03 · The inverse-square laboratory
How much weaker is gravity at twice the centre distance?
Half as strong or one quarter? Record a prediction.
- Set the initial centre radius to 7000 km and record acceleration.
- Double radius to 14000 km, reset, and compare.
- Hold radius fixed and double central mass.
Measurement
Calculate a₂/a₁ and a·r²/μ; the second should equal one.
Acceleration scales with central mass and inversely with squared centre distance. Radius is measured from the centre, not the surface. Logarithmic plots provide a useful data-analysis check: log(g) against log(r) has slope −2.
G=6.67430×10⁻¹¹ m³ kg⁻¹ s⁻²; M in kg; r in m; g in m/s².
Worked example 1
At r=7×10⁶ m, g=8.13 m/s²; at 14×10⁶ m, g=2.03 m/s².
Worked example 2
Doubling both mass and radius gives g₂/g₁=2/4=0.5.
If radius doubles, what mass factor preserves acceleration?
Assumptions
Newtonian two-body model with fixed central mass. No drag, third bodies, oblateness or relativity. Visual sizes are illustrative; initial conditions are fictional, not live ephemerides.
University Physics Volume 1, 13.2 Gravitation Near Earth’s Surface