Gravity Potential of Rings, Discs, Shells and Spheres

Newton's law gives the potential of a single point mass as V = −GM/r — but planets are spheres, galaxies are discs, and some problems involve rings and hollow shells. The gravitational potential of standard bodies means the ready-made formulas for these symmetric shapes, derived by adding up (integrating) the contributions of every bit of mass.

Real objects are extended bodies: satellite energy near Earth needs the sphere formula, while the disc formula models spiral galaxies. These results also surprise — the field vanishes entirely inside a hollow shell — which no point-mass picture gives.

Key Concepts

• Superposition: potential is a scalar, so an extended body's potential is the simple sum of −Gmᵢ/rᵢ over all its pieces.
• Ring: every bit of a thin ring of radius R is equidistant from a point on its axis, so the integral is trivial.
• Shell theorem: a hollow shell acts externally like a point mass at its centre, and gives constant potential and zero field everywhere inside.
• Solid sphere: point-like outside, but inside the potential follows a parabola and the field grows linearly from the centre.

Formulas

On the axis of a thin ring (radius R, mass M), at distance x from the centre:

V = −GM/√(R² + x²), field E = GMx/(R² + x²)^{3/2} toward the ring.

On the axis of a uniform thin disc (radius R, surface density σ), at distance x:

V = −2πGσ(√(R² + x²) − |x|). At the centre (x = 0): V = −2πGσR.

Hollow spherical shell (radius R, mass M):

• Outside (r ≥ R): V = −GM/r, E = GM/r².
• Inside (r ≤ R): V = −GM/R (constant), E = 0.

Solid uniform sphere (radius R, mass M):

• Outside (r ≥ R): V = −GM/r, E = GM/r².
• Inside (r ≤ R): V = −GM(3R² − r²)/(2R³), E = GMr/R³.

Worked Example

A thin ring of mass 10 kg and radius 1 m sits in space. Find the gravitational potential at a point 2 m from the centre along its axis, and the potential energy of a 2 kg mass placed there.

1. V = −GM/√(R² + x²) = −(6.674 × 10⁻¹¹ × 10)/√(1² + 2²).
2. √5 ≈ 2.23607; numerator = 6.674 × 10⁻¹⁰.
3. V = −6.674 × 10⁻¹⁰/2.23607 ≈ −2.99 × 10⁻¹⁰ J/kg.
4. U = mV = 2 × (−2.99 × 10⁻¹⁰) ≈ −5.97 × 10⁻¹⁰ J.

Tiny — everyday masses make feeble potentials, but the same formula with galactic masses binds stars to the disc.

Common Misconceptions

• "Inside a hollow sphere gravity pulls you to the centre." Wrong — the field is zero everywhere inside. An astronaut inside a hollow planet would float weightless.
• "The ring's centre has zero potential." Potential is never zero at finite distance — at the centre V = −GM/R. Only the field is zero there, by symmetry.
• "These formulas only work far away." The shell and sphere formulas are exact at all exterior points, even just above the surface.

Real-World Applications

• Planetary science: the sphere formulas underpin every satellite energy and orbit calculation around Earth and other planets.
• Galactic dynamics: disc-potential models explain star orbits in spiral galaxies and revealed the need for dark matter.
• Electrostatics crossover: the same mathematics gives electric potentials of charged rings and shells — learn it once, use it twice.

Rings, discs, shells and spheres each have their own clean potential formula, all built by adding up point-mass contributions. The shell theorem's twin gifts — point-like behaviour outside, zero field inside — plus the sphere's parabolic interior potential cover nearly every gravity problem you will meet.