Newton's laws and gravity
Isaac Newton explained why planets obey Kepler's laws, uniting the motion of falling apples and orbiting moons in one theory.
Newton's laws of motion
Newton's first law: an object stays at rest or moves at constant velocity unless a net force acts on it (inertia). Second law: force equals mass times acceleration, F = ma. Third law: for every action there is an equal and opposite reaction. These laws, published in the Principia in 1687, describe motion on Earth and in space.
Universal gravitation
Newton proposed that every mass attracts every other mass with a force \(F = \frac{G M m}{r^2}\), where G is the gravitational constant, M and m are the masses, and r is the distance between their centres. Gravity weakens with the square of distance: double the distance and the force falls to one-quarter.
F = \frac{GMm}{r^2}Why orbits happen
An orbiting object is constantly falling toward the body it orbits while also moving sideways fast enough to keep missing it. Newton imagined a cannon on a mountaintop firing faster and faster: at the right speed, the cannonball would fall around Earth forever. The International Space Station orbits at about 7.7 km/s, completing an orbit about every 90 minutes.
Weightlessness
Astronauts in orbit feel weightless not because there is no gravity (at the ISS's altitude, gravity is about 90% as strong as on the ground) but because they and their spacecraft are falling together. This state is called free fall.
Escape velocity
Escape velocity is the speed needed to break free of a body's gravity without further propulsion: \(v_{esc} = \sqrt{\frac{2GM}{r}}\). For Earth it is about 11.2 km/s. For the Moon it is about 2.4 km/s, which is why it is easier to launch from the Moon. A black hole has an escape velocity exceeding the speed of light.
Weighing the planets
Newton's version of Kepler's third law includes the masses involved: \(P^2 = rac{4\pi^2}{G(M+m)} a^3\). By measuring the period and distance of a moon's orbit, astronomers can calculate the mass of its planet. The same method, applied to orbiting stars, gives stellar masses, and applied to stars orbiting the centre of the Milky Way, revealed a supermassive black hole.
Tides and gravity
Because gravity weakens with distance, the Moon pulls harder on Earth's near side than its far side, creating tidal bulges. Tidal forces also slowly lengthen Earth's day and push the Moon about 3.8 cm farther away each year, measured using laser reflectors left on the Moon by Apollo astronauts.
Worked example
If the distance between two objects triples, the gravitational force becomes \(\frac{1}{3^2} = \frac{1}{9}\) as strong. If one mass doubles at the same distance, the force doubles. So doubling one mass and tripling the distance gives \(\frac{2}{9}\) of the original force.
Key takeaways
Newton's laws describe inertia, F = ma, and action-reaction. Universal gravitation is F = GMm/r^2, weakening with distance squared. Orbits are continuous free fall with sideways motion; the ISS orbits at about 7.7 km/s. Astronauts are weightless because they fall with their craft. Earth's escape velocity is about 11.2 km/s. Newton's form of Kepler's third law lets us weigh planets and stars. Tidal forces lengthen Earth's day and move the Moon away.
Flashcard
Inertia
Check your understanding
Why do astronauts on the International Space Station float?