Gravitational Field From Ring Mass

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SUMMARY

The gravitational field from a ring mass can be expressed using the formula g_{x} = -\frac{Gmx}{(x^2 + a^2)^{1.5}}. In this equation, 'm' represents the total mass of the ring, 'x' is the distance from the point mass to the ring center, and 'a' is the radius of the ring. For cases where the point mass is not located on the axis of the ring, a more general approach is required, involving integration of the force over the volume, specifically using dF = GM/r^2 dm, where dm is defined as ρdV.

PREREQUISITES
  • Understanding of gravitational fields and forces
  • Familiarity with calculus, particularly integration techniques
  • Knowledge of mass distribution and density (ρ)
  • Basic concepts of symmetry in physics
NEXT STEPS
  • Study advanced gravitational field calculations for non-axisymmetric mass distributions
  • Explore integration techniques in physics, focusing on volume integrals
  • Learn about the implications of symmetry in gravitational problems
  • Investigate the applications of gravitational fields in astrophysics
USEFUL FOR

Physics students, educators, and researchers interested in gravitational theory and applications, particularly those focusing on complex mass distributions and integration methods.

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Most introductory physics textbooks derive the following formula for the gravitational field for a point mass on the on the axis of the ring:

g_{x} = -\frac{Gmx}{(x^2 + a^2)^{1.5} }

where,

m = total mass of ring
x = distance from point mass to ring center
a = radius of ring

Is there a more general treatment for this ring when the point mass is not on the axis?
 
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yes, and for any other system as well. To solve it for an axis you need to have symmetry like the ring, or the math won't be pretty. Basically integrate dF (dF = GM/r^2 dm , dm=ρdV) over the volume to find F(r)
 
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