Why Do Equal Gravitational Forces Give an Apple and the Earth Unequal Accelerations?

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Gravitational Force: Why Equal Forces Produce Unequal Accelerations

Gravitational Force: Why Equal Forces Produce Unequal Accelerations

Apple mass m1
Planet mass m2
Release distance r0
PHYSICS INSIGHTS

One force, two bodies. m1 pulls m2 and m2 pulls m1, equally hard, in opposite directions along the line between their centers: a Newton’s third law pair. F = Gm1m2/r2 treats the two masses alike, so it cannot give them different forces. The two orange arrows are always the same length.

The same force, divided by different masses. a = F/m, so a1/a2 = m2/m1: about 6 × 1025 for a 0.10 kg apple and Earth, 81 for the Moon and Earth, 1 for equal planets. The lighter body does nearly all the moving while the center of mass stays put. Both start at rest here; the real Moon’s sideways speed keeps it in orbit instead.

Why everything falls at the same rate. a1 = F/m1 = Gm2/r2: the apple’s own mass cancels. Double it and the force doubles, but its acceleration does not change. The pull grows as the gap closes: halve r and F is 4 times as big, a third of r and 9 times, as the dots on the graph show. The apple reaches the ground with a1 = g = 9.81 m/s2.

Simulation by The Science Cube — https://www.thesciencecube.com/

Why do equal gravitational forces give an apple and the Earth such different accelerations?

In short: Gravity pulls each body with the same force, F = Gm₁m₂ ÷ r², as Newton’s third law requires. But acceleration is force divided by that body’s own mass: a₁ = F ÷ m₁ = Gm₂ ÷ r² and a₂ = F ÷ m₂ = Gm₁ ÷ r². Earth’s mass is huge, so its acceleration is tiny, while the apple’s is large. The ratio is a₁ ÷ a₂ = m₂ ÷ m₁.

Common doubts

  • “Does Earth pull harder on the apple than the apple pulls on Earth?” No. The two forces are equal in size and opposite in direction. Only the accelerations differ.
  • “Then why do we only ever see the apple fall?” Because Earth’s acceleration is far too small to notice. In the simulation, a 0.10 kg apple accelerates about 6 × 10²⁵ times more than the Earth does.
  • “What counts as r in the formula?” The distance between the centers of the two bodies, not between their surfaces. Doubling r cuts the force to one quarter.

Change the masses and the release distance in the simulation above, press RELEASE, and compare the two accelerations.

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