The Memory Trick
💡 F = Gm₁m₂/r²
Every pair of masses in the universe attracts each other with a force proportional to the product of their masses, and inversely proportional to the square of the distance between their centers: F = Gm₁m₂/r², where G = 6.674×10⁻¹¹ N·m²/kg² is the universal gravitational constant.
Why It Works
Because distance is SQUARED in the denominator, gravity weakens very quickly with distance. Double the distance between two objects, and the gravitational force between them drops to just a quarter of its original strength — not half.
Step by Step
Working With the Gravitation Formula
1
Force depends on the PRODUCT of both masses
Doubling either mass doubles the gravitational force; doubling both masses quadruples it.
Two planets each twice as massive as before would attract each other with four times the original gravitational force at the same distance.
2
Force follows an inverse SQUARE law with distance
This is the detail most often mishandled — distance isn't just inversely proportional to force, it's inversely proportional to the SQUARE of distance.
Tripling the distance between two masses reduces the gravitational force between them to 1/9 (1/3²) of its original value.
3
This is the same force at every scale
The identical equation explains an apple falling from a tree, the Moon orbiting Earth, and Earth orbiting the Sun — gravity doesn't behave differently at planetary scales, it's the same universal law.
Newton famously connected the falling apple and the orbiting Moon using this exact same formula.
🏥 Worked Example
Two objects are 4 meters apart. If the distance between them is doubled to 8 meters, what happens to the gravitational force between them?
1
Recall the relationship: F ∝ 1/r² — force is inversely proportional to the square of distance.
2
Apply the doubling: new distance is 2× the old distance, so the force becomes 1/(2²) = 1/4 of the original.
3
Conclusion: the gravitational force drops to one-quarter of its original strength, not one-half — a common point of confusion.
⚠️ Most Common Newton's Law of Gravitation Mistakes
The most common trap is treating gravity as simply inversely proportional to distance (F ∝ 1/r) instead of inversely proportional to distance SQUARED (F ∝ 1/r²) — this leads to significantly underestimating how quickly gravitational force weakens with distance.
✓ Quick Self-Test
1) Write Newton's Law of Universal Gravitation. F = Gm₁m₂/r². 2) If the distance between two masses triples, what happens to the gravitational force between them? It drops to 1/9 of its original value. 3) If one of the two masses doubles while distance stays the same, what happens to the force? It doubles. 4) What does the fact that this is the same law for a falling apple and an orbiting Moon tell us about gravity? Gravity is a universal force that behaves identically at every scale, not a different force for 'earthly' vs. 'celestial' objects. 5) Why does gravity weaken so quickly with distance? Because force is inversely proportional to the SQUARE of distance, not distance itself — small increases in distance cause large decreases in force.