TwiddleLab / Physics / Gravity and Orbits

Universal Gravitation: Inverse Square

Lesson 3 of 10 · 7 min · Free, no account needed

What you will learn

• Every two masses pull each other: F = G m1 m2 / r², with r measured centre to centre.
• Inverse square: twice as far, a quarter of the pull; three times as far, a ninth.
• The pull grows with each mass, and both bodies feel the same size of force.

Start the lesson

How the lesson goes

1. Newton compared a falling apple with the Moon. Why is the Moon's fall 3600 times gentler?

   Start smaller. Bet: at twice the distance from a planet's centre, how much pull is left?

2. Drag the satellite out and drop dots on the graph. What shape do they make?

3. Now change the masses. Solve three puzzles.

4. Newton's law of universal gravitation

   Any two masses pull each other with F = G m1 m2 / r². r is the distance between their centres.

   G = 6.67 × 10⁻¹¹ N m²/kg² is tiny, so you only notice the pull when one mass is planet-sized.

   Both bodies feel the same size of force, in opposite directions. Simplified model: we treat planets as points at their centres, which works for round planets.

5. Why a square? Shine a lamp through a window and move the screen back.

6. Somewhere between Earth and the Moon the two pulls are equal. Tap that point.

   Equal pulls need 81 / d_E² = 1 / d_M², so d_E = 9 d_M. Splitting the distance 9 to 1 puts the point 90% of the way to the Moon.

7. Game: Gravity golf. Sink the ball in two holes with as few putts as you can.

   Drag the arrow tip to aim. The planets pull the ball on the way.

8. Code: build a gravity function and a loop that prints the pull at distances 1 to 4.

   Toy units: G = 1 and both masses are 60.

   gravity() returns G m1 m2 / r². The loop calls it for r = 1, 2, 3, 4 and prints each pair, which the plot turns into dots.

9. A satellite moves 3 times as far from the planet's centre. How many times weaker is the pull?

   F goes as 1 / r². Make r 3 times bigger and r² becomes 9 times bigger, so F is 9 times smaller.

10. Newton's test: the apple and the Moon. Try both laws, then pick the one that matches.

    The Moon is 60 Earth radii from Earth's centre. The apple is 1.

    The Moon's orbit needs a = v² / r = 0.0027 m/s². 9.8 / 60² = 0.0027 m/s², so the 1 / r² law fits and 1 / r does not.

Previous lesson: Moving in Circles: a = v²/r · Next lesson: Orbit Speed and Period · Part of the course: Gravity and Orbits

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