Physics HL · Chapter 17: Gravitation
17.3 Gravitational Potential and Potential Energy
Use potential-energy and potential language to compute work, compare locations in a field, and interpret equipotential geometry.
Estimated time: 42 minutes
Why Gravitational Potential Energy Is Negative
For two masses separated by distance r, gravitational potential energy is defined relative to zero at infinite separation. Because gravity is attractive, bringing masses together from infinity can happen while an external agent removes energy from the system, so the stored potential energy becomes negative. This sign convention encodes that energy must be supplied to fully separate a bound pair.
E_p = - rac{GMm}{r}
As r increases, Ep moves upward toward 0; as r decreases, Ep becomes more negative.
Thinking in terms of a potential well is useful. Deep negative Ep means strongly bound. A trajectory can still pass through deep regions if kinetic energy is high enough, but the total energy sign determines whether the path can reach infinity or must remain bound.
Gravitational Potential and Work Calculations
Gravitational potential V is potential energy per unit mass. It is scalar, so contributions from multiple masses add algebraically. For one spherical source, V has the same inverse-distance structure as Ep but without the test mass factor. This makes V an efficient tool for calculating work per unit mass between two radii.
V = - rac{GM}{r},qquad W_{ ext{ext}} = mDelta V,qquad Delta E_p = mDelta V
External work for slow repositioning equals increase in potential energy; gravitational work is the negative of that value.
A practical exam habit is to decide first which agent's work is being asked for. If the question asks work done by gravity, reverse the sign of work done by the external repositioning force. If it asks required launch or transfer energy, you are usually computing an external positive input.
Equipotential Surfaces and Field Gradient
Equipotential surfaces connect points with equal V. Moving a mass along one equipotential needs zero work because DeltaV = 0. Field vectors are always normal to equipotentials and point toward lower potential. In one-dimensional radial notation, field strength is the negative gradient of potential, expressing that g points in the direction where V decreases most rapidly.
g = - rac{dV}{dr}
Large slope magnitude in V(r) corresponds to strong field.
Note
In two-mass systems, zero field and zero potential are different conditions. g can cancel vectorially while V remains negative.
Simulation: Potential Well and Gradient Explorer
Track V(r), g(r), and work-per-kilogram between radii to connect potential differences with external work and equipotential reasoning.
Gravitation + Orbit Lab
V at start
-5.21e+7 J/kg
V at end
-1.39e+7 J/kg
Work per kg
3.82e+7 J/kg
g at start
6.8194 N/kg
Potential-energy landscape and field gradient
Test Yourself
Around Earth, move a 1 kg probe from 2R_E to 5R_E. Enter the required external work per kg in J/kg.
Hint: Use DeltaV = -GM/(5R) - (-GM/(2R)) and GM/R approximately 6.25 x 10^7 J/kg.