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.
What to measure
Calculate a₂/a₁ and a·r²/μ; the second should equal one.
The idea underneath
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 = GM/r²; g₂/g₁ = (M₂/M₁)(r₁/r₂)²G=6.67430×10⁻¹¹ m³ kg⁻¹ s⁻²; M in kg; r in m; g in m/s².
Build the argument
- Start from F = GMm/r² and divide by satellite mass: a = μ/r².
- For circular motion set v²/r = μ/r². Then v = √(μ/r) and T = 2π√(r³/μ).
- Combine kinetic and potential energy per mass: ε = v²/2 − μ/r. Its sign separates bound and unbound trajectories.
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.
Applications
Orbital mechanics connects satellite navigation, communications and interplanetary transfers. A launch is a choice of position and velocity, not a command to follow a circle.
Common misconception
Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.
A further investigation
If radius doubles, what mass factor preserves acceleration?
Check your understanding
If satellite mass doubles, what happens to its acceleration?
- It doubles
- It stays the same
Explanation
Satellite mass cancels in a=F/m. The challenge requires four times the central mass.
Model limits
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