Earth's Mass - EnchantedLearning.com Geologic Time Chart HOW IS THE MASS OF THE EARTH DETERMINED? How is the mass of the Earth determined? Newton , Galileo, Henry Cavendish, and Eratosthenes contributed to this amazing calculation. This calculation is done using Newton's Law of Gravity, which formulates the attractive force (gravitational force) that two masses exert on each other: F=GmM/r2 In Newton's equation, F is the gravitational force, G is a constant of proportionality, M and m are the two masses exerting the forces, and r is the distance between the two objects. G was calculated by Henry Cavendish in 1798, and was determined to be 6.67 x 10-11 m3/(kg sec2). Also needed is Newton's second law of motion, F=ma, where F is the force applied to an object, m is the mass of the object, and a is its acceleration due to the force. Galileo determined that the acceleration due to the force of gravity of Earth was a constant equal to 9.8 m/sec2 near the surface of the Earth.

Lastly, you need to know the radius of the Earth; this was first calculated by the Greek Eratosthenes thousands of years ago (by comparing shadows in wells during the summer solstice about 230 B.C.). CALCULATING THE MASS OF THE EARTH 1. F = GmM/r2 = ma, where F is the gravitational force, G is the gravitational constant, M is the mass of the Earth, r is the radius of the Earth, and m is the mass of another object (near the surface of the Earth). 2. GM/r2= a (The m's canceled out.) Now solve for M, the mass of the Earth. 3. M = ar2/G, where a = 9.8m/sec2, r = 6.4 x 106m, and G = 6.67 x 10-11m3/(kg sec2). 4.

M = 9.8 x (6.4 x 106)2/(6.67 x 10-11) = 6.0 x 1024 kg THE PEOPLE BEHIND THE CALCULATION CAVENDISH, HENRY Henry Cavendish (1731-1810) was an English chemist and physicist. Cavendish discovered that hydrogen gas was a substance different from ordinary air (whose components he analyzed), described the composition of water (hydrogen and oxygen) and other important works. Cavendish was the first person to determine Newton's gravitational constant gravitational constant and accurately measured of the Earth's mass and density. ERATOSTHENES Eratosthenes (276-194 BC) was a Greek scholar who was the first person to determine the circumference of the Earth. He compared the midsummer's noon shadow in deep wells in Syene (now Aswan on the Nile in Egypt) and Alexandria. He properly assumed that the Sun's rays are virtually parallel (since the Sun is so far away ).

Knowing the distance between the two locations, he calculated the circumference of the Earth to be 250,000 stadia. Exactly how long a stasia is is unknown, so his accuracy is uncertain. He also accurately measured the tilt of the Earth's axis and the distance to the sun and moon, and devised a method for determining the prime numbers up to a given number (the Sieve of Eratosthenes). Eratosthenes made numerous contributions to the sciences and arts in many fields, including geography, mathematics, astronomy, chronography (calendars), music, and literature. Eratosthenes was a brilliant all-around scholar; although not the top expert on any topic, he was well-versed in all subjects, and therefore nicknamed "Beta" (which is the second letter of the Greek alphabet). GALILEI, GALILEO Galileo Galilei (1564-1642) was an Italian mathematician, astronomer, and physicist. He was the first person to use a telescope to observe the skies (in 1609). Galileo discovered four moons of Jupiter , observed the phases of Venus , studied sunspots, and discovered many other important phenomena.

After publishing the many discoveries he made using his telescope, including the motion of the Earth around the Sun (the Copernican System ), Galileo was accused of heresy by the Inquisition (in 1633). gravitation facts, information, pictures | Encyclopedia.com articles about gravitation Aristotle 's Model Greek philosophers of the period from the sixth to the fourth century b.c. grappled with a variety of questions concerning the fundamental nature of physical reality, and the forces that bind that reality into a whole. Among the most advanced thinkers of that period was Democritus (c. 460-370 b.c.), who put forth a hypothesis many thousands of years ahead of its time: that all of matter interacts at the atomic level. Aristotle (384-322 b.c.), however, rejected the explanation offered by Democritus, an unfortunate circumstance given the fact that the great philosopher exerted an incalculable influence on the development of scientific thought. Aristotle's contributions to the advancement of the sciences were many and varied, yet his influence in physics was at least as harmful as it was beneficial.

Furthermore, the fact that intellectual progress began slowing after several fruitful centuries of development in Greece only compounded the error. By the time civilization had reached the Middle Ages (c. 500 a.d.) the Aristotelian model of physical reality had been firmly established, and an entire millennium passed before it was successfully challenged. Wrong though it was in virtually all particulars, the Aristotelian system offered a comforting symmetry amid the troubled centuries of the early medieval period. It must have been reassuring indeed to believe that the physical universe was as simple as the world of human affairs was complex. According to this neat model, all materials on Earth consisted of four elements: earth , water, air, and fire. Each element had its natural place. Hence, earth was always the lowest, and in some places, earth was covered by water. Water must then be higher, but clearly air was higher still, since it covered earth and water. Highest of all was fire, whose natural place was in the skies above the air. Reflecting these concentric circles were the orbits of the Sun, the Moon, and the five known planets.

Their orbital paths, in the Aristotelian model of the universe—a model developed to a great degree by the astronomer Ptolemy (c. 100-170)—were actually spheres that revolved around Earth with clockwork precision. On Earth, according to the Aristotelian model, objects tended to fall toward the ground in accordance with the admixtures of differing elements they contained. A rock, for instance, was mostly earth, and hence it sought its own level, the lowest of all four elements. For the same reason, a burning fire rose, seeking the heights that were fire's natural domain. It followed from this that an object falls faster or slower, depending on the relative mixtures of elements in it: or, to use more modern terms, the heavier the object, the faster it falls. Galileo Takes Up the Copernican Challenge Over the centuries, a small but significant body of scientists and philosophers—each working independent from the other but building on the ideas of his predecessors—slowly began chipping away at the Aristotelian framework.

The pivotal challenge came in the early part of the century, and the thinker who put it forward was not a physicist but an astronomer: Nicolaus Copernicus (1473-1543.) Based on his study of the planets, Copernicus offered an entirely new model of the universe, one that placed the Sun and not Earth at its center. He was not the first to offer such an idea: half a century after Aristotle's death, Aristarchus (fl. 270 b.c.) had a similar idea, but Ptolemy rejected his heliocentric (Sun-centered) model in favor of the geocentric or Earth-centered one. In subsequent centuries, no less a political authority than the Catholic Church gave its approval to the Ptolemaic system. This system seemed to fit well with a literal interpretation of biblical passages concerning God 's relationship with man, and man's relationship to the cosmos; hence, the heliocentric model of Copernicus constituted an offense to morality.

For this reason, Copernicus was hesitant to defend his ideas publicly, yet these concepts found their way into the consciousness of European thinkers, causing a paradigm shift so fundamental that it has been dubbed "the Copernican Revolution." Still, Copernicus offered no explanation as to why the planets behaved as they did: hence, the true leader of the Copernican Revolution was not Copernicus himself but Galileo Galilei (1564-1642.) Initially, Galileo set out to study and defend the ideas of Copernicus through astronomy, but soon the Church forced him to recant. It is said that after issuing a statement in which he refuted the proposition that Earth moves—a direct attack on the static harmony of the Aristotelian/Ptolemaic model—he protested in private:" E pur si muove!" (But it does move!) Placed under house arrest by authorities from Rome , he turned his attention to an effort that, ironically, struck the fatal blow against the old model of the cosmos: a proof of the Copernican system according to the laws of physics. GRAVITATIONAL ACCELERATION.

In the process of defending Copernicus, Galileo actually inaugurated the modern history of physics as a science (as opposed to what it had been during the Middle Ages: a nest of suppositions masquerading as knowledge). Specifically, Galileo set out to test the hypothesis that objects fall as they do, not because of their weight, but as a consequence of gravitational force. If this were so, the acceleration of falling bodies would have to be the same, regardless of weight. Of course, it was clear that a stone fell faster than a feather, but Galileo reasoned that this was a result of factors other than weight, and later investigations confirmed that air resistance and friction, not weight, are responsible for this difference. On the other hand, if one drops two objects that have similar air resistance but differing weight—say, a large stone and a smaller one—they fall at almost exactly the same rate. To test this directly, however, would have been difficult for Galileo: stones fall so fast that, even if dropped from a great height, they would hit the ground too soon for their rate of fall to be tested with the instruments then available.

Instead, Galileo used the motion of a pendulum, and the behavior of objects rolling or sliding down inclined planes, as his models. On the basis of his observations, he concluded that all bodies are subject to a uniform rate of gravitational acceleration, later calibrated at 32 ft (9.8 m) per second. What this means is that for every 32 ft an object falls, it is accelerating at a rate of 32 ft per second as well; hence, after 2 seconds, it falls at the rate of 64 ft (19.6 m) per second; after 3 seconds, at 96 ft (29.4 m) per second, and so on. Newton Discovers the Principle of Gravity Building on the work of his distinguished fore-bear, Sir Isaac Newton (1642-1727)—who, incidentally, was born the same year Galileo died—developed a paradigm for gravitation that, even today, explains the behavior of objects in virtually all situations throughout the universe. Indeed, the Newtonian model reigned until the early twentieth century, when Albert Einstein (1879-1955) challenged it on certain specifics.

Even so, Einstein's relativity did not disprove the Newtonian system as Copernicus and Galileo disproved Aristotle's and Ptolemy's theories; rather, it showed the limitations of Newtonian mechanics for describing the behavior of certain objects and phenomena. However, in the ordinary world of day-to-day experience—the world in which stones drop and heavy objects are hard to lift—the Newtonian system still offers the key to how and why things work as they do. This is particularly the case with regard to gravity and gravitation. Like Galileo, Newton began in part with the aim of testing hypotheses put forth by an astronomer—in this case Johannes Kepler (1571-1630). In the early years of the seventeenth century, Kepler published his three laws of planetary motion, which together identified the elliptical (oval-shaped) path of the planets around the Sun. Kepler had discovered a mathematical relationship that connected the distances of the planets from the Sun to the period of their revolution around it. Like Galileo with Copernicus, Newton sought to generalize these principles to explain, not only how the planets moved, but also why they did.

Almost everyone has heard the story of Newton and the apple—specifically, that while he was sitting under an apple tree, a falling apple struck him on the head, spurring in him a great intuitive leap that led him to form his theory of gravitation. One contemporary biographer, William Stukely, wrote that he and Newton were sitting in a garden under some apple trees when Newton told him that "…he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion'd by the fall of an apple, as he sat in a contemplative mood. Why should that apple always descend perpendicularly to the ground, he thought to himself. Why should it not go sideways or upwards, but constantly to the earth's centre?" The tale of Newton and the apple has become a celebrated myth, rather like that of George Washington and the cherry tree. It is an embellishment of actual events: Newton never said that an apple hit him on the head, just that he was thinking about the way that apples fell. Yet the story has become symbolic of the creative intellectual process that occurs when a thinker makes a vast intuitive leap in a matter of moments.

Of course, Newton had spent many years contemplating these ideas, and their development required great effort. What is important is that he brought together the best work of his predecessors, yet transcended all that had gone before—and in the process, forged a model that explained a great deal about how the universe functions. The result was his Philosophiae Naturalis Principia Mathematica, or "Mathematical Principles of Natural Philosophy." Published in 1687, the book—usually referred to simply as the Principia —was one of the most influential works ever written. In it, Newton presented his three laws of motion, as well as his law of universal gravitation. The latter stated that every object in the universe attracts every other one with a force proportional to the masses of each, and inversely proportional to the square of the distance between them. This statement requires some clarification with regard to its particulars, after which it will be reintroduced as a mathematical formula. MASS AND FORCE. The three laws of motion are a subject unto themselves, covered elsewhere in this volume. However, in order to understand gravitation, it is necessary to understand at least a few rudimentary concepts relating to them.

The first law identifies inertia as the tendency of an object in motion to remain in motion, and of an object at rest to remain at rest. Inertia is measured by mass, which—as the second law states—is a component of force. Specifically, the second law of motion states that force is equal to mass multiplied by acceleration. This means that there is an inverse relationship between mass and acceleration: if force remains constant and one of these factors increases, the other must decrease—a situation that will be discussed in some depth below. Also, as a result of Newton's second law, it is possible to define weight scientifically. People typically assume that mass and weight are the same, and indeed they are on Earth—or at least, they are close enough to be treated as comparable factors. Thus, tables of weights and measures show that a kilogram, the metric unit of mass, is equal to 2.2 pounds, the latter being the principal unit of weight in the British system. In fact, this is—if not a case of comparing to apples to oranges—certainly an instance of comparing apples to apple pies.

In this instance, the kilogram is the "apple" (a fitting Newtonian metaphor!) and the pound the "apple pie." Just as an apple pie contains apples, but other things as well, the pound as a unit of force contains an additional factor, acceleration, not included in the kilo. BRITISH VS. SI UNITS. Physicists universally prefer the metric system, which is known in the scientific community as SI (an abbreviation of the French Système International d'Unités —that is, "International System of Units"). Not only is SI much more convenient to use, due to the fact that it is based on units of 10; but in discussing gravitation, the unequal relationship between kilograms and pounds makes conversion to British units a tedious and ultimately useless task. Though Americans prefer the British system to SI, and are much more familiar with pounds than with kilos, the British unit of mass—called the slug—is hardly a household word. By contrast, scientists make regular use of the SI unit of force—named, appropriately enough, the newton.

In the metric system, a newton (N) is the amount of force required to accelerate 1 kilogram of mass by 1 meter per second squared (m/s2) Due to the simplicity of using SI over the British system, certain aspects of the discussion below will be presented purely in terms of SI. Where appropriate, however, conversion to British units will be offered. CALCULATING GRAVITATIONAL FORCE. The law of universal gravitation can be stated as a formula for calculating the gravitational attraction between two objects of a certain mass, m 1 AND M 2: F grav = G · (m 1M 2)/R2. F grav is gravitational force, and r 2 the square of the distance between m 1 and m 2. As for G, in Newton's time the value of this number was unknown. Newton was aware simply that it represented a very small quantity: without it, (m 1m 2)/r 2 could be quite sizeable for objects of relatively great mass separated by a relatively small distance. When multiplied by this very small number, however, the gravitational attraction would be revealed to be very small as well.

Only in 1798, more than a century after Newton's writing, did English physicist Henry Cavendish (1731-1810) calculate the value of G. As to how Cavendish derived the figure, that is an exceedingly complex subject far beyond the scope of the present discussion. Even to identify G as a number is a challenging task. First of all, it is a unit of force multiplied by squared area, then divided by squared mass: in other words, it is expressed in terms of (N · m2)/kg2, where N stands for newtons, m for meters, and kg for kilograms. Nor is the coefficient, or numerical value, of G a whole number such as 1. A figure as large as 1, in fact, is astronomically huge compared to G, whose coefficient is 6.67 · 10−11—in other words, 0.0000000000667. REAL-LIFE APPLICATIONS Weight vs. Mass Before discussing the significance of the gravitational constant, however, at this point it is appropriate to address a few issues that were raised earlier—issues involving mass and weight.

In many ways, understanding these properties from the framework of physics requires setting aside everyday notions. First of all, why the distinction between weight and mass? People are so accustomed to converting pounds to kilos on Earth that the difference is difficult to comprehend, but if one considers the relation of mass and weight in outer space, the distinction becomes much clearer. Mass is the same throughout the universe, making it a much more fundamental characteristic—and hence, physicists typically speak in terms of mass rather than weight. Weight, on the other hand, differs according to the gravitational pull of the nearest large body. On Earth, a person weighs a certain amount, but on the Moon, this weight is much less, because the Moon possesses less mass than Earth. Therefore, in accordance with Newton's formula for universal gravitation, it exerts less gravitational pull. By contrast, if one were on Jupiter, it would be almost impossible even to stand up, because the pull of gravity on that planet—with its greater mass—would be vastly greater than on Earth. It should be noted that mass is not at all a function of size: Jupiter does have a greater mass than Earth, but not because it is bigger.

Mass, as noted earlier, is purely a measure of inertia: the more resistant an object is to a change in its velocity, the greater its mass. This in itself yields some results that seem difficult to understand as long as one remains wedded to the concept—true enough on Earth—that weight and mass are identical. A person might weigh less on the Moon, but it would be just as difficult to move that person from a resting position as it would be to do so on Earth. This is because the person's mass, and hence his or her resistance to inertia, has not changed. Again, this is a mentally challenging concept: is not lifting a person, which implies upward acceleration, not an attempt to counteract their inertia when standing still? Does it not follow that their mass has changed? Understanding the distinction requires a greater clarification of the relationship between mass, gravity, and weight. F = ma. Newton's second law of motion, stated earlier, shows that force is equal to mass multiplied by acceleration, or in shorthand form, F = ma. To reiterate a point already made, if one assumes that force is constant, then mass and acceleration must have an inverse relationship. This can be illustrated by performing a simple experiment.

Suppose one were to apply a certain amount of force to an empty shopping cart. Assuming the floor had just enough friction to allow movement, it would be easy for almost anyone to accelerate the shopping cart. Now assume that the shopping cart were filled with heavy lead balls, so that it weighed, say, 1,102 lb (500 kg). If one applied the same force, it would not move. What has changed, clearly, is the mass of the shopping cart. Because force remained constant, the rate of acceleration would become very small—in this case, almost infinitesimal. In the first case, with an empty shopping cart, the mass was relatively small, so acceleration was relatively high. Now to return to the subject of lifting someone on the Moon. It is true that in order to lift that person, one would have to overcome inertia, and, in that sense, it would be as difficult as it is on Earth. But the other component of force, acceleration, has diminished greatly. Weight is, again, a unit of force, but in calculating weight it is useful to make a slight change to the formula F = ma. By definition, the acceleration factor in weight is the downward acceleration due to gravity, usually rendered as g.

So one's weight is equal to mg —but on the Moon, g is much smaller than it is on Earth, and hence, the same amount of force yields much greater results. These facts shed new light on a question that bedeviled physicists at least from the time of Aristotle, until Galileo began clarifying the issue some 2,000 years later: why shouldn't an object of greater mass fall at a different rate than one of smaller mass? There are two answers to that question, one general and one specific. The general answer—that Earth exerts more gravitational pull on an object of greater mass—requires a deeper examination of Newton's gravitational formula. But the more specific answer, relating purely to conditions on Earth, is easily addressed by considering the effect of air resistance. Gravity and Air Resistance One of Galileo's many achievements lay in using an idealized model of reality, one that does not take into account the many complex factors that affect the behavior of objects in the real world. This permitted physicists to study processes that apparently defy common sense. For instance, in the real world, an apple does drop at a greater rate of speed than does a feather. However, in a vacuum, they will drop at the same rate.

Since Galileo's time, it has become commonplace for physicists to discuss specific processes such as gravity with the assumption that all non-pertinent factors (in this case, air resistance or friction) are nonexistent or irrelevant. This greatly simplified the means of testing hypotheses. Idealization of reality makes it possible to set aside the things people think they know about the real world, where events are complicated due to friction. The latter may be defined as a force that resists motion when the surface of one object comes into contact with the surface of another. If two balls are released in an environment free from friction—one of them simply dropped while the other is rolled down a curved surface or inclined plane—they will reach the bottom at the same time. This seems to go against everything that is known, but that is only because what people "know" is complicated by variables that have nothing to do with gravity. The same is true for the behavior of falling objects with regard to air resistance. If air resistance were not a factor, one could fire a cannonball over horizontal space and then, when the ball reached the highest point in its trajectory, release another ball from the same height—and again, they would hit the ground at the same time.

This is the case, even though the cannonball that was fired from the cannon has to cover a great deal of horizontal space, whereas the dropped ball does not. The fact is that the rate of acceleration due to gravity will be identical for the two balls, and the fact that the ball fired from a cannon also covers a horizontal distance during that same period is irrelevant. TERMINAL VELOCITY. In the real world, air resistance creates a powerful drag force on falling objects. The faster the rate of fall, the greater the drag force, until the air resistance forces a leveling in the rate of fall. At this point, the object is said to have reached terminal velocity, meaning that its rate of fall will not increase thereafter. Galileo's idealized model, on the other hand, treated objects as though they were falling in a vacuum—space entirely devoid of matter, including air. In such a situation, the rate of acceleration would continue to grow indefinitely. By means of a graph, one can compare the behavior of an object falling through air with that of an object falling in a vacuum. If the x axis measures time and the y axis downward speed, the rate of an object falling in a vacuum describes a 60°-angle.

In other words, the speed of its descent is increasing at a much faster rate than is the rate of time of its descent—as indeed should be the case, in accordance with gravitational acceleration. The behavior of an object falling through air, on the other hand, describes a curve. Up to a point, the object falls at the same rate as it would in a vacuum, but soon velocity begins to increase at a much slower rate than time. Eventually, the curve levels off at the point where the object experiences terminal velocity. Air resistance and friction have been mentioned separately as though they were two different forces, but in fact air resistance is simply a prominent form of friction. Hence air resistance exerts an upward force to counter the downward force of mass multiplied by gravity—that is, weight. Since g is a constant (32 ft or 9.8 m/sec2), the greater the weight of the falling object, the longer it takes for air resistance to bring it to terminal velocity. A feather quickly reaches terminal velocity, whereas it takes much longer for a cannonball to do the same.

As a result, a heavier object does take less time to fall, even from a great height, than does a light one—but this is only because of friction, and not because of "elements" seeking their "natural level." Incidentally, if raindrops (which of course fall from a very great height) did not reach terminal velocity, they would cause serious injury by the time they hit the ground. Applying the Gravitational Formula Using Newton's gravitational formula, it is relatively easy to calculate the pull of gravity between two objects. It is also easy to see why the attraction is insignificant unless at least one of the objects has enormous mass. In addition, application of the formula makes it clear why G (the gravitational constant, as opposed to g, the rate of acceleration due to gravity) is such a tiny number. If two people each have a mass of 45.5 kg (100 lb) and stand 1 m (3.28 ft) apart, m 1m 2 is equal to 2,070 kg (4,555 lb) and r 2 is equal to 1 m2 .

Applied to the gravitational formula, this figure is rendered as 2,070 kg2/1 m2. This number is then multiplied by gravitational constant, which again is equal to 6.67 · 10−11 (N · m2)/kg2. The result is a net gravitational force of 0.000000138 N (0.00000003 lb)—about the weight of a single-cell organism! EARTH , GRAVITY, AND WEIGHT. Though it is certainly interesting to calculate the gravitational force between any two people, computations of gravity are only significant for objects of truly great mass. For instance, there is the Earth, which has a mass of 5.98 · 1024 kg—that is, 5.98 septillion (1 followed by 24 zeroes) kilograms. And, of course, Earth's mass is relatively minor compared to that of several planets, not to mention the Sun. Yet Earth exerts enough gravitational pull to keep everything on it—living creatures, manmade structures, mountains and other natural features—stable and in place.

One can calculate Earth's gravitational force on any one person—if one wants to take the time to do so using Newton's formula. In fact, it is much simpler than that: gravitational force is equal to weight, or m · g. Thus if a woman weighs 100 lb (445 N), this amount is also equal to the gravitational force exerted on her. By dividing 445 N by the acceleration of gravity—9.8 m/sec2—it is easy to obtain her mass: 45.4 kg. The use of the mg formula for gravitation helps, once again, to explain why heavier objects do not fall faster than lighter ones. The figure for g is a constant, but for the sake of argument, let us assume that it actually becomes larger for objects with a greater mass. This in turn would mean that the gravitational force, or weight, would be bigger than it is—thus creating an irreconcilable logic loop. Furthermore, one can compare results of two gravitation equations, one measuring the gravitational force between Earth and a large stone, the other measuring the force between Earth and a small stone.

(The distance between Earth and each stone is assumed to be the same.) The result will yield a higher quantity for the force exerted on the larger stone—but only because its mass is greater. Clearly, then, the increase of force results only from an increase in mass, not acceleration. Gravity and Curved Space As should be clear from Newton's gravitational formula, the force of gravity works both ways: not only does a stone fall toward Earth, but Earth actually falls toward it. The mass of Earth is so great compared to that of the stone that the movement of Earth is imperceptible—but it does happen. Furthermore, because Earth is round, when one hurls a projectile at a great distance, Earth curves away from the projectile; but eventually gravity itself forces the projectile to the ground. However, if one were to fire a rocket at 17,700 MPH (28,500 km/h), at every instant of time the projectile is falling toward Earth with the force of gravity—but the curved Earth would be falling away from it at the same moment as well. Hence, the projectile would remain in constant motion around the planet—that is, it would be in orbit.

The same is true of an artificial satellite's orbit around Earth: even as the satellite falls toward Earth, Earth falls away from it. This same relationship exists between Earth and its great natural satellite, the Moon. Likewise, with the Sun and its many satellites, including Earth: Earth plunges toward the Sun with every instant of its movement, but at every instant, the Sun falls away. WHY IS EARTH ROUND? Note that in the above discussion, it was assumed that Earth and the Sun are round. Everyone knows that to be the case, but why? The answer is "Because they have to be"—that is, gravity will not allow them to be otherwise. In fact, the larger the mass of an object, the greater its tendency toward roundness: specifically, the gravitational pull of its interior forces the surface to assume a relatively uniform shape. There is a relatively small vertical differential for Earth's surface: between the lowest point and the highest point is just 12.28 mi (19.6 km)—not a great distance, considering that Earth's radius is about 4,000 mi (6,400 km).

It is true that Earth bulges near the equator, but this is only because it is spinning rapidly on its axis, and thus responding to the centripetal force of its motion, which produces a centrifugal component. If Earth were standing still, it would be much nearer to the shape of a sphere. On the other hand, an object of less mass is more likely to retain a shape that is far less than spherical. This can be shown by reference to the Martian moons Phobos and Deimos, both of which are oblong—and both of which are tiny, in terms of size and mass, compared to Earth's Moon. Mars itself has a radius half that of Earth, yet its mass is only about 10% of Earth's. In light of what has been said about mass, shape, and gravity, it should not surprising to learn that Mars is also home to the tallest mountain in the solar system.

Standing 15 mi (24 km) high, the volcano Olympus Mons is not only much taller than Earth's tallest peak, Mount Everest (29,028 ft [8,848 m]); it is 22% taller than the distance from the top of Mount Everest to the lowest spot on Earth, the Mariana Trench in the Pacific Ocean (−35,797 ft [−10,911 m]) A spherical object behaves with regard to gravitation as though its mass were concentrated near its center. And indeed, 33% of Earth's mass is at is core (as opposed to the crust or mantle), even though the core accounts for only about 20% of the planet's volume. Geologists believe that the composition of Earth's core must be molten iron, which creates the planet's vast electromagnetic field. THE FRONTIERS OF GRAVITY. The subject of curvature with regard to gravity can be both a threshold or—as it is here—a point of closure. Investigating questions over perceived anomalies in Newton's description of the behavior of large objects in space led Einstein to his General Theory of Relativity, which posited a curved four-dimensional space-time.

This led to entirely new notions concerning gravity, mass, and light. But relativity, as well as its relation to gravity, is another subject entirely. Einstein offered a new understanding of gravity, and indeed of physics itself, that has changed the way thinkers both inside and outside the sciences perceive the universe. Here on Earth, however, gravity behaves much as Newton described it more than three centuries ago. Meanwhile, research in gravity continues to expand, as a visit to the Web site < reveals. Spurred by studies in relativity, a branch of science called relativistic astrophysics has developed as a synthesis of astronomy and physics that incorporates ideas put forth by Einstein and others. The < site presents studies—most of them too abstruse for a reader who is not a professional scientist—across a broad spectrum of disciplines. Among these is bioscience, a realm in which researchers are investigating the biological effects—such as mineral loss and motion sickness—of exposure to low gravity. The results of such studies will ultimately protect the health of the astronauts who participate in future missions to outer space. WHERE TO LEARN MORE Ardley, Neil. The Science Book of Gravity. San Diego , CA: Harcourt Brace Jovanovich, 1992.

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991. Bendick, Jeanne. Motion and Gravity. New York : F. Watts, 1972. Dalton, Cindy Devine. Gravity. Vero Beach, FL: Rourke, 2001. David, Leonard. "Artificial Gravity and Space Travel." Bio-Science, March 1992, pp. 155-159. Exploring Gravity—Curtin University, Australia (Web site). < (March 18, 2001). The Gravity Society (Web site). < (March 18, 2001). Nardo, Don. Gravity: The Universal Force. San Diego, CA: Lucent Books, 1990. Rutherford, F. James; Gerald Holton; and Fletcher G. Watson. Project Physics. New York: Holt, Rinehart, and Winston, 1981. Stringer, John. The Science of Gravity. Austin, TX: Raintree Steck-Vaughn, 2000.

KEY TERMS The product of mass multiplied by acceleration. FRICTION: The force that resists motion when the surface of one object comes into contact with the surface of another. INERTIA: The tendency of an object in motion to remain in motion, and of an object at rest to remain at rest. INVERSE RELATIONSHIP: A situation involving two variables, in which one of the two increases in direct proportion to the decrease in the other. LAW OF UNIVERSAL GRAVITATION: A principle, put forth by Sir Isaac Newton (1642-1727), which states that every object in the universe attracts every other one with a force proportional to the masses of each, and inversely proportional to the square of the distance between them. MASS: A measure of inertia, indicating the resistance of an object to a change in itsmotion. TERMINAL VELOCITY: A term describing the rate of fall for an object experiencing the drag force of air resistance. In a vacuum, the object would continue to accelerate with the force of gravity, but in most real-world situations, air resistance creates a powerful drag force that causes a leveling in the object's rate off all.

VACUUM: Space entirely devoid of matter, including air. WEIGHT: A measure of the gravitational force on an object; the product of mass multiplied by the acceleration due to gravity. Cite this article Pick a style below, and copy the text for your bibliography. MLA College Physics: Newton’s Universal Law of Gravitation - VOER Newton’s Universal Law of Gravitation 0 What do aching feet, a falling apple, and the orbit of the Moon have in common? Each is caused by the gravitational force. Our feet are strained by supporting our weight—the force of Earth’s gravity on us. An apple falls from a tree because of the same force acting a few meters above Earth’s surface. And the Moon orbits Earth because gravity is able to supply the necessary centripetal force at a distance of hundreds of millions of meters. In fact, the same force causes planets to orbit the Sun, stars to orbit the center of the galaxy, and galaxies to cluster together. Gravity is another example of underlying simplicity in nature. It is the weakest of the four basic forces found in nature, and in some ways the least understood.

It is a force that acts at a distance, without physical contact, and is expressed by a formula that is valid everywhere in the universe, for masses and distances that vary from the tiny to the immense. Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. See [link] . But Newton was not the first to suspect that the same force caused both our weight and the motion of planets. His forerunner Galileo Galilei had contended that falling bodies and planetary motions had the same cause. Some of Newton’s contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph—it had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose a mechanism that caused them to follow these paths and not others.

According to early accounts, Newton was inspired to make the connection between falling bodies and astronomical motions when he saw an apple fall from a tree and realized that if the gravitational force could extend above the ground to a tree, it might also reach the Sun. The inspiration of Newton’s apple is a part of worldwide folklore and may even be based in fact. Great importance is attached to it because Newton’s universal law of gravitation and his laws of motion answered very old questions about nature and gave tremendous support to the notion of underlying simplicity and unity in nature. Scientists still expect underlying simplicity to emerge from their ongoing inquiries into nature. The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Stated in modern language, Newton’s universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Gravitational attraction is along a line joining the centers of mass of these two bodies. The magnitude of the force is the same on each, consistent with Newton’s third law.

Misconception Alert The magnitude of the force on each object (one has larger mass than the other) is the same, consistent with Newton’s third law. The bodies we are dealing with tend to be large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass (CM), which will be further explored in Linear Momentum and Collisions . For two bodies having masses m between their centers of mass, the equation for Newton’s universal law of gravitation is F size 12{F=G { { ital "mM"} over {r rSup { size 8{2} } } } } {} where is the magnitude of the gravitational force and G is a proportionality factor called the gravitational constant. G size 12{G} {} is a universal gravitational constant—that is, it is thought to be the same everywhere in the universe.

It has been measured experimentally to be G kg 2 size 12{G=6 "." "673" times "10" rSup { size 8{ - "11"} } { {N cdot m rSup { size 8{2} } } over {"kg" rSup { size 8{2} } } } } {} in SI units. Note that the units of G are such that a force in newtons is obtained from F 2 size 12{F=G { { ital "mM"} over {r rSup { size 8{2} } } } } {} , when considering masses in kilograms and distance in meters. For example, two 1.000 kg masses separated by 1.000 m will experience a gravitational attraction of 6 N size 12{6 "." "673" times "10" rSup { size 8{ - "11"} } N} {} . This is an extraordinarily small force. The small magnitude of the gravitational force is consistent with everyday experience. We are unaware that even large objects like mountains exert gravitational forces on us.

In fact, our body weight is the force of attraction of the entire Earth on us with a mass of 6 size 12{6 times "10" rSup { size 8{"24"} } `"kg"} {} . Recall that the acceleration due to gravity g 2 size 12{9 "." 8`"m/s" rSup { size 8{2} } } {} on Earth. We can now determine why this is so. The weight of an object mg is the gravitational force between it and Earth. Substituting mg for F in Newton’s universal law of gravitation gives mg size 12{ ital "mg"=G { { ital "mM"} over {r rSup { size 8{2} } } } } {} where is the mass of the object, M is the mass of Earth, and r size 12{r} {} is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). See [link] .

The mass m of the object cancels, leaving an equation for g size 12{g=G { {M} over {r rSup { size 8{2} } } } } {} Substituting known values for Earth’s mass and radius (to three significant figures), g 2 , size 12{g= left (6 "." "67" times "10" rSup { size 8{ - "11"} } { {N cdot m rSup { size 8{2} } } over {"kg" rSup { size 8{2} } } } right ) times { {5 "." "98" times "10" rSup { size 8{"24"} } " kg"} over { \( 6 "." "38" times "10" rSup { size 8{6} } " m" \) rSup { size 8{2} } } } } {} and we obtain a value for the acceleration of a falling body: g .

size 12{g=9 "." "80"" m/s" rSup { size 8{2} } } {} The distance between the centers of mass of Earth and an object on its surface is very nearly the same as the radius of Earth, because Earth is so much larger than the object. This is the expected value and is independent of the body’s mass. Newton’s law of gravitation takes Galileo’s observation that all masses fall with the same acceleration a step further, explaining the observation in terms of a force that causes objects to fall—in fact, in terms of a universally existing force of attraction between masses. Take-Home Experiment Take a marble, a ball, and a spoon and drop them from the same height. Do they hit the floor at the same time? If you drop a piece of paper as well, does it behave like the other objects? Explain your observations. Making Connections Attempts are still being made to understand the gravitational force. As we shall see in Particle Physics , modern physics is exploring the connections of gravity to other forces, space, and time. General relativity alters our view of gravitation, leading us to think of gravitation as bending space and time.

In the following example, we make a comparison similar to one made by Newton himself. He noted that if the gravitational force caused the Moon to orbit Earth, then the acceleration due to gravity should equal the centripetal acceleration of the Moon in its orbit. Newton found that the two accelerations agreed “pretty nearly.” Earth’s Gravitational Force Is the Centripetal Force Making the Moon Move in a Curved Path (a) Find the acceleration due to Earth’s gravity at the distance of the Moon. (b) Calculate the centripetal acceleration needed to keep the Moon in its orbit (assuming a circular orbit about a fixed Earth), and compare it with the value of the acceleration due to Earth’s gravity that you have just found. Strategy for (a) This calculation is the same as the one finding the acceleration due to gravity at Earth’s surface, except that r size 12{r} {} is the distance from the center of Earth to the center of the Moon. The radius of the Moon’s nearly circular orbit is 3 size 12{3 "." "84" times "10" rSup { size 8{8} } `m} {} .

Substituting known values into the expression for g is the mass of Earth not the Moon, yields g Centripetal acceleration can be calculated using either form of a } . size 12{ left none matrix { a rSub { size 8{c} } = { {v rSup { size 8{2} } } over {r} } {} ## a rSub { size 8{c} } =rω rSup { size 8{2} } } right rbrace "." } {} We choose to use the second form: a size 12{a rSub { size 8{c} } =rω rSup { size 8{2} } } {} where is the angular velocity of the Moon about Earth.

Solution for (b) Given that the period (the time it takes to make one complete rotation) of the Moon’s orbit is 27.3 days, (d) and using 1 d = 86,400 s size 12{ω= { {Δθ} over {Δt} } = { {2π" rad"} over { \( "27" "." "3 d" \) \( "86,400 s/d" \) } } =2 "." "66" times "10" rSup { size 8{ - 6} } { {"rad"} over {s} } } {} we see that s . size 12{ω= { {Δθ} over {Δt} } = { {2π" rad"} over { \( "27" "." "3 d" \) \( "86,400 s/d" \) } } =2 "." "66" times "10" rSup { size 8{ - 6} } { {"rad"} over {s} } } {} The centripetal acceleration is m/s.

2 alignl { stack { size 12{a rSub { size 8{c} } =rω rSup { size 8{2} } = \( 3 "." "84" times "10" rSup { size 8{8} } " m" \) \( 2 "." "66" times "10" rSup { size 8{ - 6} } " rad/s" \) rSup { size 8{2} } } {} # " "=2 "." "72" times "10" rSup { size 8{ - 3} } " m/s" rSup { size 8{2} } {} } } {} The direction of the acceleration is toward the center of the Earth. Discussion The centripetal acceleration of the Moon found in (b) differs by less than 1% from the acceleration due to Earth’s gravity found in (a). This agreement is approximate because the Moon’s orbit is slightly elliptical, and Earth is not stationary (rather the Earth-Moon system rotates about its center of mass, which is located some 1700 km below Earth’s surface).

The clear implication is that Earth’s gravitational force causes the Moon to orbit Earth. Why does Earth not remain stationary as the Moon orbits it? This is because, as expected from Newton’s third law, if Earth exerts a force on the Moon, then the Moon should exert an equal and opposite force on Earth (see [link] ). We do not sense the Moon’s effect on Earth’s motion, because the Moon’s gravity moves our bodies right along with Earth but there are other signs on Earth that clearly show the effect of the Moon’s gravitational force as discussed in Satellites and Kepler's Laws: An Argument for Simplicity . (a) Earth and the Moon rotate approximately once a month around their common center of mass. (b) Their center of mass orbits the Sun in an elliptical orbit, but Earth’s path around the Sun has “wiggles” in it. Similar wiggles in the paths of stars have been observed and are considered direct evidence of planets orbiting those stars. This is important because the planets’ reflected light is often too dim to be observed. Tides Ocean tides are one very observable result of the Moon’s gravity acting on Earth. [link] is a simplified drawing of the Moon’s position relative to the tides.

Because water easily flows on Earth’s surface, a high tide is created on the side of Earth nearest to the Moon, where the Moon’s gravitational pull is strongest. Why is there also a high tide on the opposite side of Earth? The answer is that Earth is pulled toward the Moon more than the water on the far side, because Earth is closer to the Moon. So the water on the side of Earth closest to the Moon is pulled away from Earth, and Earth is pulled away from water on the far side. As Earth rotates, the tidal bulge (an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits) keeps its orientation with the Moon. Thus there are two tides per day (the actual tidal period is about 12 hours and 25.2 minutes), because the Moon moves in its orbit each day as well). The Moon causes ocean tides by attracting the water on the near side more than Earth, and by attracting Earth more than the water on the far side. The distances and sizes are not to scale. For this simplified representation of the Earth-Moon system, there are two high and two low tides per day at any location, because Earth rotates under the tidal bulge.

The Sun also affects tides, although it has about half the effect of the Moon. However, the largest tides, called spring tides, occur when Earth, the Moon, and the Sun are aligned. The smallest tides, called neap tides, occur when the Sun is at a 90º size 12{"90" rSup { size 8{ circ } } } {} angle to the Earth-Moon alignment. (a, b) Spring tides: The highest tides occur when Earth, the Moon, and the Sun are aligned. (c) Neap tide: The lowest tides occur when the Sun lies at 90º size 12{"90" rSup { size 8{ circ } } } {} to the Earth-Moon alignment. Note that this figure is not drawn to scale. Tides are not unique to Earth but occur in many astronomical systems. The most extreme tides occur where the gravitational force is the strongest and varies most rapidly, such as near black holes (see [link] ). A few likely candidates for black holes have been observed in our galaxy. These have masses greater than the Sun but have diameters only a few kilometers across.

The tidal forces near them are so great that they can actually tear matter from a companion star. A black hole is an object with such strong gravity that not even light can escape it. This black hole was created by the supernova of one star in a two-star system. The tidal forces created by the black hole are so great that it tears matter from the companion star. This matter is compressed and heated as it is sucked into the black hole, creating light and X-rays observable from Earth. ”Weightlessness” and Microgravity In contrast to the tremendous gravitational force near black holes is the apparent gravitational field experienced by astronauts orbiting Earth. What is the effect of “weightlessness” upon an astronaut who is in orbit for months? Or what about the effect of weightlessness upon plant growth? Weightlessness doesn’t mean that an astronaut is not being acted upon by the gravitational force. There is no “zero gravity” in an astronaut’s orbit. The term just means that the astronaut is in free-fall, accelerating with the acceleration due to gravity. If an elevator cable breaks, the passengers inside will be in free fall and will experience weightlessness. You can experience short periods of weightlessness in some rides in amusement parks.

Astronauts experiencing weightlessness on board the International Space Station. (credit: NASA) Microgravity refers to an environment in which the apparent net acceleration of a body is small compared with that produced by Earth at its surface. Many interesting biology and physics topics have been studied over the past three decades in the presence of microgravity. Of immediate concern is the effect on astronauts of extended times in outer space, such as at the International Space Station. Researchers have observed that muscles will atrophy (waste away) in this environment. There is also a corresponding loss of bone mass. Study continues on cardiovascular adaptation to space flight. On Earth, blood pressure is usually higher in the feet than in the head, because the higher column of blood exerts a downward force on it, due to gravity. When standing, 70% of your blood is below the level of the heart, while in a horizontal position, just the opposite occurs. What difference does the absence of this pressure differential have upon the heart? Some findings in human physiology in space can be clinically important to the management of diseases back on Earth. On a somewhat negative note, spaceflight is known to affect the human immune system, possibly making the crew members more vulnerable to infectious diseases.

Experiments flown in space also have shown that some bacteria grow faster in microgravity than they do on Earth. However, on a positive note, studies indicate that microbial antibiotic production can increase by a factor of two in space-grown cultures. One hopes to be able to understand these mechanisms so that similar successes can be achieved on the ground. In another area of physics space research, inorganic crystals and protein crystals have been grown in outer space that have much higher quality than any grown on Earth, so crystallography studies on their structure can yield much better results. Plants have evolved with the stimulus of gravity and with gravity sensors. Roots grow downward and shoots grow upward. Plants might be able to provide a life support system for long duration space missions by regenerating the atmosphere, purifying water, and producing food. Some studies have indicated that plant growth and development are not affected by gravity, but there is still uncertainty about structural changes in plants grown in a microgravity environment. The Cavendish Experiment: Then and Now As previously noted, the universal gravitational constant G size 12{G} {} is determined experimentally.

This definition was first done accurately by Henry Cavendish (1731–1810), an English scientist, in 1798, more than 100 years after Newton published his universal law of gravitation. The measurement of G size 12{G} {} is very basic and important because it determines the strength of one of the four forces in nature. Cavendish’s experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at most), using apparatus like that in [link] . Remarkably, his value for G differs by less than 1% from the best modern value. One important consequence of knowing G size 12{G} {} was that an accurate value for Earth’s mass could finally be obtained. This was done by measuring the acceleration due to gravity as accurately as possible and then calculating the mass of Earth M from the relationship Newton’s universal law of gravitation gives mg can be calculated because all quantities on the right, including the radius of Earth r size 12{r} {} , are known from direct measurements.

We shall see in Satellites and Kepler's Laws: An Argument for Simplicity that knowing G size 12{G} {} also allows for the determination of astronomical masses. Interestingly, of all the fundamental constants in physics, G size 12{G} {} is by far the least well determined. The Cavendish experiment is also used to explore other aspects of gravity. One of the most interesting questions is whether the gravitational force depends on substance as well as mass—for example, whether one kilogram of lead exerts the same gravitational pull as one kilogram of water. A Hungarian scientist named Roland von Eötvös pioneered this inquiry early in the 20th century. He found, with an accuracy of five parts per billion, that the gravitational force does not depend on the substance. Such experiments continue today, and have improved upon Eötvös’ measurements. Cavendish-type experiments such as those of Eric Adelberger and others at the University of Washington, have also put severe limits on the possibility of a fifth force and have verified a major prediction of general relativity—that gravitational energy contributes to rest mass.

Ongoing measurements there use a torsion balance and a parallel plate (not spheres, as Cavendish used) to examine how Newton’s law of gravitation works over sub-millimeter distances. On this small-scale, do gravitational effects depart from the inverse square law? So far, no deviation has been observed. Cavendish used an apparatus like this to measure the gravitational attraction between the two suspended spheres ( m ) and the two on the stand ( M size 12{M} {} ) by observing the amount of torsion (twisting) created in the fiber. Distance between the masses can be varied to check the dependence of the force on distance. Modern experiments of this type continue to explore gravity. Section Summary Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form, this is F

Question: Which British scientist discovered and studied hydrogen, was the first to measure Newton's gravitational constant, and calculated the mass and mean density of the Earth?
Answer: HENRY CAVENDISH