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Weaker with distance

Gravity and orbits · level 4, lesson 2 of 6

A question to start

A torch shines on a wall 1 m away and makes a bright small patch. Move the torch to 2 m away. The patch gets bigger and dimmer. How much dimmer?

The idea

The light from the torch spreads out. At twice the distance it is spread over a patch twice as wide and twice as tall: four times the area. The same light on four times the area means each bit is a quarter as bright.

1: 1 square2: 4 squares3: 9 squares

Gravity works the same way. It spreads out in every direction from a planet, and gets weaker with the square of the distance from the planet's middle. This is the inverse square rule:

Distance from the middle (in planet radii) 1 2 3 4 10 60
Pull, compared with the surface 1 1/4 1/9 1/16 1/100 1/3600

The distance that counts is from the middle of the planet, not from its surface. Standing on the surface, you are already one radius away.

Near the surface, the pull hardly changes. A tall mountain is only 9 km high; the Earth's radius is 6,371 km. Climbing it changes your distance from the middle by about 0.1%, and your weight by about 0.3%. That is why we can treat \(g\) as fixed for everyday falls.

High up, it does change. At the space station, 400 km up, you are \(6771 \div 6371 \approx 1.06\) radii from the middle, so the pull is \(1 \div 1.06^2 \approx 0.89\): still 89% of the pull on the ground.

Worked examples

1. Twice as far from Mars. A satellite 3,390 km above the surface of Mars is 2 Mars radii from the middle. The pull there is \(3.71 \div 4 \approx 0.93\) m/s².

2. At Phobos's distance. Phobos orbits 9,376 km from the middle of Mars, about 2.77 Mars radii. The pull of Mars there is \(3.71 \div 2.77^2 \approx 3.71 \div 7.7 \approx 0.48\) m/s².

Try it

  1. How much weaker is gravity at 3 planet radii from the middle than on the surface?
  2. At 5 radii?
  3. Something is moved from 2 radii to 4 radii from a planet's middle. By how much does its weight change?
  4. The Moon is 60 Earth radii away. What fraction of \(g\) pulls on it?
  5. Why do we say "from the middle", not "from the surface"?
Answers
  1. 9 times weaker (1/9).
  2. 25 times weaker (1/25).
  3. It becomes a quarter of what it was (twice the distance).
  4. \(1/3600\) of \(g\), about 0.0027 m/s².
  5. A planet pulls as though all its mass were at its middle, so distances count from there. On the surface you are already one radius away.

Common mistake

"Twice as far, half the pull." It is a quarter, because the pull spreads over an area that grows with the square of the distance.

Where it leads