[Paper Review] Gravitational potential and energy of homogeneous rectangular parallelepiped

Z. F. Seidov, P. I. Skvirsky|arXiv (Cornell University)|Feb 28, 2000

Dynamics and Control of Mechanical Systems2 references7 citations

TL;DR

This paper presents exact analytical expressions for the gravitational potential and energy of a homogeneous rectangular parallelepiped (RP) using symbolic integration. It derives closed-form solutions in elementary functions via Mathematica, enabling precise computation at any point inside or outside the body, with key results including the potential at the cube's center and vertex, and exact potential energy formulas for cubes, thin rods, and plates.

ABSTRACT

Gravitational potential and gravitational energy are presented in analytical form for homogeneous rectangular parallelepiped.

Motivation & Objective

• To derive exact analytical expressions for the gravitational potential of a homogeneous rectangular parallelepiped (RP) in closed form using elementary functions.
• To compute the gravitational potential energy of the RP by integrating the potential over its volume, yielding exact formulas.
• To examine limiting cases such as cubes, thin rods, and thin plates to validate the general solution and explore scaling behaviors.
• To compare the WUM ratio (potential energy / (potential at center × mass)) across different shapes, including ellipsoids and RPs, revealing universal trends.

Proposed method

• Formulates the Newtonian gravitational potential as a triple integral over the RP’s volume with fixed integration limits.
• Uses symbolic computation (Mathematica) to evaluate the integral step-by-step: first over z, then y, then x, yielding a symmetric final expression.
• Employs the function $ v(x',y',z') = y'z'\ln[x' + \sqrt{x'^2 + y'^2 + z'^2}] - \frac{x'^2}{2}\arctan\left(\frac{y'z'}{x'\sqrt{x'^2 + y'^2 + z'^2}}\right) $ as the core component of the potential.
• Derives the potential energy via a triple integral of the potential over the body’s volume, simplified using symmetry and verified through limiting cases.
• Applies scaling invariance to express results in dimensionless form for universal comparison across shapes.
• Validates results using known limits: line segment, rectangle, and ellipsoid, and compares the WUM ratio to the ellipsoid’s known value of $ 2/5 $.

Experimental results

Research questions

• RQ1 What is the exact analytical expression for the gravitational potential of a homogeneous rectangular parallelepiped at an arbitrary point in space?
• RQ2 How does the gravitational potential energy of the RP depend on its dimensions and density, and can it be expressed in closed form?
• RQ3 What are the values of the gravitational potential at key symmetric points (center, vertex, face center) for a homogeneous cube?
• RQ4 How does the ratio of potential energy to the product of central potential and mass (WUM) vary across different RP geometries, and how does it compare to the ellipsoid case?
• RQ5 Can the derived formulas for the RP be reduced to known results for limiting cases such as thin rods and thin plates?

Key findings

• The gravitational potential of a homogeneous RP at the center is $ U(0,0,0) = 24\ln\left(\frac{1+\sqrt{3}}{\sqrt{2}}\right) - 2\pi \approx 9.52017 $ in units of $ a^2 G \rho $.
• The potential at the vertex of a cube is exactly half the central potential: $ U(1,1,1) = \frac{1}{2} U(0,0,0) $.
• The potential energy of a homogeneous cube with edge length $ 2a $ is $ W_C = 30.117 G \rho^2 a^5 $, derived from the general formula (31).
• For a thin long stick (with $ a \gg b = c $), the potential energy scales as $ W_{\text{stick}} = \frac{32}{3} G \rho^2 a b^4 \ln\left(\frac{a}{b}\right) $.
• For a thin square plate (with $ a \to 0 $, $ b = c $), the potential energy is $ W_S = 47.5714 G \rho^2 b^3 a^2 $, with a WUM ratio of approximately 0.421673.
• The WUM ratio (potential energy / (central potential × mass)) reaches a minimum of ~0.39544 for the cube, tends to 0.5 for thin rods, and to ~0.421673 for thin plates, showing a non-monotonic behavior distinct from the ellipsoid’s $ WUM = 2/5 = 0.4 $.