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Defining Gravitational Potential Energy Newton's Law of Universal Gravitation states that the force of gravitational attraction between two masses, m, and m2, at any separation distance, R, is given by F =- Gm,my . Now suppose we want to increase the separation distance from R1 to R2. R2 m1 F . m2 m2 R1 R2 To increase the separation of the two masses from R1 to R2 requires work to be done to overcome the force of attraction, similar to stretching a spring. When this work is done, the gravitational potential energy increases. To derive a mathematical equation for gravitational potential energy, we need to remember the relationship between gravitational force and separation. A force versus separation graph for this situation is like the one shown below. F2 R1 R2 From our work on work and energy, we learned that the work done by a varying force is simply the area under a force-displacement graph for the interval. To find the area under an inverse square curve requires doing an integration using calculus. The resulting change in potential energy is APE, = When written in this way, the first term in the expression depends on R2 and the second term on R1. Thus, each term is an expression for the gravitational potential energy at that separation. Therefore, at any separation distance R, the gravitational potential energy, PEg, between two masses M, and M2 is given by PE =- Gm,mz R&7 The Gravitational Potential Well "he equation PE =_M always produces a negative value. As R increases, that is, as the masses get farther apart, PE increases by becoming less g 1egative. As R increases to the point of approaching infinity, R R, The escape velocity of the rocket is thus Vg = ZGA R P The binding energy is the amount of kinetic energy an object needs it needs to escape the gravitational potential well. For a rocket at rest on the surface of a planet, Eyjnging = G'"P_mk P For a satellite in orbit around a planet, its total mechanical energy is - Gm,m Euoal = 3 PE, = 1\"*;{, 5 The binding gnergy of the satellite in orbit around a planet is My M Ebinding = %MRL( 3 The energy required to launch the satellite to place it in orbit around a planet is AEigta1 = Eqotar (in 0rbit) - Eyorz) (0N planet) Question(s): The physics energy and Earth's moon as a satellite. The moon is an Earth satellite of mass 7.35 x 1022 kg, whose average distance from the centre of Earth is 3.85 x 108 m. 1. What is the gravitational potential energy of the moon with respect to Earth? 2. What is the kinetic energy and the velocity of the moon in Earth's orbit? 3. What is the binding energy of the moon to Earth? 4. What is the total mechanical energy of the moon in its orbit