Connecting with mgh near the surface

The two formulas for potential energy — the universal $-GM_Tm/d$ and the familiar textbook $mgh$ — do not contradict each other: the second is the limiting case of the first for small heights. Seeing this explicitly reassures us that we are using the same physics at two levels of approximation.

Near the Earth’s surface we set $d = R_T + h$ with $h \ll R_T$. Expanding the universal formula to first order in $h/R_T$:

$\begin{aligned} E_\text{pot,G} &= -\frac{GM_Tm}{R_T+h} \approx -\frac{GM_Tm}{R_T}\!\left(1 - \frac{h}{R_T}\right) \\ &= -\frac{GM_Tm}{R_T} + \frac{GM_T}{R_T^2}\,mh = -\frac{GM_Tm}{R_T} + mgh \end{aligned}$

The first term is a constant (the potential energy at the surface), the second is precisely $mgh$, having recognised $g = GM_T/R_T^2$.

Why the constant does not matter

The two formulas differ only by an additive constant. Since in physics it is always the changes $\Delta E_\text{pot}$ that matter, not the absolute value, the constant cancels out: using $mgh$ or the universal formula gives the same results for small height differences.

This explains why the “zero” of potential energy is conventional: with $mgh$ we place it at ground level, with the universal formula at infinity. The measurable physics — the speeds, the heights reached — does not depend on this choice.

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Topics: Gravitazione Concepts: Energia potenziale gravitazionale universale · Energia potenziale gravitazionale Skills: Impostazione simbolica

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