Gravity: from falling apples to weighing the Earth
Newton's law of gravitation, why g does not depend on mass, the history behind it, and a pendulum experiment to measure g yourself.
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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.[5]
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.[5][6]
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.[10][11]
Newton's apple has a written source too: William Stukeley recorded Newton telling him the story over tea in 1726, about the mid-1660s, when Newton was home during a plague outbreak. Newton wondered why an apple always falls straight down. Physics textbooks call the story folklore that may well be based in fact.[9][5]
Newton could not measure G. In 1798 Henry Cavendish used a torsion balance (small lead spheres on a rod hanging from a wire, with two large lead spheres brought close) to measure the tiny attraction between masses. From it he found the Earth's density to be about 5.48 times that of water. From the same kind of measurement we can work out the mass of the Earth.[8]
Because of the inverse-square law, gravity weakens only slowly with height. At about 400 km, where the International Space Station orbits, it is still about 89% of its surface value. Astronauts are weightless because they and their station are in free fall around the Earth.[4]
Surface gravity on other worlds (NASA): Moon 1.6, Mars 3.7, Jupiter 23.1 and Pluto 0.7 m/s². A dropped object starting at rest falls a height h in t = √(2h/g), from h = ½·g·t². From 20 m that takes about 2.0 s on Earth and 5.0 s on the Moon.[2][6]
Try it: Measure g with a pendulum
You need
- Strong thread, about 1.2 m
- A small heavy bob (a large nut or a few washers)
- A clamp stand, or a secure hook in a door frame
- A metre stick
- A stopwatch (or a phone)
- A protractor
Safety
Make sure the stand cannot tip over (clamp or weigh down the base). Keep faces away from the swinging bob.
What to do
- 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.
- Pull the bob aside by a small angle (less than 15°) and let go without pushing.
- Time 20 complete swings (there and back). Start the stopwatch as the bob passes the middle and count 'zero'. Repeat 3 times.
- Work out the period T = (average time) ÷ 20.
- Repeat for L = 0.60, 0.80 and 1.00 m.
- Plot T² (y-axis) against L (x-axis). Draw the best straight line through the origin and find its gradient k.
- Since T = 2π√(L/g), T² = (4π²/g)·L, so g = 4π² / k. Compare your result with 9.8 m/s².
What happens, and why?
For small swings, a pendulum's period depends only on its length and on g, not on the mass of the bob or the size of the swing. Timing 20 swings instead of 1 makes your reaction-time error 20 times smaller per swing, and the graph averages out the rest. Typical class results lie within a few percent of 9.8 m/s².[7][5]
Play: Supply drop
A drone flies over the target at constant speed. Release the crate at the right moment so it lands on the target. Each level is another world with its own g. Tip: t = √(2h/g).
Your best score is kept only on this device. Nothing is sent anywhere.
Quiz
Other ages: Ages 6–7 · Ages 10–11