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Gravitation 1 Topics Covered • From The Horse’s Mouth: “I deduced that the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly.” -Isaac Newton • Gravitational Force: The magnitude of the gravitational force acting between two objects is given by Fg = Gm1 m2 /r2 , where G is the gravitational constant (6.67 × 10−11 Nm2 /kg2 ), m1 and m2 are the masses, and r is the distance between them. You should be able to figure out the direction! • Gravitational Force: The formula above is technically only valid for point masses. However, Newton invented calculus specifically to show that gravity treats an extended body like a point mass located at its center of mass. • Gravitational Potential Energy: Potential energy is given by U = −Gm1 m2 /r. Don’t be fooled by the minus sign; this energy works just like the old one (mgh) in that it increases with increasing distance. (Do you see why?) • Escape Velocity: p An object of any mass can escape from a planet of mass m and radius r if its initial velocity is Vesc = 2Gm/r. p • Satellite Speed: The speed of an object in orbit around a planet of mass m is given by V = Gm/r, where r is the distance between the object and the center of the planet. • Kepler’s Laws: 1. The orbit of every planet is an ellipse with the Sun at a focus. 2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. (A fancy expression of conservation of angular momentum) 3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. (In fact, T 2 = 4π 2 a3 /Gm, where T is the period, a is the semi-major axis of the orbit, and m is the mass of the central body, i.e., the sun.) 2 Problems 1. What is the magnitude of the gravitational force at the center of the Earth? Hint: split the Earth in half. 2. In an episode of Family Guy, Peter Griffin becomes so massive that a book, a glass, and a television begin to orbit him with period T . If Peter eats more food and doubles his mass, what will the new orbital period be in terms of the old one? 3. Four planets of mass 4 × 1023 kg sit at the corners of a square with side length 1000 km. What is the force on one of the planets due to the others? Find the magnitude and the direction. 4. We know from before that the gravitational force near the Earth’s surface is given by Fg = mg. Use your two versions of the gravitational force to find an expression for g. 1 5. A satellite circles planet Roton every 2.8 h in an orbit having a radius of 1.2 × 107 m. If the radius of Roton is 5.0 × 106 m, what is the escape velocity from the surface of Roton? 6. The United States government wants to place a surveillance satellite in orbit around the Earth so that it is directly above your room at all times. Will this work? Hint: use Kepler’s First Law. 7. A communications company wants to place a satellite in orbit around the equator so that it appears stationary from the Earth. What altitude should it be at? The mass of the Earth is 5.975 × 1024 kg and the radius of the Earth is 6371 km. 2