Universal gravitational potential energy

Near the Earth’s surface we use $E_\text{pot,g} = mgh$, valid for small height differences. But if the body moves far away — to reach a geostationary satellite, or to escape the Earth entirely — its distance from the centre changes appreciably and $g$ is no longer constant. We then need the general formula, which accounts for the force’s dependence on $1/d^2$.

Universal gravitational potential energy

$\ev{E_{\text{pot,G}} = -G\,\frac{m_1\,m_2}{d}}$ where $d$ is the distance between the centres of the two masses.

Three crucial observations

1. The formula contains $d$, not $d^2$: the square appears only in the force law.
2. The potential energy is always negative and tends to zero only as $d\to\infty$. The bodies are “bound” because positive energy is needed to separate them all the way to infinity.
3. When one body falls towards the other, $d$ decreases and $E_\text{pot,G}$ becomes more negative: the system “loses” potential energy, which is converted into kinetic energy.

The choice of zero at infinity is not arbitrary: it is the point where the two bodies no longer interact. From there, as they are brought closer, the potential energy drops below zero. The negative value therefore measures how deep the system sits “in the well”: the deeper the well, the more energy is needed to escape it.

Gravitational potential energy is a negative function that tends to zero at infinity. Two bodies are “bound” if their total energy is negative.

Links

Topics: Gravitazione Concepts: Energia potenziale gravitazionale universale · Energia potenziale gravitazionale

Related exercises: Escape velocity from Earth (energy) · True or false on energy and orbits · Speed of a comet at perihelion