Where the pulls cancel.

Two masses M and nM on a line; field arrows and a g(x) graph show where the pulls cancel.

Place the test mass where the net pull from M and 4M (6R apart) is zero (2R from M).

• Newton compared the Moon with a falling apple. The Moon (orbit radius 3.84 × 10⁸ m, period 27.3 days) has centripetal acceleration a_m = 4π²R_m/T² ≈ 2.7 × 10⁻³ m/s², far less than g.
• If gravity falls off as 1/r², then g/a_m = (R_m/R_E)² ≈ 60² = 3600, and 9.8/0.00272 is indeed about 3600. The same force explains the apple and the Moon.
• Law: every pair of bodies pulls on each other. The pull grows with the product of the two masses and falls as 1/(distance)²: F = G m₁ m₂ / r².
• In vector form the force on m₂ is F = −G m₁ m₂ r̂ / r², with r̂ pointing from m₁ to m₂; the minus sign means attraction. The force on m₁ is −F, so F₁₂ = −F₂₁ (Newton's third law).
• Superposition: the force on one mass from several others is the vector sum of the separate pair forces, each unchanged by the presence of the rest.
• Three equal masses at the corners of an equilateral triangle exert zero net force on a mass at the centroid, as symmetry predicts. If the mass at one corner is doubled, the net force points towards that corner.
• Shell results: a uniform spherical shell attracts a point outside it as if all its mass sat at its centre; on a point anywhere inside it, the shell's net pull is zero.
• The law is for point masses; for an extended body the forces from all its parts are added as vectors. Gravity cannot be shielded: a shell does not block the pull of bodies outside it.

Teaching this idea? Put the simulation on your own page or LMS with this code.