The nature of gravitational potential energy
Albert Dai
Introduction
Gravitational potential energy (GPE) is the potential energy a massive object has in relation to another massive object due to gravity. 1 It is the potential energy associated with the gravitational field. Its effect in everyday life can be seen when an object falls to the ground. The object gains speed because its gravitational potential energy is partly converted to kinetic energy, through work done by gravity. The formula GPE=mgh comes from the principle that the work done=force*distance. 'mg' is the weight of the object, and 'h' is the distance. However, it is only applicable on earth, where gravitational attraction of the object from the earth is 'mg'. It is a plausible approximation because 'g' does not change very much under the height of 100km, since earth has an average radius of 6371km. In this essay, we shall explore the general formula for gravitational potential energy, and discuss several important applications.
Derivation for gravity
An accurate mathematical description of the force is needed for its related potential energy. Gravity was first considered as the force that attracted various planets to the sun. The orbits are nearly circular, therefore the attracting force from the sun to the planet must satisfy the property of
Because gravity does not depend on one's velocity, linking 'v' with the planet's mass and the radius of its orbit is crucial. By looking at the data collected by Brahe about planetary obits in our solar system, Johannes Kepler suggested that
where T was the period of obit, and K was a constant that was only related to the centre of rotation, the sun. We also have 𝑣 = 2𝜋𝑟/𝑇 , then we can eliminate both 'v' and 'T' (the two variables that we deemed as irrelevant):
Newton's third law tells us the sun must also experience F'=F. Hence, the sun has the same status relating to the existence and properties of gravity. 2 Given that 𝐹 ∝ 𝑚 , there must be 𝐹′ ∝ 𝑀 , in which 'M' is the mass of the sun. Reminding ourselves that K is not related to planets, only the sun. By looking at the previous equation, we must have 𝐾 ∝ 𝑀 (i.e. K/M is a constant). 3 Therefore,
1 Wikipedia, n.d. 2 Zhao Kaihua 2006, 107. 3 Zhao Kaihua 2006: 107.
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