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Physics 218 Lecture 10 Dr. David Toback Physics 218, Lecture X 1 Physics 218, Lecture X 2 Overview • Newton’s gravitational law • Dynamics and Gravity • Example problems • Connection with Uniform Circular Motion • Kepler’s Laws Physics 218, Lecture X 3 Gravitation Newton’s law of Universal Gravitation “Every particle in the universe attracts every other particle” Physics 218, Lecture X 4 Large number of scales Kinda amazing! • Gravity covers the attraction between – – – – – – – – An apple near the earth The earth and the moon The earth and the sun The sun and our galaxy Our galaxy and the universe Every particle in the universe and an apple The Earth and you Bevo and Reveille Physics 218, Lecture X 5 Newton’s Law “Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the particles” • Gravity has a magnitude and direction Gravity is a force Physics 218, Lecture X 6 The Magnitude of the Force of Gravity m1m 2 Force G 2 r Distance between the masses G 6.67 10 11 N m /kg 2 2 Physics 218, Lecture X 7 Vector Form Force ON particle 2 due to particle 1 m1m 2 F21 G 2 rˆ12 r21 Physics 218, Lecture X 8 More on Vector Form r̂21 r̂12 By Newtons Laws : F21 F12 Physics 218, Lecture X 9 Bevo and Reveille Bevo and Reveille have masses m1 and m2 and are standing R meters apart. Despite what you might like to believe, what is the attraction between them? Hint: Assume a spherical cow Perhaps this explains why we’ve never observed any attraction… Physics 218, Lecture X 10 A “Spherical” Earth It takes some fancy integration, but one can show that we can “model” the Earth as if all the mass were concentrated at its center • One can model a sphere as a point • This is why we like to model things as spheres in the first place. • “spherical cow” = “point-like cow” Physics 218, Lecture X 11 Calculate the Magnitude of g • Calculate the magnitude of g • Use 24 –ME = 5.97*10 kg -11 2 –G = 6.67*10 N*M/kg –R = REarth = 6.38*106 m Physics 218, Lecture X 12 Space craft in Orbit A space craft, with mass m, is circling the earth at radius R = 2rEarth. What is the force on the space craft in terms of g and m? Model the Earth and space craft as single points at their center Physics 218, Lecture X 13 Forces and Circular Motion Tying this stuff together • Use the force of gravity along with other forces in force diagrams • Circular motion is the motion pointed towards the center of a circle • The Earth is a good “center” acceleration Physics 218, Lecture X 14 Satellites/Orbiting Problems A satellite problem is a good example of the substantive problems we need to be able to solve. Predict the outcome of the experiment Physics 218, Lecture X 15 How to solve these types of problems • Some thoughts – What keeps a satellite up? Its speed – Accel = v2/r – Force = ma = mv2/r • The trick is going to be to ask the question – What are the forces? Is it in uniform circular motion? If so, we can use Newton’s law: FGravity = FUniform Circular Motion Physics 218, Lecture X 16 Geosynchronous Satellite A satellite is in orbit around the earth and its speed is such that it always stays above the same point on the earth throughout the day. Assuming a spherical Earth with mass ME: • What is the period of the satellite? • Determine the height of the satellite in terms of the period, ME and G • Determine its speed • Compare this speed to the speed at a height twice as far from the center of the earth Physics 218, Lecture X 17 This Week • Next Lecture: –Kepler’s Laws –Examples • HW due Monday Physics 218, Lecture X 18 Weightlessness • What is the weight of the person in the figure? • What is the difference between being in “free fall” and being “out of the reach of large gravitational forces?” Physics 218, Lecture X 19 Throwing a Baseball A person throws a baseball at 100 km/hr, but it is attracted back towards him because of gravity. • Estimate the force 1 Meter away? • Assuming constant acceleration (Bad assumption), how long does it take to turn around? Physics 218, Lecture X 20