Published on January 22, 2008
GRAVITATION: GRAVITATION Slide3: 05CO, p.74 Historical Perspective to Gravitation:: Historical Perspective to Gravitation: Early View that the Earth is a Flat surface Sailors prove this to be wrong Aristotle’s view of Earth the center of Universe with all other heavenly bodies moving around it. Copernicus predicted from his Observation that Sun is the center of the Universe and all other heavenly bodies move around it even the Earth itself Copernicus prosecuted by the Church Galileo came up with same idea Slide5: Galileo imprisoned force to track back his statement regarding celestial motion Johannes Kepler formulates his observation as three famous laws for the motion of celestial bodies Sir Isaac Newton observes Apple falling from Tree Formulates the Famous Law of Gravitation Universal Constant for Gravitation calculated Albert Einstein founds how path of Light travel is influenced by gravity Slide6: p.76 Johannes Kepler’s Observations:: Johannes Kepler’s Observations: Kepler believed in Galileo and Copernicus theory of Sun centered Universe. He made some observation related to the motion of celestial bodies and laid down Laws regarding their motion Planets move around Sun because of an Interaction between them and the Sun. Planets move in an elliptical orbits around the Sun. Planets do not have constant speed during their motion rather they speed up as they come near to Sun. Slide8: This were Wonderful observation and the Laws laid down have no violation known till date. This laws were given in 16th century. Just imagine what sought of a Genius Kepler was.. !!!! Sir Isaac Newton and Gravitation:: Sir Isaac Newton and Gravitation: Slide10: Its is said that the Falling apple on Isaac Newton’s Head gave him the idea of Gravitation. (just kidding.. But for one instant thing if it would have been something else instead of an apple ??) Newton’s Gravity: Newton’s Gravity Newton hypothesized that the Moon’s acceleration was due to the force of gravity—the same gravity that causes an apple to fall from a tree. After Newton got his bachelor’s degree , he returned to the family farm at Woolsthorpe in England. Legend has it that Newton conceived his law of universal gravitation while observing an apple fall from a tree in his yard. The Demonstration—Part 1Newton’s Gravity: The Demonstration—Part 1 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 0.00272 m/s2. This is a very small acceleration. In 1 second the Moon moves about 1 km along its orbit but falls only 1.4 mm (about 1/20 inch in 0.6 miles). The Demonstration—Part 2Newton’s Gravity: The Demonstration—Part 2 Newton’s Gravity In contrast to the Moon’s acceleration, the apple has an acceleration of 9.80 m/s2 and falls about 5 meters in its first second of flight. Why are these two accelerations so different? As we saw in Chapter 2, free-falling objects all have the same acceleration independent of their masses. Newton reasoned that the Moon’s acceleration is smaller because Earth’s gravitational attraction is smaller at larger distances; it is “diluted” by distance. Interlude: Inverse-Square LawsNewton’s Gravity: Interlude: Inverse-Square Laws Newton’s Gravity Newton may not have known, but we know today that inverse-square relationships are common in nature. Light also behaves this way, and so does sound. Suppose a can of spray paint 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. If we repeat the experiment with the same gun but with a sphere that is 2 meters in radius, the paint will be only 1/4 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 1/9 as thick. The thickness of the paint decreases as the square of the radius of the sphere increases. A force reaching into space could be diluted in a similar manner. Interlude: Inverse-Square LawsNewton’s Gravity: Interlude: Inverse-Square Laws Newton’s Gravity Interlude: Inverse-Square LawsNewton’s Gravity: Interlude: Inverse-Square Laws Newton’s Gravity Question: What happens to the other quantity in an inverse-square relationship if the first quantity is cut in half? Answer: The other quantity becomes four times larger. The ConclusionNewton’s Gravity: The Conclusion Newton’s Gravity Newton now knew how gravity changed with distance: The force of Earth’s gravity exists beyond Earth and gets weaker the farther away you go. He already knew that the force of gravity depended on the object’s mass. *Remember: F = m x a. Newton’s third law of motion said that the force exerted on the Moon by Earth was equal in strength to that exerted on Earth by the Moon. They attracted each other. This symmetry indicated that both masses should be included in the same way. The gravitational force is proportional to each mass. The Law of Universal Gravitation: The Law of Universal Gravitation Having made the connection between celestial motion and motion near Earth’s surface, Newton took another, even bolder, step. He stated that the force of gravity existed between all objects, that it was 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. The Law of Universal Gravitation: The Law of Universal Gravitation Putting everything together: The proportionality of gravity to each mass, The inverse-square relationship of gravity to distance, we arrive at an equation for the gravitational force: 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. Shouldn’t it be nearly impossible to calculate…The Big Problem: Shouldn’t it be nearly impossible to calculate… The Big Problem Newton arrived at this conclusion when he was 24 years old, but he didn’t publish his results for more than 20 years. This was partly due to one unsettling aspect of his work. Shouldn’t it be nearly impossible to calculate…The Big Problem: Shouldn’t it be nearly impossible to calculate… The Big Problem The distance r 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 might seem like 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 (Figure 5-3). It seems less intuitive that all this would somehow act like a very compact mass located at Earth’s center. But that is just what happens. A (Mathematical) Way OutThe Big Problem: A (Mathematical) Way Out The Big Problem 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. Planetary Observations: Planetary Observations 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. Now these patterns had an explanation and Newton had more proof that his theory was correct. Planetary Observations: Planetary Observations When Uranus was discovered in 1781, its orbit was found to be anomalous. Factoring in the Sun and the other planets’ gravitational influence, it was still a little off. By this time, scientists were so confident of Newton’s work that the deviations were explained in terms of the influence of an unknown planet. This led to the discovery of Neptune in 1846. Further discrepancies led to the discovery of Pluto in 1930, and suggest that yet other undiscovered planets may lie beyond Pluto. The Value of G: The Value of G Newton never got to fill in a value for the constant G in his own equation. Therefore he wasn’t able to calculate the force between two objects. The way to find G is to measure the force between two known masses separated by a known distance. 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: The Value of G or 0.000 000 000 066 7. In other words, putting this value into the equation for the gravitational force tells us that the force between two 1-kg masses separated by 1 meter is only 0.000 000 000 066 7 newton. On Earth’s surface, a 1-kg mass weighs 9.8 newtons! The Value of G: 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” the Earth. 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. The Value of G: 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 km, Earth’s radius. Plugging in the numerical values yields g = 9.8 m/sec2, as expected. Because Earth orbits the Sun, the Sun’s mass can also be calculated with the Cavendish results, if we assume that the value of G is valid throughout the Solar System. The results are consistent with such an assumption. Working It Out: Gravity: Working It Out: Gravity Let’s calculate the gravitational force between two friends. Assuming that the friends are spheres [to simplify the calculation of r], have masses of 70 and 86 kg (about 154 and 189 lb, respectively), and are standing 2 m apart, we have: = 1.00 × 10−7 N. How Much Do You Weigh?: How Much Do You Weigh? According to Newton’s law of universal gravitation, a person’s weight on a planet depends on the mass and radius of the planet, as well as the mass of the person. Consider what your weight might be on Jupiter. If Jupiter were the same size as Earth only with 318 times as much mass, you would weigh 318 times as much on Jupiter as on Earth. But actually, Jupiter’s diameter is 11.2 times that of Earth’s. Because the law of universal gravitation contains the radius squared in the denominator, your weight is reduced by a factor of 11.2 squared, or 125. And its mass really is 318 × Earth’s. Combining these two factors means that you would tip a Jovian bathroom scale at 318/125, or 2½, times your weight on Earth. Your weight on each of the planets is given on the next slide. How Much Do You Weigh?: How Much Do You Weigh? Gravity Near Earth’s Surface: Gravity Near Earth’s Surface In earlier chapters we assumed that the gravitational force on an object was constant near Earth’s surface. Now you are ready to know the truth: Near Earth’s surface the gravitational force decreases by a millionth for every 3 meters (~10 feet) of gain in elevation. An individual with a mass of 50 kilograms has a weight of 500 newtons (110 pounds) in New York City; this person would weigh about 0.25 newton (1 ounce) less in mile-high Denver. Slide34: Variations in the gravitational force also 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 slightly. If an object is raised 1 kilometer (about 5/8 mile), the distance changes from 6378 km to 6379 km, and g changes from 9.800 m/s2 to 9.797 m/s2. Satellites: Satellites Replace “the Moon” with “the Apollo program” and Newton is now predicting the flight of artificial satellites—orbiting spacecraft! 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 that carried the first woman, Valentina Tereskova, into Earth orbit in June 1963. Its orbit varied in height from 170 to 210 kilometers and had a period of a little over 88 minutes. Satellites and Orbits: Satellites and Orbits The higher a satellite’s orbit, the longer it takes to complete one orbit. Vostok 6 took 88 minutes; the Moon takes 27.3 days. A satellite with an altitude of 36,000 km (5½ Earth radii) takes exactly 1 day, and its orbit is geosynchronous. If a geosynchronous satellite is positioned above the Equator, it will appear to remain fixed directly above one spot on Earth. Weather satellites can continuously observe 1 particular area. This is also a popular orbit for communications & TV satellites, because you know where to point the dish. Brochures for some of these services may tell you, “you need a clear view of the southern sky” to subscribe—i.e., the sky above the Equator. “Cable TV” in the desert. Satellites to Other Planets: Satellites to Other Planets NASA’s computers calculate the trajectories for all 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 and masses of the other bodies in the Solar System. The net force on it produces an acceleration of the spacecraft, changing its velocity. From this the computer calculates a new position for the spacecraft. Then it 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. The Tides: The Tides Before Newton’s work with gravity, no one was able to explain why we have tides. We knew what they were, though—the tides are bulges in the surface of Earth’s oceans. There are two bulges, one on each side of Earth. 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. Part 1The Reason for Tides: Part 1 The Reason for Tides Newton claimed tides were due to the Moon’s gravity. Earth exerts a gravitational force on the Moon that causes the Moon to orbit it, but also 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. (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, as shown in the figure on the next slide) Slide40: Because Earth has an orbital motion, Earth is continually falling toward the Moon, just as the Moon is Continually falling toward Earth. This is the key to understanding tidal bulges. Part 2The Reason for Tides: Part 2 The Reason for Tides Earth’s acceleration toward the Moon 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. 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. Part 3The Reason for Tides: Part 3 The Reason for Tides If there are different-sized forces at different spots on Earth, there are different accelerations for different parts. Material on the side of Earth facing the Moon tries to get ahead, while the material on the other side lags behind. We end up with a stretched-out Earth at these two points—and this accounts for the high tide. Part 3—continuedThe Reason for Tides: Part 3—continued The Reason for 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 cm (9 inches). Because the entire area moves up and down together, we don’t notice this effect. Non-Lunar Tides?: Non-Lunar Tides? We also expect to observe solar tides, because the Sun also exerts a gravitational pull on Earth and Earth is “falling” toward the Sun. These occur, but their heights are less than ½ those due to the Moon. Yet 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. When reckoning from the Sun, the difference in distance between one or the other sides of Earth is small. How Far Does Gravity Reach?: How Far Does Gravity Reach? The law of gravitation has been thoroughly tested within the Solar System. What about tests outside the Solar System? We haven’t sent any probes out that far! Voyager, the farthest man-made object, may make it out of the Solar System by 2020. However, nature has provided us with ready-made probes. 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. Slide46: 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. Slide47: Photographs of star clusters show that the gravitational interaction occurs between stars. In fact, measurements show that all the stars in our galaxy, the Milky Way, 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—giving us a model for artists’ pictures! What Is Gravity?: What Is Gravity? 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. What’s moving these objects—? Gravitational effects are evident even in situations in which there is a vacuum between the masses. No one knows what gravity is or why it exists. The Field Concept: The Field Concept However, 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: 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. Slide51: 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. 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 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 m/sec2. Gravity or No Gravity ??: Gravity or No Gravity ??