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The law of gravity

Gravity and orbits · level 6, lesson 1 of 6

A question to start

You are pulled by the Earth, but you also pull on the Earth. You pull on the person sitting next to you, and they pull on you. Why don't we feel any of that? How strong is the pull between two ordinary things?

The idea

Newton's law of universal gravitation says that any two masses \(M\) and \(m\), with their centres a distance \(r\) apart, pull on each other with a force

\[F = \frac{G M m}{r^2},\]

where \(G = 6.674 \times 10^{-11}\ \text{N m}^2/\text{kg}^2\) is the gravitational constant, the same everywhere in the universe.

The law packs together everything from earlier levels:

  • Both masses on top. Double either mass and the pull doubles. The pull on you is your mass times something.
  • \(r^2\) on the bottom. The inverse square rule from level 4.
  • Each pulls the other equally. The Earth pulls you with your weight, and you pull the Earth up with exactly the same force. The Earth just has so much mass that the same force barely moves it.
  • \(G\) is tiny. That is why we never notice the pull between people.

Measuring \(G\). Newton could not. In 1798 Henry Cavendish hung a rod with two small lead balls from a thin wire, brought two big lead balls near them, and measured how much the tiny pull twisted the wire. From the twist he found the density of the Earth, and from that we get \(G\). It is still the least precisely known of the basic constants of physics.

Pair Masses Distance Pull
Two people 60 kg and 60 kg 1 m \(2.4 \times 10^{-7}\) N, about the weight of a grain of salt
Earth and a 60 kg person \(5.97 \times 10^{24}\) kg and 60 kg 6,371 km 589 N
Mars and the same person \(6.42 \times 10^{23}\) kg and 60 kg 3,390 km 224 N
The Sun and Mars \(1.99 \times 10^{30}\) kg and \(6.42 \times 10^{23}\) kg 228 million km \(1.6 \times 10^{21}\) N

Worked examples

1. Two people. \(F = 6.674\times10^{-11} \times 60 \times 60 / 1^2 = 2.4 \times 10^{-7}\) N.

2. A person on Mars. \(F = 6.674\times10^{-11} \times 6.417\times10^{23} \times 60 / (3.3895\times10^6)^2 = 2.570 \times 10^{15} / 1.149 \times 10^{13} = 224\) N. Divide by 60 kg: 3.73 m/s², the gravity of Mars (lesson 2).

3. Earth and Mars at their closest. About 56 million km apart: \(F = 6.674\times10^{-11} \times 5.97\times10^{24} \times 6.42\times10^{23} / (5.6\times10^{10})^2 \approx 8 \times 10^{16}\) N. Big, but about 20,000 times smaller than the Sun's pull on Mars, so it hardly changes Mars's orbit.

Try it

  1. Two 1,000 kg cars are parked 5 m apart. Find the pull between them.
  2. If the distance between two masses is tripled, what happens to the pull?
  3. You weigh 500 N on Earth. With what force do you pull the Earth?
  4. Find the pull of Mars on Phobos (\(m = 1.07 \times 10^{16}\) kg, \(r = 9.376 \times 10^6\) m).
  5. Why can the Earth pull you with 589 N without you moving it noticeably, even though you pull it back with 589 N?
Answers
  1. \(6.674\times10^{-11} \times 10^6 / 25 = 2.7 \times 10^{-6}\) N.
  2. It becomes a ninth.
  3. 500 N, upwards.
  4. \(6.674\times10^{-11} \times 6.417\times10^{23} \times 1.07\times10^{16} / (9.376\times10^6)^2 = 4.58\times10^{29} / 8.79\times10^{13} \approx 5.2 \times 10^{15}\) N.
  5. The same force gives an acceleration of \(F/m\); the Earth's mass is about \(10^{23}\) times yours, so its acceleration is about \(10^{23}\) times smaller.

Common mistake

Using the distance between surfaces. \(r\) is between centres. For a person on a planet, it is the planet's radius, not zero.

Where it leads