Gravitational potential energy
Derivation of gravitational potential energy
If the earth is like a particle floating very far from the sun, then it barely feels any attraction from the sun (equation (1.2)). The earth is pulled and, as the distance between earth and sun decreases, the force of gravity increases. The earth finally comes to a position where it is now 1.5×10 8 km from the sun. 7 During the whole process, gravity is pulling the earth towards the direction of gravity, therefore the work done by gravity is positive ( 𝑊 = 𝐹 ×𝑥 , and F and x are in the same direction). Hence, the change in earth's gravitational potential energy is negative, because force is always in the direction of decreasing potential energy. To give an example of the above statement, we may consider a massless spring that obeys Hooke's Law, hanging freely from the wall. As we stretch it, we feel the tension caused by its elasticity pulling it upwards, while it gains elastic potential energy ( 𝐸 = 𝑘𝑥 2 /2 ). If we decrease its extension, we decrease its energy. So the direction of decreasing energy is upwards, in the same direction with the tension. Alternatively, as we compress it, we feel tension acting downwards, towards the direction of decreasing extension (i.e. decreasing energy). Kinetic energy on the other hand does not have this property, because it is not a form of potential energy. Therefore, if we define the GPE of the earth distanced infinitely far from the sun to be zero, the GPE of its final position will be negative (because, as explained above, its GPE has decreased in the process). We can use simple calculus to work out its final GPE. First, since force is in the direction of decreasing energy, and work done equals energy transfer, we have:
(2.1)
in which 'W' is work done by gravity, ' ∆𝐸 ' is the change in earth's GPE relating to the sun. Now, by the definition of work done:
That is the work done by gravity to pull the earth from infinitely far to distance 'R' from the sun. Therefore,
(2.2) The situation we are discussing here is very general, therefore formula (2.2) is indeed the formula for one's GPE when it is distanced 'R' from another object.
Principle of energy conservation
Before we deal with the application of GPE, it is important to note a principle of energy conservation. The work energy principle, which stated that the change in a system's mechanical energy is equal to the sum of work done to it from outside the system ( 𝑊 1 ), plus the work done by the forces in the system that is not contributing to the system's potential energy ( 𝑊 2 ). 8 As shown below
7 Solarsystem 2020. 8 Zhao Kaihua 2006:141.
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