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Universal Gravitation Motion in the Heavens and on Earth Objectives Relate Kepler's Law to Newton's Law. Calculate the periods & speeds of orbiting objects. Describe the method Cavendish used to measure G. Motion in the Heavens and on Earth Because of the work of early scientists (Galileo, Kepler, Newton, etc..) we know that planets, stars, comets and other bodies follow the same laws as objects do on Earth. Observed Motion Kepler, only an assistant to Tycho Brahe in the 1600's, was convinced that our universe was sun centered and that mathematics could explain the number, distance and motion of the planets. Kepler discovered laws that describe the motion of astronomical bodies after careful analysis of Brahe’s data Observed Motion These 1. 2. 3. are Keppler’s laws: The paths of the planets are ellipses with the sun at one focus. Planets move faster when they are closer to the sun. The square of the ratio of the periods of any two planets revolving about the sun ( TA/TB )2, is equal to the cube of the ratio of their average distances from the sun ( RA/RB)3. Universal Gravitation Newton used Kepler's 1st law, (paths of planets are ellipses) to determine that the magnitude of the force (F) on the planet resulting from the sun must vary inversely with the square of the distances between the center of the planet and the center of the sun. Universal Gravitation (F is proportional to 1/d2) (∝)=means proportional to d = distance between the centers of the two bodies. Universal Gravitation Newton later wrote that the apple falling straight down made him wonder if the same force extended beyond to the clouds, moon and even beyond. The force of attraction is proportional to their masses, (the apple to the Earth, and the Earth to the apple). This attractive force between all objects is Gravitational Force. Universal Gravitation Newton was confident that the laws governing Earth would work anywhere in the universe on any two masses, (Ma and Mb). This is the Law of Universal Gravitation. G is a universal constant. F= G (MaMb/d2) Universal Gravitation If the mass of a planet near the sun were doubled, the force of the attraction would be doubled. If a planet were near a star having twice the mass of the sun, the force between the two bodies would be twice as great. If a planet were twice the distance from the sun, the gravitational force would be only one quarter as strong. Using Newton's Law of Universal Gravity Mp = Mass of planet ac = Centripetal acceleration Ms = Mass of sun Fg = Gravitational Force r = radius of the planet's orbit F = ma F = Mp ac Using Newton's Law of Universal Gravity Assume circular orbits and use ac = 4π2r/T2 and substitute for ac, so...... F = Mpac now becomes F= Mp(4π2r/T2) Using Newton's Law of Universal Gravity Now, set this equal to Newton's Law of Universal Gravitation and arrive at: G(MSMP/r2) = MP4π2r/T2 or T2=(4π2/GMS)r3 This is Kepler's 3rd law of planetary motion, or the period of an planet orbiting the Sun!! Weighing the Earth Cavendish (1798) invented the equipment to measure the gravitational force between two objects, G. He used a rod, 4 lead spheres, wire, a mirror and a light source. Weighing the Earth He substituted values for force, mass and distance into Newton's law and found a value for G. G = 6.67 x 10-11 Nm2/Kg2 Weighing the Earth Ex: Find the gravitational force between two objects where mass is 7.26 Kg, and their centers are separated by .30 m. Fg = G(MAMB/r2 ) Fg = 6.67×10-11Nm2/kg2 (7.26kg × 7.26kg/(.30m)2) Fg = 3.9×10-8 N Weighing the Earth Determine the force of gravitational attraction between the Earth (m=5.98 x 1024kg) and a 70 kg physics student if the student is standing at sea level, a distance of 6.37 x 106 m from Earth's center. Fg = G(MAMB/r2 ) Fg = 6.67×10-11Nm2/kg2 (5.98 x 1024kg × 70kg/(6.37x106 m)2) Fg = 688.09 N Weighing the Earth Ex: use To find the weight (force) of the Earth, Fg=GME/r2 We or ME=gr2/G know that the Earth's radius is 6.38 x 106m and that ag or g is 9.8 m/s2, and that G is 6.67 x 10-11Nm2/Kg2