Topic #13: Universal Gravitation and Satellites (Teacher)

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Spencer Snow
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1 The Development of the Theory of Universal Gravitation Before Newton developed the theory of universal gravitation, there were two separate notions of gravity. Terrestrial gravity was thought to be the force holding things to the surface of the Earth. It was not clear that this force was the same as astronomical gravity which was responsible for the motion of the planets, moons and comets. Newton showed that these two forces were actually the same; he unified terrestrial and astronomical gravity. There is a popular story that Newton was sitting under an apple tree, an apple fell on his head, and he suddenly came up with the theory of universal gravitation. This is almost certainly not true. What is thought to have actually happened is that Newton, upon observing an apple fall from an apple tree, thought: if the apple accelerates toward the ground, there must be a force that acts on it. Let's call this force "the force due to gravity", and its acceleration we ll call the acceleration due to gravity". Also, if the apple tree was twice as high, the apple would still accelerate toward the ground, so the force due to gravity must reach to the top of the tallest apple tree. Perhaps, the force of gravity reaches even further, perhaps all the way to the Moon! This would mean that the orbit of the Moon around the Earth is caused by the force of gravity. Page 1 of 14

2 This idea can be illustrated with the thought experiment that follows. If we fire a cannonball horizontally from a tall building, it will eventually fall to earth because of the force due to gravity, as indicated by the trajectory marked 1 in the figure. As we increase the muzzle velocity of the cannon, the projectile will travel further and further before returning to earth. Finally, Newton reasoned that if the cannon projected the cannonball with exactly the right velocity, the cannonball would travel completely around the Earth, always falling in the gravitational field but never reaching the Earth. That is, the cannonball is put into orbit around the Earth. Newton concluded that the orbit of the Moon was of exactly the same nature: the Moon continuously "fell" in its path around the Earth because of the acceleration due to gravity, thus producing its orbit. By such reasoning, Newton concluded that terrestrial and astronomical gravity were in fact the same and that any two objects in the Universe exert gravitational attraction on each other, with the force having a universal form: (1.11) Note that G is the same everywhere hence the name Newton s Law of Universal Gravitation Page 2 of 14

3 Weight and the Gravitational Force In everyday conversation, the terms mass and weight are often used to mean the same thing. In reality they are quite different. Weight is defined as: the gravitational force exerted on an object of a certain mass by another mass. Mathematically, the weight is: (1.11) where one of the masses is the mass of the Earth. We can also define the acceleration due to gravity as: The acceleration due to gravity is approximately the product of the universal gravitational constant G and the mass of the Earth M, divided by the radius of the Earth, r, squared. (We assume the Earth to be spherical and neglect the radius of the object relative to the radius of the Earth in this discussion.) The measured gravitational acceleration at the Earth's surface is found to be about 9.80 m/second/second. So the weight of a mass m at the surface of the Earth is obtained by multiplying the mass m by the acceleration due to gravity, g, at the surface of the Earth. This means that the mass of a specific object is a constant for that object but its weight depends on its location. If we transport an object of mass m to the surface of the Moon, the gravitational acceleration would change because the radius and mass of the Moon both differ from those of the Earth. Page 3 of 14

1.13.3 Using Universal Gravitation to Weigh the Earth Towards the latter half of the 18th century, spurred by his interest in the structure and composition of the interior of the earth, Henry

4 Using Universal Gravitation to Weigh the Earth Towards the latter half of the 18th century, spurred by his interest in the structure and composition of the interior of the earth, Henry Cavendish in a 1783 letter to his friend Rev. John Michell discussed the possibility of devising an experiment to "weigh the earth." Borrowing an idea from the French scientist Coulomb who had investigated the electrical force between small charged metal spheres, Michell suggested using a torsion balance to detect the tiny gravitational attraction between metal spheres and set about constructing an appropriate apparatus. He died in 1793, however, before conducting experiments with the apparatus. Cavendish, Henry ( ) The apparatus eventually made its way to Cavendish's home/laboratory, where he rebuilt most of it. His balance was constructed from a 2 metre wooden rod suspended by a metal fiber, with 5 cm. diameter lead spheres mounted on each end of the rod. These were attracted to 160kg. lead spheres brought close to the enclosure housing the rod, roughly as shown below. He began his experiments to "weigh the world" in 1797 at the age of 67, and published his result in 1798 that the average density of the earth is 5.48 times that of water. His work was done with such care that this value was not improved upon for over a century. The modern value for the mean density of the earth is 5.52 times the density of water. Cavendish's extraordinary attention to detail and to the quantification of the errors in this experiment has lead C. W. F. Everitt to describe this experiment as the first experiment in modern physics. Page 4 of 14

5 Apparent Weight In our daily experience, our apparent weight is equal to the Normal force applied by the floor upwards on our feet (This is what a bathroom scale measures). When you ride in an elevator, the normal force you feel varies which you interpret as your apparent weight. Three situations are worth examining here: 1. The elevator is at rest or moving at a constant speed In this case, the normal force you feel is exactly equal to your weight and you feel normal 2. The elevator accelerates upward The normal force in this case is greater than it usually is so you feel heavier 3. The elevator accelerates downward The normal force in this case is less than it usually is so you feel lighter 4. The cable snaps The normal force in this case is zero. Any situation in which we feel no normal (supporting) force, we feel as though our weight were zero, aka weightless. This sensation has also been called zero gravity and micro gravity however these terms are misleading since the force due to gravity is not zero. Page 5 of 14

6 Eg.#1 Determine the apparent weight of a 50 kg. person standing in an elevator, a) that is motionless b) that accelerates upward at 2.0 m/s 2 c) that moves downward at 1.5 m/s d) That is in free-fall Satellites and Space Stations in Geosynchronous Orbits A satellite in geosynchronous orbit circles the earth once each day. In other words, its orbital period is 24 hours. Eg.#2 What distance above the Earth s surface must a satellite be in order to be geosynchronous (the radius of the Earth is 6378 km.)? Page 6 of 14

For a satellite's orbit period to be one Earth day, it must be approximately 36000 kilometers above the earth's surface. That is a lot higher than the Shuttle ever goes (usually about 300 kilometers).

Otherwise, from the earth the satellite would appear to move in a north-south line every day. We call that "orbiting in the equatorial plane.

7 For a satellite's orbit period to be one Earth day, it must be approximately kilometers above the earth's surface. That is a lot higher than the Shuttle ever goes (usually about 300 kilometers). We calculate this height using, what are today, common geometric formulas. To stay over the same spot on earth, a geostationary satellite also has to be directly above the equator. Otherwise, from the earth the satellite would appear to move in a north-south line every day. We call that "orbiting in the equatorial plane." The Shuttle's orbit is always inclined to the equator by at least 28.5 degrees. Given this and the Shuttle's relatively low orbit, getting a satellite from its deployment orbit to its final geosynchronous orbit takes an Inertial Upper Stage (IUS) for a boost, and something called a Hohmann transfer. A Hohmann transfer is a fuel efficient way to transfer from one circular orbit to another circular orbit that is in the same plane (same inclination), but a different altitude. Inertial Upper Stage Boeing corp. To change from a lower orbit (A) to a higher orbit (C), an engine is first fired in the opposite direction from the direction the vehicle is traveling. This will add velocity to the vehicle causing its orbit to become elliptical (B). This elliptic orbit is carefully designed to reach the desired final altitude of the higher orbit (C). In this way the elliptic orbit or transfer orbit is tangent to both the original orbit (A) and the final orbit (C). This is why a Hohmann transfer is fuel efficient. When the target altitude is reached the engine is fired in the same manner as before but this time the added velocity is planned such that the elliptic transfer orbit is circularized at the new altitude of orbit (C). The Geometry of a Hohmann Transfer Page 7 of 14

1.13.6 Global Positioning System GPS, which stands for Global Positioning System, is the only system today able to show you your exact position on the Earth anytime, in any weather, anywhere to

8 Global Positioning System GPS, which stands for Global Positioning System, is the only system today able to show you your exact position on the Earth anytime, in any weather, anywhere to within about 10 metres. Even greater accuracy, usually within less than one metre, can be obtained with corrections calculated by a GPS receiver at a known fixed location. GPS has 3 parts: the space segment (the satellites), the user segment (the handheld or car mounted receiver), and the control segment (5 ground stations to ensure the satellites are working properly). The Three Parts of the G.P.S. System Page 8 of 14

9 1.13.6a GPS Satellites The complete GPS space system includes 24 satellites, 20,000 km. above the Earth, which each take 12 hours to orbit the Earth. They are positioned so that signals from six of them can be received nearly 100 percent of the time at any point on Earth. Six signals are required in order to get the best position information. G.P.S. satellites are equipped with very precise clocks that keep accurate time to within three nanoseconds. This precise timing is important because the receiver must determine exactly how long it takes for signals to travel from each GPS satellite. The receiver uses this information to calculate its position. The first GPS satellite was launched in The first 10 satellites were developmental satellites, called Block I. From 1989 to 1993, 23 production satellites, called Block II, were launched. The launch of the 24th satellite in 1994 completed the system. GPS Nominal Constellation Twenty Four Satellites in Six Orbital Planes Four Satellites in Each Orbital Plane Page 9 of 14

1.13.6b GPS Ground Control Stations The GPS ground control segment consists of five unmanned monitor stations located around the world: Hawaii in the Pacific Ocean Kwajalein in the Pacific Ocean;

10 1.13.6b GPS Ground Control Stations The GPS ground control segment consists of five unmanned monitor stations located around the world: Hawaii in the Pacific Ocean Kwajalein in the Pacific Ocean; Diego Garcia in the Indian Ocean; Ascension Island in the Atlantic Ocean; and Schriever Air Force Base in Colorado Springs, Colorado (the master station) These ground stations broadcast signals to the satellites and also track and monitor them c GPS Receivers GPS receivers detect, decode, and process GPS satellite signals. Many different receiver models are already in use. The typical hand-held receiver is about the size of a cellular telephone, and the newer models are even smaller. Page 10 of 14

11 Worksheet Four masses are located on a plane as shown below. What is the net gravitational force on m due to the other three masses? 1 2. Do you agree with the statement, There is no location anywhere in the universe where a body can exist with no force acting on it? Explain your answer. Given that, there can never be a location where since this would never happen unless. Page 11 of 14

3. At a certain point between earth and the moon, the net gravitational force exerted on an object by Earth and the Moon is zero. The Earth Moon centre to centre distance is 384000 km.

12 3. At a certain point between earth and the moon, the net gravitational force exerted on an object by Earth and the Moon is zero. The Earth Moon centre to centre distance is km. and the mass of the moon is 1.2% the mass of the Earth a) Where is this point located? b) What is the physical meaning of the root of the quadratic equation whose value exceeds the Earth Moon distance? This is the location where the forces are equal but are in the same direction. Page 12 of 14

13 4. The International Space Station follows an orbit that is, on average, 450 km. above the surface of the Earth. a) Determine the speed of the ISS b) Determine the orbital period of the ISS Page 13 of 14

14 5. Astronomers have identified a black hole at the centre of the galaxy M87. From the properties of the light observed, they have measured material orbiting at a distance of x 10 m. from centre of the black hole, travelling at an estimated speed of 7.5 x 5 10 m./s. a) Assuming that the material is in a circular orbit, calculate the mass of the black hole. b) What is the ratio of the mass of the black hole to the mass of the Sun (1.99 x kg.) Page 14 of 14

Unit #1: Dynamics. Introduction. Introduction

Unit #1: Dynamics. Introduction. Introduction Unit #1: Dynamics Lesson #14: Universal Gravita7on 1/26 Introduction Before Newton developed the theory of universal gravita7on, there were two separate no7ons of gravity. Terrestrial gravity was thought