Gravitational Potential Energy

Gravitational Potential Energy

The concept of gravitational potential energy is central to understanding how objects interact in a gravitational field. While you have encountered $mgh$ as gravitational potential energy near the Earth's surface, that formula is only an approximation valid when the height $h$ is small compared to the Earth's radius. The full treatment requires a more general expression that works for any separation between two masses.

Consider two point masses $m_1$ and $m_2$ separated by a distance $r$. The gravitational force between them is attractive and given by Newton's law of gravitation:

This force is conservative, meaning the work done by it depends only on the initial and final positions, not on the path taken. For a conservative force, we can define a potential energy function $U(r)$ such that the work done by the gravitational force equals the negative of the change in potential energy:

The gravitational force is directed along the line joining the two masses. If we move one mass radially outward (increasing $r$), the force does negative work because it opposes the displacement. The work done by the gravitational force when the separation changes from $r_i$ to $r_f$ is:

The negative sign appears because the force is attractive (toward the other mass) while the displacement $dr$ is outward. Evaluating this integral:

Since $W = U_i - U_f$, we have:

This tells us that the potential energy function must satisfy:

The constant is arbitrary — only differences in potential energy matter physically. By convention, we choose the constant so that the potential energy is zero when the masses are infinitely far apart ($r \to \infty$). This gives:

This is the gravitational potential energy of a system of two point masses separated by distance $r$. The negative sign is crucial: it means the potential energy decreases (becomes more negative) as the masses come closer together, and increases (toward zero) as they move apart.

Properties of Gravitational Potential Energy

The textbook lists several important properties of this potential energy function. Each one follows directly from the definition and the nature of the gravitational force.

Property (I): The gravitational potential energy is always negative for finite separations.

This is evident from the formula $U(r) = -G m_1 m_2 / r$. Since $G$, $m_1$, $m_2$, and $r$ are all positive quantities, the expression is always negative. The only way to get $U = 0$ is to have $r \to \infty$, which is a limiting case, not a finite separation.

Property (II): The gravitational potential energy increases (becomes less negative) as the separation increases, and decreases (becomes more negative) as the separation decreases.

›Proof

Consider the derivative of $U$ with respect to $r$:

$\frac{dU}{dr} = \frac{d}{dr}\left(-\frac{G m_1 m_2}{r}\right) = \frac{G m_1 m_2}{r^2}$

This derivative is positive for all $r > 0$. A positive derivative means that $U$ increases as $r$ increases. Since $U$ is negative, "increases" means it becomes less negative (moves toward zero). Conversely, as $r$ decreases, $U$ decreases (becomes more negative). This matches our physical intuition: you must do work to pull the masses apart, increasing their potential energy.

Property (III): The gravitational potential energy of a system of more than two particles is the sum of the potential energies of all pairs.

This is the principle of superposition for gravitational potential energy. For a system of $n$ particles with masses $m_1, m_2, \ldots, m_n$, the total gravitational potential energy is:

where the sum is over all distinct pairs $(i, j)$ with $i < j$, and $r_{ij}$ is the distance between masses $m_i$ and $m_j$.

Property (IV): The gravitational potential energy of a system depends only on the positions of the masses, not on how they arrived at those positions.

This is a direct consequence of gravity being a conservative force. The work done by gravity depends only on the initial and final configurations, not on the path taken. Therefore, the potential energy is a state function — it has a unique value for any given arrangement of masses.

Gravitational Potential Energy Near the Earth's Surface

The general formula $U = -GMm/r$ must reduce to the familiar $mgh$ for small heights near the Earth's surface. Let's verify this.

Consider a mass $m$ at a height $h$ above the Earth's surface. The Earth has mass $M_E$ and radius $R_E$. The distance from the Earth's centre to the mass is $r = R_E + h$. The potential energy is:

We want to compare this to the potential energy at the Earth's surface ($h = 0$): …