AP® Physics C: Mechanics · Unit 2 · Topic 2.6
Newton's Law of Universal Gravitation
Every pair of masses pulls on each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Newton's insight was that the force pulling an apple down and the force holding the Moon in orbit are the same force, just evaluated at different distances. He wrote this as F = G m1 m2 / r2, where G = 6.674 x 10−11 N m2/kg2 is a fixed constant of nature, m1 and m2 are the two masses, and r is the distance between their centers. The force is attractive, acts along the line joining the two masses, and by Newton's third law, each mass pulls on the other with equal and opposite force even if their masses differ wildly.
The inverse square part is what makes distance so punishing. Double r and the force drops to a quarter; triple it and the force drops to a ninth. This comes from the geometry of how influence spreads out over a sphere of area 4 π r2 as you move away from the source. It also means that near Earth's surface, where r barely changes, gravity looks constant at 9.8 m/s2, but far from Earth, small changes in distance matter a lot.
This law connects directly to orbital motion and circular motion: setting gravitational force equal to the required centripetal force, G M m / r2 = m v2 / r, gives orbital speed and period. It also sets up gravitational field strength, g(r) = G M / r2, which is just force per unit mass and explains why g shrinks with altitude but never actually reaches zero until r is infinite.
Key terms
- Gravitational constant (G)
- The fixed proportionality constant 6.674 x 10−11 N m2/kg2 that sets the strength of gravity between any two masses.
- Inverse square law
- A relationship where a quantity falls off in proportion to 1 divided by the square of distance from the source.
- Gravitational field strength
- The gravitational force per unit mass at a point in space, g(r) = G M / r2, measured in m/s2.
- Free fall
- Motion under gravity alone with no other forces acting, which is why orbiting objects feel weightless despite gravity still acting on them.
Watch it explained
- How it worksHow does the inverse square make distance matter so violently in the gravitation law?
- Worked exampleFind the gravitational force between two 5000 kg trucks parked 3 m apart.
- Worked exampleFind the acceleration due to gravity at a distance of two Earth radii from Earth's center.
- Common mistakeWhy isn't there zero gravity at the space station, 400 km up from the ground?
Check yourself
End of topic.