than the gravitational constant G may be calculated directly. The value of G is found by squaring the radius of the earth, R, then multiplying the result by g and dividing by the mass of the earth, M (G = R²g/M). This result is a constant for a given location, which implies that g should be the same also for any location.

The technical problems of the measurement of G were solved by Henry Cavendish in 1798. Cavendish devised a sensitive torsion balance composed of a light rod supported at its center by a thin wire approximately 1 meter long, with lead balls about 5 centimeters in diameter placed at the ends of the rod. If a force were applied to each lead ball in opposing directions and at right angles to both the wire and rod, the wire is subjected to a rotation that may be measured as an angular displacement. Cavendish initially applied small forces, measuring the amount of twisting that resulted. Carefully shielding the experimental equipment from air currents, Cavendish placed two large lead balls about 20 centimeters in diameter nearly in contact with the small lead balls but on opposing sides. Gravitational force between both sets of balls caused a twist in the wire, and from the angle displaced by the wire, Cavendish was able to measure the forces between the large and small balls. The force turned out, as expected, to be very small, only one two-millionth of a newton. The value of G could then be calculated directly, since Cavendish now knew the force involved as well as the masses of the lead balls and their distance of separation. The results of Cavendish, as well as later determinations, have established that G has the same value, whatever the composition of the masses or the location; the constant is truly universal in nature.

The gravity pendulum has been used to measure the differences in gravitational force on Earth. Modern gravity pendulums are governed by the principle relating to the period of oscillation discovered by the Dutch scientist Christiaan Huygens. The period of the pendulum, as he noted, varies directly with the square root of the length and inversely with the square root of the local value for the acceleration of gravity g. Gravity pendulums are built nearly friction-free, supported on knife-edge jewel bearings, and swing in chambers from which air has been evacuated. The period of oscillation is timed with precise chronometers enabling determinations to within a few parts per million. Unfortunately, gravity pendulums are unwieldy and difficult to transport; consequently, now most gravity measurements are made with portable instruments called gravimeters.

Gravimeters make use of the principle of a spring balance--that the distortion or strain is directly proportional to the applied stress or force, provided that the measurements are made within the elastic limits of the material. A small quartz fiber is distorted in the local gravitational field at an observed station with results compared to a measured pendulum station. The readings are generally so precise that distortions to one part in 10 million can be recorded. Through development of such instruments, variations in gravity over large areas can now be measured.

Some instruments have been adapted to operate from aircraft in flight for aerial surveys of the earth and for use on surface vessels at sea in regions not suitable for gravity pendulums because of wave disturbances and motions.

A torsion balance employs an arrangement similar to a Cavendish balance, but as opposed to measuring the deflection produced by large masses, the period of oscillation is measured when the large masses are placed perpendicular to the equilibrium positions of the small masses and next when the large masses are rotated another 90 degrees. When the large masses are located at the first position, the period is less as a result of the additional restoring force. In the second position, the period is increased by the large masses, pulling the small ones away from the equilibrium position. The difference in the periods from both of these measurements gives the value of G. The advantage of this method is that the period may be measured more precisely than a corresponding deflection; the precision in error is estimated as 0.5 percent with this technique.

Although measurement of the gravitational constant G has improved, the value is not known with nearly the precision of other physical constants; the value itself is given to only four places of accuracy. The laboratory measurement of G is difficult because of the extremely small forces between the masses. Planetary sized objects are much larger, but the problem is not resolved because the product of G and the mass of the attracting planet both appear in the equation. Planetary observations alone cannot determine the individual values of G or mass.

A torsion balance may also be used for ascertaining the equivalence between inertial mass and gravitational mass, known also as the principle of equivalence. For these experiments, one body called the inertial mass is defined with respect to a standard mass of 1 kilogram. The body and the standard will either accelerate toward or away from each other. The gravitational mass of the body is defined in terms of this acceleration and the distance between the objects.

Experiments have tested a variety of materials, obtaining a ratio between the masses. Results indicate that various types of energy contribute to the inertial mass of a system to the same degree that they would contribute to the gravitational mass.

A gravity gradiometer is an instrument designed to measure local tidal fields. The instrument is portable and designed for use on an airplane or satellite and permits precise mapping of anomalies in the earth's gravitational field. The instrument, in the shape of a Greek cross, has four masses at the ends of each arm, which are held together at the center by a torsional spring. When pressed together, the arms oscillate with a frequency of 32 hertz (oscillations per second). If placed in a tidal field at right angles, the cross will be deformed.

Rotated with an angular frequency of 16 hertz in the reference frame of the cross, a tidal driving force would appear at 32 hertz. Since the oscillation frequency matches the natural vibration frequency of the arms, a resonance condition is established, producing large amplitudes. Very small tidal fields have been detected with this instrument.

Detectors of gravitational radiation, or waves, were first built in 1966 at the University of Maryland by Joseph Weber. This type of detector consists of a large aluminum cylinder placed inside a vacuum tank and suspended on a wire. The cylinder is supported on rubber blocks as an insulation to external mechanical vibrations. Any oscillations of the cylinder in the fundamental or longitudinal mode are detected by piezoelectric strain transducers bonded to the outside middle section. The ability of the device to detect radiation at resonance is termed its cross section, which turned out to be quite small. Limiting the ability to detect this type of radiation is the thermal motion of the individual molecules and oscillations of the cylinder that interfere with observations. Experimental results obtained by Weber and his group have not been duplicated elsewhere, casting doubt on the reliability of this technique.

More sensitive detectors are now under construction and involve strains in solid bodies employing resonance; they are more sensitive to radiation of a given frequency and reject all other frequencies.

Applications

Measurement of the gravitational field over Earth demonstrates that the field is not uniform. The Earth departs from being a perfect spheroid because of rotational effects and topographic variations. Regions that are topographically higher than a datum surface are located farther from the Earth's center and experience a smaller gravitational force. Other regions located below the surface, although closer to the Earth's center, may experience compensating effects from mass concentrations, thereby increasing the strength of the field over its expected value.

Gravity data indicate that the field is increased near mountain ranges because of the greater concentration of rock. Closer observations show that mountain masses do not deflect the field as much as expected if the mountain were a load resting on top of a uniform crust. If the mountain were merely a load on the rigid crust, the force of gravity (corrected for the effect of additional altitude) should be larger on top of the mountain than on the surrounding plains as a result of the increased gravitational pull of the mountain mass beneath the crust. Such observations led geophysicists to conclude that a rigid crust is not responsible for supporting the load of the mountains but is instead buoyed up by floating on a denser deformable interior. The interior of the earth must yield and be subject to lateral flow to compensate for loads on equal-size regions. Areas of depression in the crust, as oceanic trenches, show lower values of the gravitational field, as there is less mass near the surface.

Newtonian mechanics, traditionally used to describe the behavior of bodies at the surface of the earth, tend to break down outside the range of normal observable motion. The theory fails when gravitational fields become very intense near collapsed objects such as neutron stars or black holes.

In 1915, Albert Einstein completely changed the understanding of gravitation with his general theory of relativity. According to theory, gravity is not a force in the usual sense but is the result of the curvature of space-time. Bodies then follow the easiest course through space-time, which is manifested in the shape of their orbits. Einstein theorized that gravity may be explained by geometry. Mercury's orbit could not be explained adequately by ordinary mechanics but only by the warping of space near the Sun. Time warps at the Earth's surface may be detected by using very precise clocks.

Gravitational time dilation--the slowing down of clocks in a gravitational field--may be used as a direct test of the curvature of space-time. For these experiments, cesium-beam atomic clocks are used. Small frequency shifts are measured in the clocks placed in a potential and are calibrated against clocks at rest in a stationary gravitational field. The pulses of the clocks are monitored to rule out the possibility of a frequency loss during the light beam propagation. The clock tick rate is found to depend upon the strength of the gravitational field and, therefore, space-time geometry is dependent upon the gravitational field. The possibilities that strong tidal forces would have an effect on the clock may be ruled out because it is known that the atomic forces are stronger and resist tidal distortions.

Gravitational redshifts have been measured on light emitted from the atoms on stellar surfaces. It has been difficult to obtain reliable results for these measurements because of strong convection currents in the stellar photosphere masking the spectral lines, which are Doppler shifted by gaseous motion. Measurements made above the photosphere of the sun have given more definitive results. Gravitational redshifts should be very prominent in light emitted by white dwarf stars because these stars have about 1 solar mass contained in a much smaller radius than the sun and would consequently have very intense gravitational fields. Past observations on white dwarf stars has been difficult because these stars are so small that many of them cannot be observed.

Context

All of nature's events and activities can be explained in terms of four fundamental forces. In the historical context, gravity was the first of these four forces that was investigated scientifically. Although scientists have had an awareness of gravity and the direction in which it acts, the role of gravity as a force was not appreciated fully until Newton's law of universal gravitation was published. The importance of gravity is its universal nature--everything in the cosmos is affected by it and every particle of matter is a source of gravity. The force of gravity as observed is always attractive, tending to pull matter together.

One of the surprising facts concerning gravity as the dominant universal force is its extreme weakness. Gravity is so weak that physicists generally ignore its effects completely when dealing with masses on the level of the subatomic particle. Gravity's strength on the atomic scale is vastly overwhelmed by the nuclear and electrical forces at that distance.

The law of universal gravitation, which was adequate for more than two hundred years, was not effective in the twentieth century in explaining discrepancies in observations near very massive objects. In this respect, the law of universal gravitation conflicted with the relativity theory. In Newton's theory, gravitational force between two bodies should be transmitted instantaneously across space, but Einstein's theory rejects physical effects that travel faster than the speed of light. Gravitational fields around objects as massive as the Sun appear to distort space and time to a degree that is detectable. Observing stars near the Sun during solar eclipses indicates that they are not observed in their true positions; that is, the light from these stars has been bent or deflected noticeably toward the sun by its gravitational field. Black holes appear to distort severely the space and time surrounding them to unimaginable degrees. They are the final state of very massive stars that have collapsed into nothing with a gravity so overpowering that not even light can escape.

For all of its success, the theory of general relativity is at odds with the quantum theory, which describes subatomic particles on a statistical basis, and is at odds with the theory of superstrings, which treats subatomic particles as very tiny vibrating loops. Physicists are now searching for a more comprehensive theory of quantum gravity that perhaps will be more useful in mapping out the very early history of the universe very near the moment of creation.

Some physicists now postulate the existence of a fifth force in nature that may diminish the effectiveness of the gravitational force out to a limited range. Experiments performed in mines, for example, seem to show that the measured gravitational force does not agree with predicted values from theory. These observations as well as others, however, have not established conclusively the existence of a previously unknown repulsive force.

The nature of just how gravitation is transmitted at a distance has not been resolved as to whether it has a wave or particle nature, or both. If gravity has a particle nature, then this particle must be extremely tiny because gravity as a force is very weak.

Principal terms

CROSS SECTION: the fraction of the total energy of a wave that is absorbed per unit area

DOPPLER SHIFT: a frequency or wavelength change caused by the relative motion between the source of waves and an observer

GRAVITATIONAL CONSTANT, G: an empirically determined number independent of time, place, and composition

GRAVITATIONAL FIELD: the strength of the gravitational force that a given mass experiences per location

GRAVITATIONAL FORCE: an attraction that acts on all masses, causing weight and planetary motion

GRAVITATIONAL MASS: the ratio, expressed in kilograms, between the acceleration of a given mass and that of a standard mass, defined at a specific distance

INERTIAL MASS: acceleration ratio, as for gravitational mass, expressed in kilograms, excluding position or distance

NEWTON: the basic unit of measure for force or weight

PRINCIPLE OF EQUIVALENCE: for a specific gravitational field, all bodies fall with the same acceleration

TIDAL FIELD: refers to the detection of gravitational fields by tidal effects on Earth

Bibliography

Asimov, Isaac. THE HISTORY OF PHYSICS. New York: Walker and Company, 1985. Gravitation is presented in a format and style comprehensible to the lay reader of physics. The chapter on the gravitational constant discusses Cavendish's experiment, the method, and its significance.

Davies, Paul. SUPERFORCE. New York: Simon & Schuster, 1984. Written in a style that the average reader will appreciate, Davies presents the symmetry and beauty of the universe in terms of gravity, electromagnetism, and the weak and strong forces. The existence of a superforce is speculated upon and the possibility of a universe of eleven dimensions is discussed.

Hawking, Stephen W., and William Israel. THREE HUNDRED YEARS OF GRAVITATION. New York: Cambridge University Press, 1987. A comprehensive treatise spanning the development of gravitation from Newton to concepts of quantum gravity and time asymmetry. Additional chapters discuss gravitational radiation, gravitational interaction of cosmic strings, inflationary cosmology, quantum cosmology, and superstring unification. An important reference in gravitational physics.

Holton, Gerald. INTRODUCTION TO CONCEPTS AND THEORIES IN PHYSICAL SCIENCE. Princeton, N.J.: Princeton University Press, 1985. A unique perspective on the historical development of physical theories with emphasis on the geometric derivation of Newton's law of universal gravitation. A section is also included on the discovery of planets using this law. Photographs, diagrams, and tables abound with minimal use of mathematics.

Ohanian, Hans C. GRAVITATION AND SPACETIME. New York: W. W. Norton, 1976. Intended for the student with a background in physics. Discusses the emission and detection of gravitational waves, gravity in and around rotating and nonrotating black holes, curved space-time, gravitational time dilation, and tidal forces associated with gravitation.

Parker, Sybil P., ed. MCGRAW-HILL ENCYCLOPEDIA OF PHYSICS. New York: McGraw-Hill, 1983. An excellent reference for the nontechnical as well as for the technical reader. Topics under gravitation include Newton's law of universal gravitation, gravitational constant, mass and weight, gravity, gravitational potential energy, application and accuracy of Newtonian gravitation, relativistic theories, supergravity, and gravitational waves.

Cavendish's torsion balance experiment

Grand Unification Theories and Supersymmetry

Couplings and Resonances in Planetary Orbits