Rongzhou: Created page with "Gravitational potential energy, commonly noted $U$, refers to the energy incorporated in a system as a result of the gravitational attraction between its elements. == Zero reference point == Generally, we define the zero reference point: {{#content:Q1779}} Since there is no gravitational interaction when objects are at an infinite distance from each other. Then, the gravitational potential energy of a system measures the amount of work done by gravitational forces t..."

2026-09-24T13:42:55Z

Created page with "Gravitational potential energy, commonly noted $U$, refers to the energy incorporated in a system as a result of the gravitational attraction between its elements. == Zero reference point == Generally, we define the zero reference point: {{#content:Q1779}} Since there is no gravitational interaction when objects are at an infinite distance from each other. Then, the gravitational potential energy of a system measures the amount of work done by gravitational forces t..."

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Gravitational potential energy, commonly noted $U$, refers to the energy incorporated in a system as a result of the gravitational attraction between its elements.

== Zero reference point ==

Generally, we define the zero reference point:

{{#content:Q1779}}

Since there is no gravitational interaction when objects are at an infinite distance from each other.

Then, the gravitational potential energy of a system measures the amount of work done by gravitational forces to move all masses of the system away from each other until the distance between them is infinite.

== Basic case: 2 point masses ==

For two point masses $m$, $M$, we can calculate the gravitational potential energy of the system mostly easily by constructing a cartesian coordinate system with one of the point masses, $M$, at the origin.

Then, if $m$ has position vector $ \mathbf r_1$:

$$\mathbf F_g(m)=-G\frac{Mm}{r_1^2}\hat{\mathbf r}_1$$

The potential energy of the system is:

$$U=-\int_{\infty}^{r_1}\left(-G\frac{Mm}{r^2}\right) \hat{\mathbf r} \cdot dr=-G\frac{Mm}{r}$$

<blockquote>
$$\hat{\mathbf r} \cdot dr = dr$$
</blockquote>

== Many point masses ==

Recall the gravitational potential energy between 2 point masses is

{{#content:Q1780}}

Applying the formula to all pairs of point masses in the system:

{{#content:Q1781}}

We can similarly derive a formula for the gravitational potential energy of a point mass external to the system:

{{#content:Q1782}}

== Continuous mass distribution ==

Any generic mass system can be modelled by a continuous mass distribution, with $\rho=0$ where there is no mass.

If the density at point with position vector $\mathbf{r}$ is $\rho(\mathbf{r})$:

{{#content:Q1783}}

<blockquote>
$dV=dr^3$: infinitesimal volume
</blockquote>

== Gravitational potential ==

If a test mass $m$ has gravitational potential energy $U(\mathbf{r})$ at a point, then the gravitational potential at that point is

$$\Phi(\mathbf{r})=\frac{U(\mathbf{r})}{m}$$

For another test mass $m_1$, the gravitational potential energy at the same point is then simply:

{{#content:Q1784}}

Gravitational potential energy - Revision history