Gravitational potential energy
Note that subscript 'p' means 'the previous curve', and subscript 'n' means 'the new curve'.
Therefore, 𝐴1= 𝐴2 (note the symmetry of curves in case 1), which means the work done by gravity in both cases, is the same. Thus, the final gravitational potential energy of both cases is equal. The general result for gravitation is that the potential difference of the system does not depend on how the system got to that configuration; it only depends on the configuration of the system. Hence we call gravity a conservative force.
Gravitational potential energy is linear
Finally, the total gravitational potential energy of a three-body system can be considered. To simplify matters, let us consider a system of three particles, all have mass m, located in the corners of an equilateral triangle of length R. 10 In the equation of gravity (equation (1.2)), there is no reference to a third body, because gravity is the force only based between two objects. We call this characteristic of gravity 'linear', that it can be added by simple algebra. Because gravitational potential energy is derived through work done (i.e. integration), and
Therefore, GPE is also linear. Because a system of three particles is three pairs of two ( 3𝐶2=3 ), the total potentil energy is the sum of the potential energy of each pair:
3𝐺𝑚 2 𝑅
𝐸 123 = 𝐸 12 +𝐸 13 +𝐸 23 =−
Conclusion
This essay gave a derivation for gravity using observations of planetary orbits in our solar system; introduced the link between change in energy and work done, and thus deduced a formula for gravitational potential energy. The Work Energy Principle was introduced, before looking at some applications. By analysing a system with two objects with similar mass, it is realized that gravitational potential energy only applies to at least two masses, and they and the system must 'possess' the same amount – the work done to pull them infinitely far away. Importantly, gravity as a conservative force was demonstrated by both a visual method and a mathematical method. Finally, the reason why gravity and gravitational potential energy are linear was briefly discussed, and applied to a three-body system.
Bibliography
British Physics Olympiad. 2019. BPhO Round 1 . 10. Richard P. Feynman. 2011. Six Easy Pieces . 108. Penguin Books NASA Science. n.d. ‘ How big is the Solar System ’ . Page Updated on 28 July, 2021. https://solarsystem.nasa.gov/news/1164/how-big-is-the-solar-system/
10 Quora 2019.
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