13.10 Absolute potential energy

There is a disconcerting aspect of potential energy. In the earlier module titled “Potential energy”, we defined “change in potential energy” – not the potential energy itself!

We assigned zero gravitational potential reference for Earth’s gravitation to the ground level and zero elastic potential energy to the neutral position of the spring. The consideration of zero reference potential energy enabled us to define and assign potential energy for a unique position – not to the difference of positions. This was certainly an improvement towards giving meaning to absolute value of potential energy of a system. In this module, we shall broaden the reference and aim to define absolute potential energy for a particular configuration of a system in general.

Reference at infinity

The references to ground for gravitation or a neutral position for a spring are essentially local context. For example, gravitation is not confined to Earth system only. What if we want to refer potential energy value to an object on the surface of our moon? Would we refer its potential energy in reference to Earth’s ground?

We may argue that we can have moon’s ground as reference for the object on its surface. But this will also not serve purpose as there might be occasions (as always is in the study of the motions of celestial bodies) where we would need to compare potential energies of systems belonging to Earth and moon simultaneously. The point is that the general concept of potential energy can not be bounded to a local reference. We need to expand the meaning of reference, which is valid everywhere.

Now, we have seen that change in potential energy is equal to negative of work by conservative force. So existence of potential energy is related to existence of conservative force. Can we think a situation in which this conservative force is guaranteed to be zero. There is no such physical reference, but there is a theoretical possibility of such eventuality. Let us have a look at the Newton’s law of gravitation (this law will be discussed subsequently). The force of gravitation between two particles, “ $m_1$ ” and “ $m_1$ ” is given by :

$$F=\frac{G{m}_1{m}_2}{r^2}$$

As $r\to \infty ,F\to 0$ . As there is no force on the particle, there is no work involved. Hence, we can conclude that a system of two particles at a large (infinite) distance has zero potential. As infinity is undefined, we can think of system of particles at infinity, which are separated by infinite distances and thus have zero potential energy.

Theoretically, it is also considered that kinetic energy of the particle at infinity is zero. Hence, mechanical energy of the system of particles, being equal to the sum of potential and kinetic energy, is also zero at infinity.

Infinity appears to serve as universal zero reference. The measurement of potential energy of any system with respect to this zero reference is a unique value for a specific configuration of the system. Importantly, this is valid for all conservative force system and not confined to a particular force type like gravitation.