Multiple Definition For Gravitational Potential Energy?

Question

This may just be a simple Misconception Question, here goes:

Definition for Gravitational Potential Energy:

1. The work done by gravity to pull an object to the ground.

   $E=-(\frac{GMm}{r}-\frac{GMm}{R})$
   Where, $r =$ Distance from Centre of Mass , $R =$ Radius of Earth

   Example:
   Object of 1 kg released from 1 meter on the surface of Earth
   $E=-(\frac{GMm}{r}-\frac{GMm}{R})$
   $E = -(\frac{( 6.67 x 10^{-11}Nm^2kg^{-2} )(6.0x10^{24}kg)(1kg)}{6.4x10^6m + 1m}-\frac{( 6.67 x 10^{-11}Nm^2kg^{-2} )(6.0x10^{24}kg)(1kg)}{6.4x10^6m})$
   $E = 9.77J$

   Which is quite consistent from the formula unless $h$ is too big
   $E = mgh$
   $E = (1kg)(9.81ms^{-1})(1m)$
   $E = 9.81J$

2. The work done by gravity to pull an object from infinity to point 'r'.

   $E=-\frac{GMm}{r}$
   Where, $r =$ Distance from Centre of Mass

So,
Question is why are they using the same name if they don't represent the same thing?

Answer 1

As per the formal definition: the potential energy of a field $\mathbf{v}$ is any function $f$ such that $\mathbf{v} = - \textrm{grad} f$ anywhere in the domain of definition of $\mathbf{v}$ (or wherever it makes sense). According to the above if $f$ is a potential for the field $\mathbf{v}$ so is $f + c, c$ being any constant; therefore if a field admits one potential energy, it admits infinite thereof and you can span some of them with at least an additional constant. To directly answer your question:

Question is why are they using the same name if they don't represent the same thing?

The definition of potential energy is unique but the collection of functions satisfying it is infinite.

Furthermore the potential energies are such that the work done by the field between two points can be expressed as the difference in potential energy between the two, i. e. $$ W_{A\to B}(\textbf{v}) = f(A) - f(B). $$ One can now easily see that no matter which of the functions in the collection you have chosen, the additional constant always drops off when subtracting the contributions on the right hand side. Consequently, it does not really matter which one you choose as long as you look at differences and work done (which is what one does in physics).

This said, it is very often conventional to choose the additional constant to be zero just for the sake of avoiding to append one more term that by no means contributes. Since, given the fields at hand, the potential energy scales as $1/r$, it reaches zero when $r\to\infty$, so that one reads that as to be the work done by the field to move a unit (mass) charge from where it stands to infinity.

Answer 2

When working with potential energy in classical mechanics we always compare the difference in energy between different points. It's only this difference that matters and not the absolute value of the energy (which is not observable). We are therefore free to take the zero-point wherever we want. This is the difference between the two formulas: in the second formula the zero point is at infinity and in the first the zero-point is at $r=R$.

Since the choice of zero point does not matter for the dynamics both formulas give rise to the same physics. That being said the second formula is the most natural choice to make (as is does not have the unneccesary reference to the Earth in it).