Gravitational potential
Section: Gravitational Fields | Syllabus: Cambridge A Level Physics (9702)
Why we use Potential
The standard formula Δ E_P = mgΔ h only works when g is constant (near the surface). For large changes (like launching satellites), we need a system approach considering work done against the variable field.
Examiner DefinitionGravitational Potential (phi): The work done per unit mass in bringing a small test mass from infinity to that point.
φ = -(GM/r)Units: J kg^-1. It is a scalar quantity.
Infinity and the Negative Sign
By convention, we set Potential = 0 at Infinity (r = ∞). Since gravity is attractive, work is done by the field as you approach Earth. This means energy decreases as you get closer, resulting in negative values. Potential is most negative at the Earth's surface (~-62.5 MJ kg^-1).
Figure 13.10: Earth's Gravitational Potential GraphShow phi on y-axis (negative MJ/kg) vs r on x-axis. Curve starts at -62.5 at R_E and curves upward, asymptotically reaching 0 at infinity.
Potential Energy of a System (E_P)
For two masses M and m separated by distance r:
E_P = -(GMm/r) Worked Example: Launch EnergyQuestion: Determine the minimum energy required to move a 1650 kg satellite from the surface to an orbit of altitude 2250 km.
Solution1. Energy at surface E_1 = -(GMm/R_E) ≈ -1.03 × 10^11 J.
2. Energy in orbit E_2 = -(GMm/R_E + 2250km) ≈ -7.62 × 10^10 J.
3. Δ E_P = E_2 - E_1 = 2.69 × 10^10 J.
Equipotentials
Equipotential surfaces are regions where the potential has the same value. In 2D, these are circles centered on the mass.
Figure 13.11: Field Lines vs EquipotentialsShow field lines (radial arrows) and equipotentials (concentric circles). Label circles with values like -50 MJ kg^-1, -40 MJ kg^-1. Increasing separation between circles indicates a weakening field.
- No work is done when a satellite moves along an equipotential surface (perpendicular to field lines).
- The gradient of a potential-distance graph is equal to the gravitational field strength
(g = -Δ φ / Δ r).
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