Gravitational potential and energy (5.4.4)
On this page
The gravitational potential at a point is the work done per unit mass to bring a test mass from an infinite distance to a chosen point inside a body’s gravitational field.
At an infinite distance away from an object, the gravitational potential is defined as zero:
A mass at this point feels no force due to the object’s gravitational field.
Gravitational potential is always negative because:
- Gravitational forces are attractive. Energy is required to move a mass away from another mass. Therefore, the work done to move a mass from infinity towards another mass is negative.
- At any point within a gravitational field, the potential is lower than at infinity (where ).
Gravitational potential can be thought of as a ‘gravitational well’. Moving a mass to infinity is like climbing out of the well – energy is needed to reach the ‘zero’ level. A larger mass has a stronger gravitational field, corresponding to a larger ‘gravitational well’.
Gravitational potential energy is the energy a mass has due to its position in a gravitational field. It can be found from by multiplying by the object’s mass :
Gravitational potential at a point is the gravitational potential energy per unit mass.
Gravitational field strength is given by:
Where:
- is Newton’s gravitational constant ,
- is the mass of the body producing the gravitational field (kg),
- is the distance (m) from the centre of the mass to the point in the field, and
- is measured in
Comparing this to the equation for gravitational potential shows that
Where:
- is the energy per unit mass at a point in a gravitational field.
- is the force per unit mass acting on an object in a gravitational field.
It is important to note that and are directly proportional to each other, but they are distinctly different quantities. Moreover, both quantities are generally measured from the body’s centre of mass. However, for practical applications, we often express these quantities at different distances from the centre, including the surface.
The zero point for gravitational potential is at a distance of infinity. Hence, all distances closer than infinity have negative gravitational potential – this reflects that work is required to escape a gravitational field.
The gravitational potential at a distance from a mass is given by:
Where:
- is Newton’s gravitational constant ,
- is the mass of the body producing the gravitational field ,
- is the distance in from the centre of the mass to the point in the field.
is negative because gravitational forces are attractive, and has units of
The closer a point is to the mass the more negative the gravitational potential. As increases, becomes less negative, approaching zero at infinity.
Work is required to move a unit mass away from a planet or mass This work increases as the object moves further away.
Two points at different distances from a mass have different gravitational potentials because increases (becomes less negative) with distance.
The gravitational potential difference between two points is equal to:
Where:
- is the final gravitational potential,
- is the initial gravitational potential,
- is the change in gravitational potential.
All quantities are in
The change in potential between two distances (initial) and (final) from a mass is:
Where:
- is the gravitational constant
- is the mass of the object creating the field ()
- is the initial distance from the mass ()
- is the final distance from the mass ()
The change in potential is measured in
Question walkthrough
Work Done Moving a Satellite in a Gravitational Field
Calculate the work done moving a satellite between two distances from a planet's centre using the change in gravitational potential.
Newton’s law of gravitation states that the force between two masses and is given by:
Where:
- is Newton’s gravitational constant
- and are the masses of the larger body and small body, respectively (),
- is the distance between the two masses ().
This equation shows gravitational force is inversely proportional to This means a force–distance graph will be a curve (not a straight line). As increases, decreases rapidly and approaches zero at large
Work done is the product of force and distance
In a force–distance graph, the area under the curve represents the work done (or energy transferred) to move an object between two points.
When a satellite (mass is moved from one point (A) to another (B) in a gravitational field:
- Gravity is attractive – work is required to move the satellite away from the source of the field.
- The work done increases the satellite’s gravitational potential energy.
- The area under the curve between points A and B represents the satellite’s change in gravitational potential energy.
Question walkthrough
Estimating Work Done Using a Constant-Force Approximation
Estimate the work done moving a spacecraft a small distance in a gravitational field by treating the gravitational force as approximately constant.
The gravitational potential energy of an object in a gravitational field is given by:
Where:
- is the object’s mass (kg),
- is the gravitational potential at that point
The equation for is:
Substituting this into the equation for gravitational potential energy gives:
Where:
- is the universal gravitational constant ,
- is the mass creating the gravitational field (),
- is the distance from the centre of mass to the object of mass (), and
- is measured in Joules ().
Gravitational potential energy is negative because the gravitational force is attractive, and energy is required to move a mass away from the source of the field. At infinity, gravitational potential energy is zero.
Work is required to move a mass against a gravitational field.
The work done or energy transferred ) to move an object between two points in the gravitational field, with a change in gravitational potential ) is given by:
Where:
- is the change in gravitational potential between two points in the field with gravitational potentials and
- is the object’s mass .
This work is equal to the change in gravitational potential energy of the mass. If there is no change in potential there is no change in This means no work is done moving an object between points with the same potential.
When a mass is moved from one point to another point in the gravitational field of a mass , the change in gravitational potential energy is:
Which can be written as:
Where:
- is the mass creating the gravitational field ()
- is the mass moving in the field ()
- and are the initial and final distances from the centre of mass ()
Moving a mass further from the source of the gravitational field (increasing results in a positive change in potential energy, meaning energy is required to move the mass.
Conversely, moving a mass closer to the source (decreasing results in a negative change in potential energy, releasing energy in the process.
Near the Earth’s surface, gravitational potential energy can be approximated using:
Where:
- is the mass of the object ()
- is the gravitational field strength
- is the height above a reference level often taken as the ground ().
The approximation is valid because Earth’s gravitational field is nearly uniform near its surface. This means remains approximately constant.
The gravitational potential energy is defined as zero at Earth’s surface. As an object is lifted, work is done against gravity, and its gravitational potential energy increases.
It is useful to note that the choice of zero at the surface is a convention that simplifies calculations when dealing with heights relative to the ground.
The gravitational field strength can be approximated as constant () near the Earth’s surface. The gravitational field strength on the surface of other celestial bodies can generally be considered constant, too:
However, the simplified equation above only applies only near the Earth’s surface because at greater distances, the field is no longer uniform but radial, and varies with distance.
Use the following formula for cases where the object is far from the massive body and the gravitational field is radial:
Question walkthrough
Change in Gravitational PE Moving Away From Earth
Calculate the change in gravitational potential energy of a spacecraft moving away from Earth to a specified distance from its centre.
Escape velocity is the minimum speed an object must travel to escape a gravitational field completely without any further energy input.
Escape velocity depends only on the mass creating the gravitational field and the position of the escaping object in the field; it does not depend on escaping the object’s mass.
The equation for escape velocity is:
Where:
- is the universal gravitational constant (measured in
- is the mass () creating the gravitational field,
- () is the distance from the centre of mass to the escaping object.
Whether a tennis ball or a spaceship, all objects have the same escape velocity in the same gravitational field (at the same position).
Remember that escape velocity is the minimum speed an object must travel to completely escape a gravitational field without any further energy input.
Lighter gas molecules (e.g. hydrogen and helium) can escape planetary atmospheres if their speeds exceed escape velocity.
The Maxwell–Boltzmann distribution gives the average speed of gas molecules, where is the root mean square speed. The is the square root of the average of the square speeds of all molecules in the system.
Where:
- is the Boltzmann constant,
- is the temperature in Kelvin,
- is the mass of the gas molecule in ()
If the average molecular speed approaches the escape velocity, the planet will gradually lose that gas.
For example:
- Earth retains oxygen and nitrogen (heavier gases move more slowly).
- Small planets (e.g. Mars) and moons have lost most of their atmospheres.
- Due to their high escape velocities, Jupiter and Saturn retain light gases like hydrogen.
Question walkthrough
Deriving the Escape Velocity Formula
Derive the escape velocity formula from the kinetic and gravitational potential energy equations.
Question walkthrough
Comparing Escape Velocity and Gas Particle Speed
Compare an exoplanet's escape velocity with the Maxwell–Boltzmann RMS speed of helium atoms to determine whether the atmosphere is retained.