Gravitation

Drop a phone and it falls. That much is obvious.

Newton’s leap was realizing that the same force pulling the phone to the floor also keeps the Moon in orbit around the Earth, and the Earth in orbit around the Sun, and every other planet, comet, and star in motion. Gravity isn’t just a local “things-fall-down” effect near Earth’s surface — it’s a universal force acting between every pair of masses in the universe. The equation describing it is one of the most consequential in physics.

Newton’s Law of Universal Gravitation

Four key features:

• Gravity is always attractive. There’s no gravitational repulsion. This makes it different from the electric force, which can attract or repel.
• It follows an inverse square law. Double the distance and the force drops to $\frac{1}{4}$. Triple it and the force drops to $\frac{1}{9}$. (Same pattern as Coulomb’s law for electric charges.)
• It acts between every pair of masses, no matter how small. Two coffee cups on a table actually attract each other gravitationally — but the force is mind-bogglingly weak (gravity is the weakest of the four fundamental forces by many orders of magnitude).
• The "$r$" is the center-to-center distance, not the surface-to-surface distance. For the Earth and a falling object, $r$ is the Earth’s radius (~6400 km), not the few meters between the object and the ground.

Gravitational Field Strength (g)

Near any planet or star, we can define a gravitational field strength — the gravitational force per unit mass:

Crucially: $g$ decreases with altitude. An astronaut at twice Earth’s radius from the center ($r = 2R$) experiences $g = GM/(2R)^2 = g_{surface}/4$. They weigh ¼ as much, but their mass hasn’t changed.

Notice that Mt. Everest barely changes $g$ — Earth’s radius is so big that another 9 km up is a tiny fractional change. But the Moon, with much less mass, has $g$ about ⅙ that of Earth’s. That’s why astronauts could leap several feet on the lunar surface.

Weightlessness = Free Fall

Astronauts on the International Space Station float around as if there’s no gravity. But gravity at 400 km altitude is still about 89% of the gravity at sea level. They’re not “above” gravity — they’re in free fall.

Here’s the trick: the ISS and everything inside it are falling toward Earth at the same rate. Astronaut, spacecraft, lab equipment — all accelerating at the same $g$ together. Since nothing is accelerating relative to anything else, there’s no normal force between the astronauts and the floor. No normal force = no felt weight. They float.

This is the same physics as the brief weightless feeling you get at the top of a roller-coaster drop, or in a plummeting elevator, or on those “vomit comet” parabolic-flight planes that NASA uses to train astronauts. All of them are in free fall — and all of them feel weightless during it, regardless of how strong gravity actually is.

Orbits as Continuous Free Fall

An orbit is just the right combination of forward speed and gravitational pull so that the object continuously falls toward the planet but always misses it. The curvature of the fall matches the curvature of the planet — falling and missing forever.

For a circular orbit, gravity provides the centripetal force:

$F_{gravity} = F_{centripetal} \;\Rightarrow\; \dfrac{Gm_1 m_2}{r^2} = \dfrac{m_2 v^2}{r}$

The orbiting mass $m_2$ cancels, leaving:

Two big takeaways:

• Orbital speed depends only on the central body’s mass and the orbital radius, not on the orbiting object. That’s why all satellites at the same altitude move at the same speed.
• Higher orbits → slower orbital speed. Counter-intuitive but real: a satellite far from Earth moves slower than one close in. (And it has a longer orbital period, by Kepler’s third law below.)

Kepler’s Laws (Brief Overview)

The MCAT occasionally references Kepler’s three laws of planetary motion:

1. Law of Ellipses: Planets orbit in ellipses with the Sun at one focus. (Circular orbit is a special case where the ellipse is a perfect circle.)
2. Law of Equal Areas: A planet sweeps out equal areas in equal times. Practically: it moves faster when closer to the Sun and slower when farther.
3. Law of Periods: $T^2$ is proportional to $r^3$. Bigger orbits → longer years.