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Newton's law of universal gravitation says that any two masses attract each other with a force F = G·m₁·m₂ / r², where r is the distance between their centres and G = 6.674 × 10⁻¹¹ N·m²/kg². Doubling the distance makes the force four times smaller: an inverse-square law.

Near the Earth's surface the force on a mass m is F = G·M·m / R². Since a = F/m, the mass m cancels: g = G·M / R² ≈ 9.8 m/s² for every object. A heavier object is pulled harder, but it is also harder to accelerate by exactly the same factor. That is why, without air resistance, all objects fall with the same acceleration.

History is messier than the textbook. Galileo is said to have dropped balls from the Leaning Tower of Pisa, but the only record is a biography written by his student Viviani in 1654, and historians doubt it happened. A documented drop was published in 1586 by the Flemish engineer Simon Stevin, who with Jan Cornets de Groot dropped two lead balls of different weights from the church tower in Delft and found they landed almost together.

Experiment: Measure g with a pendulum

1. Hang the bob so it can swing freely. Measure L from the pivot to the centre of the bob. Start with L = 0.40 m.
2. Pull the bob aside by a small angle (less than 15°) and let go without pushing.
3. Time 20 complete swings (there and back). Start the stopwatch as the bob passes the middle and count 'zero'. Repeat 3 times.
4. Work out the period T = (average time) ÷ 20.
5. Repeat for L = 0.60, 0.80 and 1.00 m.
6. Plot T² (y-axis) against L (x-axis). Draw the best straight line through the origin and find its gradient k.
7. Since T = 2π√(L/g), T² = (4π²/g)·L, so g = 4π² / k. Compare your result with 9.8 m/s².

L (m)	20 swings, run 1 (s)	Run 2 (s)	Run 3 (s)	T (s)	T² (s²)
0.40
0.60
0.80
1.00

Questions

1. Write Newton's law of gravitation and name each symbol with its unit.

2. Show that g does not depend on the mass of the falling object.

3. Calculate how long a stone takes to fall 45 m on Earth (g = 9.8 m/s²) and on Mars (g = 3.7 m/s²).

4. Your value of g from the pendulum: ____ m/s². Give two sources of error and how to reduce them.

Sources for every fact on this sheet: echoesof.science/school/gravity/sources/

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