Gravity.

Gravity

Introduction In the physicist’s view of the world, there are four fundamental forces: gravitational, electromagnetic, weak, and strong. We begin our studies of forces with the most familiar force in our everyday lives.

The Concept of Gravity Every school-age child knows that objects fall because of gravity. But what is gravity? Saying that it is what makes things fall doesn’t tell us much. Is gravity a material like a fluid or a fog? Or is it something more ethereal? No one knows. Because we have given something a name doesn’t mean we understand it completely. We do understand gravity in the sense that we can precisely describe how it affects the motion of objects.

The Concept of Gravity We can do more.
By looking carefully at the motions of certain objects, we can develop an equation that describes this attractive force between material objects and explore some of its consequences. On the other hand, we cannot answer questions such as, “What is gravity?” or “Why does gravity exist?”

The Concept of Gravity The concept of gravity hasn’t always existed.
It was conceived when changes in our world view required a new explanation of why things fell to Earth. When Earth was believed to be flat, gravity wasn’t needed. Objects fell because they were seeking their natural places. A stone on the end of a string hung down because of its tendency to return to its natural place. “Up” and “down” were absolute directions.

The Concept of Gravity The realization that Earth was spherical required a change in perspective. What happens to the unfortunate people on the other side of Earth who are upside down? But the change in thinking was made without gravity. The center of Earth was at the center of the universe, and things naturally moved toward this point. “Up” and “down” became relative, but the location of the center of the universe became absolute.

The Concept of Gravity Gravity was also not needed to understand the motion of the heavenly bodies. The earliest successful scheme viewed Earth as the center of the universe, with the celestial bodies going around Earth in circular orbits. Perpetual, circular orbits were considered quite natural for celestial motions; little attention was given to the causes of these motions. Aristotle did not recognize any connections between what he saw as perfect, heavenly motion and imperfect, earthly motion. He stated that circular motion with constant speed was the most perfect of all motions, and thus the natural heavenly motion needed no further explanation.

The Concept of Gravity This changed slightly with a new view of heavenly motion by Nicholas Copernicus, a 16th-century Polish scientist and clergyman. He proposed that the planets (including Earth) go around the Sun in circular orbits and that the Moon orbits Earth. This is essentially the scheme taught in schools today. A hint of a concept of gravity appears in Copernicus’s work. He believed that the Sun and Moon would attract objects near their surfaces—each would have a local gravity—but he had no concept of that attractive influence spreading throughout space.

The Concept of Gravity A hundred years later, Johannes Kepler, a German mathematician and astronomer, suggested that the planets move because of an interaction between them and the Sun. Kepler also moved us away from the assumption that the planets traveled in circular paths. After many years of trial and error, Kepler correctly deduced that the orbits of the planets were ellipses—but ellipses that are close to being circles. Furthermore, the planets do not have constant speeds in their journeys around their elliptical paths but speed up as they approach the Sun and slow down as they move farther away.

The Concept of Gravity Influenced by early, important work on magnetism, Kepler postulated that the planets were “magnetically” driven along their paths by the Sun. Because this work occurred shortly before the acceptance of the idea of inertia, Kepler did not realize that a force was not needed to drive the planets along their orbits but to cause the orbits to be curved. Kepler postulated an interaction reaching from the Sun to the various planets and driving the planets, but he didn’t consider the possibility of any interaction between the planets themselves; the Sun reigned supreme, a metaphor for his god, from which everything else gained strength.

The Concept of Gravity Newton developed our present view of gravity. He started by saying there was nothing special about the rules of nature that he had developed for use on Earth. They should also apply to heavenly motions. As we learned in the previous chapter, anything traveling in a circle must be accelerating. The acceleration must be toward the center of the circle, and the object must therefore have a net force acting on it. Newton went searching for this force.

Newton’s Gravity Creativity often involves bringing together ideas or things from seemingly unrelated areas. After an artist or scholar has done it, the connection often seems obvious to others. Newton made such a synthesis between motions on Earth and motions in the heavens. Legend has it that he made his intellectual leap while contemplating such matters and seeing an apple fall. Newton believed that the laws of motion that worked on Earth’s surface should also apply to motion in the heavens.

Newton’s Gravity Because the Moon orbits Earth in a nearly circular orbit, it must be accelerating toward Earth. According to the second law, any acceleration requires a force. Newton believed that if this force could be shut off, the Moon would no longer continue to move along its circular path but would fly off in a straight line like a stone from a sling. The genius of Newton was in relating the cause of this heavenly motion to earthly events. Newton felt that the Moon’s acceleration was due to the force of gravity—the same gravity that caused the apple to fall from the tree.

Newton’s Gravity How could he demonstrate this?
First, he calculated the acceleration of the Moon. Because the distance to the Moon and the time it took the Moon to make one revolution were already known, he was able to calculate that the Moon accelerated (meter per second) per second. This is a very small acceleration. In 1 second the Moon moves about 1 kilometer along its orbit but falls only 1.4 millimeters (about 1 20 inch in 0.6 mile). In contrast to the Moon’s acceleration, the apple has an acceleration of 9.80 (meters per second) per second and falls about 5 meters in its first second of flight.

Newton’s Gravity We can compare these two accelerations by dividing one by the other: Why are these two accelerations so different? The mass of the Moon is certainly much larger than that of the apple. But that doesn’t matter, free-falling objects all have the same acceleration independent of their masses. However, the accelerations of the apple and the Moon weren’t equal.

Newton’s Gravity Could Newton’s idea that both motions were governed by the same gravity be wrong? Or could the rules of motion that he developed on Earth not apply to heavenly motion? Neither. Newton reasoned that the Moon’s acceleration is smaller because Earth’s gravitational attraction is smaller at larger distances; it is “diluted” by distance.

Newton’s Gravity How did the force decrease with increasing distance?
Retracing Newton’s reasoning is impossible because he didn’t write about how he arrived at his conclusions, but he may have used the following kind of reasoning: Many things get less intense the farther you are from their source. Imagine a paint gun that can spray paint uniformly in all directions. Suppose the gun is in the center of a sphere of radius 1 meter, and at the end of 1 minute of spraying, the paint on the inside wall of the sphere is 1 millimeter thick.

Newton’s Gravity If we repeat the experiment with the same gun but with a sphere that is 2 meters in radius, the paint will be only ¼ millimeter thick because a sphere with twice the radius has a surface area that is four times the original. If the sphere has three times the radius, the surface is nine times bigger, and the paint is (1/3)2 = 1/9 as thick. The thickness of the paint decreases as the square of the radius of the sphere increases. This is known as an inverse-square relationship. A force reaching into space could be diluted in a similar manner.

On the Bus Q: If the sphere were 4 meters in radius, how thick would the paint be? A: It would be (1/4)2 × 1 millimeter = 1/16 millimeter thick.

Newton’s Gravity Newton may also have received encouragement for this explanation by working backward from observational results on the motion of the planets that had been developed by Kepler. Kepler found a relationship that connected the orbital periods of the planets with their average distances from the Sun. Using Kepler’s results and an expression for the acceleration of an object in circular motion, we can show that the force decreases with the square of the distance. This means that if the distance between the objects is doubled, the force is only one-fourth as strong. If the distance is tripled, the force is one-ninth as strong, and so on.

Newton’s Gravity What, exactly does it mean to say that the force is one-ninth as strong? We cannot be referring to the gravitational forces acting on different objects; the gravitational force on an automobile is obviously much larger than the gravitational force on a person. We must compare the gravitational forces acting on a single object, such as our apple, at different distances. When the apple is moved three times as far away from Earth’s center, the gravitational force on the apple is one-ninth as strong.

Newton’s Gravity The obvious test of the notion of gravity was to see whether the relationship between distance and force gave the correct relative accelerations for the apple on Earth and the orbiting Moon. Newton could use this rule and make the comparison. The distance from the center of Earth to the center of the Moon is about 60 times the radius of Earth. The Moon is therefore 60 times farther away from the center of Earth than the apple. Thus, the force at the Moon’s location—and its acceleration—should be 602, or 3600, times smaller. This is in agreement with the previous calculation.

Newton’s Gravity The data available in Newton’s time were not as good as those we have used here, but they were good enough to convince him of the validity of his reasoning. Modern measurements yield more precise values and agree that the gravitational force is inversely proportional to the square of the distance.

On the Bus Q: What happens to the force of gravity if the distance between the two objects is cut in half? A: The force becomes four times as strong.

Newton’s Gravity Newton now knew how gravity changes with distance:
the force of Earth’s gravity exists beyond Earth and gets weaker the farther away you go. But there are other factors. He already knew that the force of gravity depends on the object’s mass. His third law of motion says that the force exerted on the Moon by Earth is equal in strength to that exerted on Earth by the Moon; they attract each other. This symmetry indicates that both masses should be included in the same way. The gravitational force is proportional to each mass.

The Law of Universal Gravitation
Having made the connection between celestial motion and motion near Earth’s surface with a force that reaches across empty space and pulls objects to Earth, Newton took another, even bolder, step. He stated that the force of gravity exists between all objects, that it is truly a universal law of gravitation. The boldness of this assertion becomes apparent when one realizes that the force between two ordinary-sized objects is extremely small. Clearly, as you walk past a friend, you don’t feel a gravitational attraction pulling you together. But that is exactly what Newton was claiming.

Any two objects have a force of attraction between them; his rule for gravity is a law of universal gravitation. Putting everything together, we arrive at an equation for the gravitational force: where m1 and m2 are the masses of the two objects, r is the distance between their centers, and G is a constant that contains information about the strength of the force.

Although Newton arrived at this conclusion when he was 24 years old, he didn’t publish his results for more than 20 years. This was partly due to one unsettling aspect of his work. The distance that appears in the relationship is the distance from Earth’s center. This means that Earth’s mass is assumed to be concentrated at a point at its center. This may seem to be a reasonable assumption when considering the force of gravity on the Moon; Earth’s size is irrelevant when dealing with these huge distances. But what about the apple on Earth’s surface?

In this case the apple is attracted by mass that is only a few meters away and mass that is 13,000 kilometers away, as well as all the mass between. It seems less intuitive that all of this mass would somehow act like a very compact mass located at Earth’s center. But that is just what happens. Newton was eventually able to show mathematically that the sum of the forces due to each cubic meter of Earth is the same as if all of them were concentrated at its center.

This result holds if Earth is spherically symmetric. It doesn’t have to have a uniform composition; it need only be composed of a series of spherical shells, each of which has a uniform composition. In fact, 1 cubic meter of material near Earth’s center has almost four times the mass of a typical cubic meter of surface material. Newton applied the laws of motion and the law of universal gravitation extensively to explain the motions of the heavenly bodies. He was able to show that three observational rules developed by Kepler to describe planetary motion were a mathematical consequence of his work. Kepler’s rules were the results of years of work reducing observational data to a set of simple patterns.

By the 18th century, scientists were so confident of Newton’s work that when a newly discovered planet failed to behave “properly,” they assumed that there must be other, yet to be discovered, masses causing the deviations. When Uranus was discovered in 1781, a great effort was made to collect additional data on its orbit. By going back to old records, scientists determined additional times and locations of its orbit. Although the main contribution to Uranus’s orbit is the force of the Sun, the other planets also have their effects on Uranus. In this case, however, the calculations still differed from the actual path by a small amount. The deviations were explained in terms of the influence of an unknown planet. This led to the discovery of Neptune in 1846.

This still didn’t completely account for the orbits of Uranus and Neptune; a search began for yet another planet. The discovery of Pluto in 1930 still left some discrepancies. Although the search for new planets continues, analysis of the paths of the known planets indicates that any additional planets must be very small and/or very far away.

The Value of G Even though Newton had an equation for the gravitational force, he couldn’t use it to actually calculate the force between two objects; he needed to know the value of the constant G. The way to get this is to measure the force between two known masses separated by a known distance. However, the force between two objects on Earth is so tiny that it couldn’t be detected in Newton’s time. It was more than 100 years after the publication of Newton’s results before Henry Cavendish, a British scientist, developed a technique that was sensitive enough to measure the force between two masses.

The Value of G Modern measurements yield the value:
Putting this value into the equation for the gravitational force tells us that the force between two 1-kilogram masses separated by 1 meter is only newton. This is minuscule compared with a weight of 9.8 newtons for each mass. The small value of G explains why two friends don’t feel their mutual gravitational attraction when standing near each other.

The Value of G Cavendish referred to his experiment as one that “weighed” Earth, although it would have been more accurate to claim that it “massed” Earth. His point, though, was important. By measuring the value of G, Cavendish made it possible to accurately determine Earth’s mass for the first time. The acceleration of a mass near Earth’s surface depends on the value of G and Earth’s mass and radius. Because he now knew the values of all but Earth’s mass, he could calculate it. Earth’s mass is 5.98 × 1024 kilograms; that’s about a million million million million times as large as your mass.

Working it Out: Gravity
Let’s calculate the gravitational force between two friends. To make the calculation of this force easier, we make one unrealistic assumption: we assume that the friends are spheres! This allows us to use the distance between their centers as their separation and still get a reasonable answer. Assuming that the friends have masses of 70 and 86 kg (about 154 and 189 lb, respectively) and are standing 2 m apart, we have This very tiny force is about one ten-billionth (1010) of either friend’s weight.

Flawed Reasoning In the blockbuster movie Armageddon, the heroes land their space shuttle on a Texas-sized comet that is careening toward Earth. They then walk around the comet like construction workers here on Earth. What is wrong with this picture? ANSWER A comet is not massive enough to provide the gravity required to walk around normally. The astronauts would have to tether themselves to keep from flying away because of the smallest exertion. The physics was much more accurate in the other comet-coming-to-destroy-Earth movie Deep Impact. In this movie, the spaceship is tethered to the comet and one of the astronauts is thrown into space by an explosion on the surface of the comet.

The Value of G Once Earth’s mass is known, we can use the law of universal gravitation to calculate the acceleration due to gravity near Earth’s surface: where ME is Earth’s mass and RE = 6370 kilometers is Earth’s radius. Plugging in the numerical values yields g = 9.8 (meters per second) per second, as expected.

The Value of G Because Earth orbits the Sun, the Sun’s mass can also be calculated with the Cavendish results. In making these computations, the value of G measured on Earth is assumed to be valid throughout the Solar System. This cannot be proved. On the other hand, there is no evidence to the contrary, and this assumption gives consistent results. Newton made this same claim more than 100 years earlier.

Flawed Reasoning You read in a comic book that the gravity on the Moon is not as strong as on Earth because there is no atmosphere on the Moon. This doesn’t seem right, so you do a little research. What do you find? ANSWER The source of the gravitational attraction is mass, not air. The gravitational attraction on the Moon is less than on Earth because the Moon’s mass is so much less than Earth’s. Actually, the argument in the comic book is completely backward. The Moon does not have an atmosphere because its gravity is too weak to hold one.

Gravity Near Earth’s Surface
In earlier chapters we assumed that the gravitational force on an object is constant near Earth’s surface. We were able to do this because the force changes so little over the distances in question. In fact, to assume otherwise would have unnecessarily complicated matters. Near Earth’s surface the gravitational force decreases by one part in a million for every 3 meters (about 10 feet) of gain in elevation. Therefore, an object that weighs 1 newton at Earth’s surface would weigh newton at an elevation of 3 meters. An individual with a mass of 50 kilograms weighs 500 newtons (110 pounds) in New York City; this person would weigh about 0.25 newton (1 ounce) less in mile-high Denver.

The variations in the gravitational force result in changes in the acceleration due to gravity. The value of the acceleration—normally symbolized as g —is nearly constant near Earth’s surface. As long as one stays near the surface, the distance between the object and Earth’s center changes very little. If an object is raised 1 kilometer (about 58 mile), the distance changes from 6378 kilometers to 6379 kilometers, and g only changes from (meters per second) per second to (meters per second) per second.

However, even without a change in elevation, g is not strictly constant from place to place. Earth would need to be composed of spherical shells, with each shell being uniform; this is not the case. Underground salt deposits have less mass per cubic meter and give smaller values of g than average, whereas metal deposits produce larger g values. Therefore, measurements of g can be used to locate large-scale underground ore deposits. By noting variations, geologists can map regions for further exploration.

On the Bus Q: Given the fact that water has less mass per cubic meter than soil and rock, would you expect the value of g to be smaller or larger than average over a lake? A: A cubic meter of water would provide less attraction than a cubic meter of soil and rock. Therefore, the value of g would be smaller.

Satellites Newton’s theory also predicts the orbits of artificial satellites orbiting Earth. By knowing how the force changes with distance from Earth, we know what accelerations—and, consequently, other orbital characteristics—to expect at different altitudes. For instance, a satellite at a height of 200 kilometers should orbit Earth in 88.5 minutes. This is close to the orbit of the satellite Vostok 6, which carried the first woman to enter outer space, Valentina Tereshkova, into Earth orbit in June 1963. Its orbit varied in height from 170 to 210 kilometers and had a period of a little more than 88 minutes.

Satellites The higher a satellite’s orbit, the longer it takes to complete one orbit. The Moon takes 27.3 days; Vostok 6 took 88 minutes. It is possible to calculate the height that a satellite would need to have a period of 1 day. With this orbit, if the satellite were positioned above the equator, it would appear to remain fixed directly above one spot on Earth—an orbit called geosynchronous. Such geosynchronous satellites have an altitude of 36,000 kilometers, or about 512 Earth radii, and are useful in establishing worldwide communications networks.

Satellites Backyard satellite dishes that pick up television signals point to geosynchronous satellites. The first successful geosynchronous satellite was Syncom II, launched in July 1963. Some geosynchronous satellites are used to monitor the weather on Earth.

Satellites Any space probe requires the same computations as those done for satellites; the computers at the National Aeronautics and Space Administration (NASA) calculate the trajectories for space flights, using Newton’s laws of motion and the law of gravitation. The forces on the spacecraft at any time depend on the positions of the other bodies in the Solar System.

Satellites These can be calculated with the gravitation equation by inserting the distance to and the mass of each body. The net force produces an acceleration of the spacecraft, which changes its velocity. From this the computer calculates a new position for the spacecraft. It also calculates new positions for the other celestial bodies, and the process starts over. In this manner the computer plots the path of the spacecraft through the Solar System.

On the Bus Q: If you could spot a geosynchronous satellite in the sky, how could you distinguish it from a star? A: The satellite remains in the same location in the sky, whereas the stars drift westward as Earth rotates under them.

Tides Before Newton’s work with gravity, no one was able to explain why we have tides. Some things were known: The tides are caused by bulges in the surface of Earth’s oceans. There are two bulges, one on each side of Earth. The occurrence of tides at a given location is due to Earth’s rotation. Exaggerated ocean bulges. As Earth rotates, the bulges appear to move around Earth’s surface

Tides Imagine for simplicity that the bulges are stationary—pointing in some direction in space—and that Earth is rotating. Each point on Earth passes through both bulges in 24 hours, and we have high tides at these times. Low tides occur halfway between the bulges. So we have two low and two high tides each day. What wasn’t known was why Earth had these bulges.

Tides Newton claimed that they were due to the Moon’s gravity.
Earth exerts a gravitational force on the Moon that causes the Moon to orbit it. But the Moon exerts an equal and opposite force on Earth that causes Earth to orbit the Moon. Actually, both Earth and the Moon orbit a common point located between them. This point is the center of mass of the Earth–Moon system.

Tides Because Earth is so much more massive than the Moon, the center of mass is much closer to Earth. In fact, its location is inside Earth. Earth’s orbital motion would look more like a wobble to somebody viewing its motion from high above the North Pole. But it is an orbit.

Tides Because Earth has an orbital motion, we can use the conclusions developed for the Moon’s motion to help us understand Earth’s tides. Namely, because we concluded that the Moon is continually falling toward Earth, Earth then is continually falling toward the Moon. This centripetal acceleration toward the Moon is the key to understanding tidal bulges.

Tides Forget momentarily that Earth is moving along its orbit and just consider Earth falling toward the Moon. This acceleration is the major contributor to the tides. Because the strength of the Moon’s gravity gets weaker with increasing distance, the force on different parts of Earth is different. For example, on the side nearest the Moon, 1 kilogram of ocean water feels a stronger force than an equal mass of rock at Earth’s center. Similarly, 1 kilogram of material on the far side of Earth feels a smaller force than both the kilogram on the near side and the one at the center.

On the Bus Q: Are the forces between Earth and the Moon a Newtonian third-law pair? A: Yes, one of the forces is the force of Earth on the Moon, and the other is the force of the Moon on Earth.

Tides If there are different-sized forces at different spots on Earth, there are different accelerations for different parts. Parts of Earth outrace other parts in their fall toward the Moon. Material on the side of Earth facing the Moon tries to get ahead, while the material on the other side lags behind. Of course, Earth has internal forces keeping it together that eventually balance these inequalities. But we do end up with a stretched-out Earth.

Tides Although this reasoning accounts for the occurrence of the two high tides each day, it is too simple to get the details right. We observe that high tides do not occur at the same time each day. This happens because the Moon orbits the rotating Earth once a month. The normal time interval between successive high tides is 12 hours and 25 minutes. High tides do not occur when the Moon is overhead but later—as much as 6 hours later. This is due to such factors as the frictional and inertial effects of the water and the variable depth of the ocean.

Tides Although the height difference between low and high tides in the middle of the ocean is only about 1 meter, the shape of the shoreline can greatly amplify the tides. The greatest tides occur in the Bay of Fundy, on the eastern seaboard between Canada and the United States; there the maximum range from low to high tide is 16 meters (54 feet)!

Tides We would also expect to observe solar tides because the Sun also exerts a gravitational pull on Earth and Earth is “falling” toward the Sun. These do occur, but their heights are a little less than one-half those due to the Moon. This value may seem too small, taking into consideration that the Sun’s gravitational force on Earth is about 180 times as large as the Moon’s. The solar effect is so small because it is the difference in the force from one side of Earth to the other that matters and not the absolute size. The tides due to the planets are even smaller, that of Jupiter being less than one ten-millionth that due to the Sun.

Tides The continents are much more rigid than the oceans.
Even so, the land experiences measurable tidal effects. Land areas may rise and fall as much as 23 centimeters (9 inches). Because the entire area moves up and down together, we don’t notice this effect.

On the Bus Q: Is the height of the high tide related to the phase of the Moon? That is, is it higher when the Sun and Moon are on the same side of Earth (new moon), when they are on opposite sides (full moon), or when they are at right angles to each other (first- or third-quarter moon)? A: The highest high tides and the lowest low tides occur near new and full moons, when Earth, the Moon, and the Sun are in a line.

How Far Does Gravity Reach?
The law of gravitation has been thoroughly tested within the solar system. It accounts for the planets’ motions, including their irregularities due to the mutual attraction of all the other planets. What about tests outside the solar system? We haven’t sent probes out there. We are fortunate, however, because nature has provided us with ready-made probes. Astronomers observe that many stars in our galaxy revolve around a companion star. These binary star systems are the rule rather than the exception. These pairs revolve around each other in exactly the way predicted by Newton’s laws.

Occasionally, a star is spotted that appears to be alone yet is moving in an elliptical path. Our faith in Newton’s laws is so great that we assume a companion star is there; it is just not visible. Some of these invisible stars have later been detected because of signals they emit other than visible light.

Photographs of star clusters show that the gravitational interaction occurs between stars. In fact, measurements show that all the stars in the Milky Way Galaxy are rotating about a common point under the influence of gravity. This has been used to estimate the total mass of the galaxy and the number of stars in it. The Milky Way Galaxy is very similar in size and shape to our neighboring galaxy, the Andromeda Galaxy.

Such successes are a remarkable witness to Newton’s genius. For more than two centuries, scientists applied his laws of motion and the law of gravitation without discovering any discrepancies. However, some exceptions to the Newtonian world view were eventually discovered. It should not take away from his fame to admit these exceptions. They occur only when we venture very far from the realm of our ordinary senses.

In the world of very high velocities and extremely large masses, we must replace Newton’s ideas with the theories of special and general relativity (Chapter 10). In the world of the extremely small, we must use the theories of quantum mechanics (Chapter 24). It should be noted, however, when these newer theories are applied in the realm where Newton’s laws work, the new theories give the same results.

We also do not know whether the value of G varies with time. No such variation has been detected, but a small variation with time could exist. Because the measurements of G are still limited in accuracy, it has been suggested that NASA orbit two satellites about each other. Accurate knowledge of the satellites’ masses as well as their orbital data would give a more accurate value for G.

The Field Concept Implicitly, we have assumed the force between two masses to be the result of some kind of direct interaction—sort of an action-at-a-distance interaction. This type of interaction is a little unsettling because there is no direct pushing or pulling mechanism in the intervening space. Gravitational effects are evident even in situations in which there is a vacuum between the masses.

The Field Concept It is both conceptually and computationally useful to separate the gravitational interaction into two distinct steps using the field concept. First, one of the objects modifies, by virtue of its mass, the surrounding space; it produces a gravitational field at every point in space. Second, the other object interacts, by virtue of its mass, with this gravitational field to experience the force. The field concept divides the task of determining the force on a mass into two distinct parts: determining the field from the first mass, and then calculating the force that this field exerts on the second mass.

The Field Concept If this were the only purpose of the field idea, it would play a minor role in our physics world view. In fact, it probably seems as if we are trading one unsettling idea for another. However, as we continue our studies, we will find that the field takes on an identity of its own and is a valuable aid in understanding these and many other phenomena.

The Field Concept By convention the value of the gravitational field at any point in space is equal to the force experienced by a 1-kilogram mass if it were placed at that point. Then the gravitational force on any other object is the product of its mass and the gravitational field at that point. If you hold a 1-kilogram block near Earth’s surface, it feels a gravitational force of 10 newtons. Therefore, the gravitational field has a magnitude of 10 newtons per kilogram at this point. If you replace this block with a 5-kilogram block, the gravitational force changes to 50 newtons; this is just the product of (5 kilograms) and (10 newtons per kilogram).

The Field Concept We can also see that we do not need to use a 1-kilogram block to find the gravitational field. We could just as easily use the 5-kilogram block and divide the resulting force by the mass of the block: (50 newtons)/(5 kilograms) =10 newtons per kilogram.

The Field Concept Because force is a vector quantity, the gravitational field is a vector field; it has a magnitude and a direction at each point in space. It is often convenient to talk about the gravitational field rather than the gravitational force. The strength of the gravitational force depends on the object being considered, whereas the strength of the gravitational field is independent of the object. We saw in the preceding chapter that the gravitational force W could be expressed through Newton’s second law as where g is the acceleration due to gravity, 9.8 (meters per second) per second.

The Field Concept Because the gravitational force is equal to the mass times either the gravitational field or the gravitational acceleration, these two must be numerically the same. Indeed, we use the symbol g for both the gravitational field and the acceleration due to gravity. We can also show that newtons per kilogram may be rewritten as (meters per second) per second.

Summary Although no one knows what gravity is or why it exists, we can accurately describe how gravity affects the motions of objects. The same laws of motion work on Earth and in the heavens. Newton’s universal law of gravitation states that a gravitational attraction exists between every pair of objects and is given by where m1 and m2 are the masses of the two objects, r is the distance between their centers, and G is the gravitational constant. The value of G was first determined by Cavendish and is believed to be constant with time and space.

Summary The higher a satellite’s orbit, the longer it takes to complete one orbit. A satellite with a period of 1 day and positioned above the equator would appear to remain fixed in the sky. The Moon, a natural satellite, takes 27.3 days to complete one orbit around Earth. The force of gravity can be considered constant when the motion occurs over short distances near Earth’s surface. However, small variations occur in the acceleration due to gravity with latitude, elevation, and the types of surface material.

Summary At larger distances the force decreases as the square of the distance. Stars in binary systems revolving around each other and the motion of stars within galaxies support this idea. The value of the gravitational field at any point in space is equal to the force experienced by a 1-kilogram mass placed at that point.

Gravity.

Similar presentations

Presentation on theme: "Gravity."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Gravity.

Similar presentations

Presentation on theme: "Gravity."— Presentation transcript:

Similar presentations

About project

Feedback