Circular Motion and Gravitation: Orbits and Kepler's Laws

A car going around a roundabout at a steady 30 km/h has a constant speed, but it is still accelerating. Velocity includes direction, and the car's direction changes all the time. By Newton's first law, anything whose velocity changes must have a net force on it.

At every moment the velocity points along the circle, at a tangent. The change in velocity points toward the center. This inward acceleration is called centripetal acceleration:

a = v² ÷ r

Here v is the speed and r the radius of the circle. Double the speed and the acceleration is four times bigger; make the turn twice as tight and it doubles. The speed is also linked to the period: one trip around covers the circumference, so v = 2πr ÷ T.

Newton's second law, F = ma, turns the centripetal acceleration into the net inward force needed to keep a mass m on the circle:

F = mv² ÷ r

Centripetal force isn't a separate kind of force; it's whatever real force pulls the object inward. For a ball on a string, the tension does it. For a car on a flat curve, it's friction between the tires and the road; on ice there isn't enough friction, and the car slides off in a straight line. For the Moon, gravity does it.

When a car turns sharply, you feel thrown outward against the door. Nothing is pushing you out. Your body is trying to keep going straight, and the door pushes you inward to turn you with the car. That outward feeling is sometimes called centrifugal force, but seen from the road there's no outward force at all.

In 1687 Newton proposed that every mass in the universe attracts every other mass. The force between two masses m₁ and m₂ whose centers are a distance r apart is

F = Gm₁m₂ ÷ r²

The force grows with either mass, and it's an inverse-square law: double the distance and the force falls to a quarter; triple it and the force falls to a ninth. The pull never quite reaches zero, however far apart the masses are.

The two pulls are equal and opposite, as Newton's third law says. You pull on Earth exactly as hard as Earth pulls on you; Earth just has so much mass that its acceleration is far too small to notice. Because G is so tiny, gravity between everyday objects is too weak to feel. Two 1 kg masses 1 m apart attract each other with less than a ten-billionth of a newton. Gravity only becomes strong when at least one of the masses is the size of a moon, a planet or a star.

Your weight is the gravitational pull of the planet you're standing on. Put Earth's mass M and radius R into Newton's law, divide by your mass, and you get the force on each kilogram, the gravitational field strength:

g = GM ÷ R²

With M = 5.97 × 10²⁴ kg and R = 6.37 × 10⁶ m, this gives 9.8 N/kg, the value used for falling objects in Unit 1. Weight is then W = mg. A 60 kg student weighs 588 N on Earth. On the Moon, where g is 1.62 N/kg, the same student weighs 97 N but still has a mass of 60 kg.

Gravity gets weaker with height, but slowly at first. The International Space Station (ISS) orbits about 400 km up, where g is still about 8.7 N/kg, close to 89% of its value on the ground. Astronauts there float because they and the station are falling together, not because gravity has gone away.

Newton pictured a cannon on top of a very tall mountain. Fire a ball slowly and it curves down to the ground nearby. Fire it faster and it lands farther away. Fire it fast enough and the ground curves away beneath it as fast as it falls, so it never lands: it's in orbit.

For a circular orbit, gravity supplies exactly the centripetal force. Setting GMm ÷ r² equal to mv² ÷ r gives the orbital speed:

v = √(GM ÷ r)

Here r is measured from the center of the planet, so it's the planet's radius plus the altitude. The satellite's own mass cancels out, so a bolt and a space station at the same height move at the same speed. The ISS travels at about 7.7 km/s and goes around Earth in about 92 minutes. Farther out, orbits are slower and longer. At 35,786 km up, a satellite takes 23 hours 56 minutes, the time Earth takes to spin once, so it stays over the same spot. TV and weather satellites in this geostationary orbit seem to hang still in the sky. The Moon, about 384,400 km away, takes 27.3 days.

Fire the cannonball even faster, at 11.2 km/s from Earth's surface (ignoring air), and it has the escape speed: it never comes back.

Decades before Newton, Johannes Kepler spent years working through Tycho Brahe's careful measurements of Mars. He found three rules that describe how planets move.

First law: each planet moves in an ellipse, with the Sun at one focus. An ellipse is a stretched circle with two focus points. Most planet orbits are very nearly circles: Earth is only about 3% closer to the Sun in early January than in early July.

Second law: a line from the Sun to a planet sweeps out equal areas in equal times. A planet moves fastest when it's closest to the Sun and slowest when it's farthest away.

Third law: the square of a planet's period is proportional to the cube of its average distance from the Sun, T² ∝ a³. With T in years and a in astronomical units (AU, Earth's average distance from the Sun), T² = a³ for every planet. Jupiter is 5.2 AU out, so its year lasts about √(5.2³) ≈ 11.9 Earth years.

Kepler found these rules from data but couldn't say why they hold. Newton showed that all three follow from his law of gravitation and his laws of motion.

For most of history, people thought the heavens followed different rules from things on Earth. Kepler described the planets' paths with simple laws, and Newton then showed that the force pulling an apple down is the same one holding the Moon and planets in their orbits.