Teacher sheet · Ages 15–16 · about 45–50 minutes
Gravity: from falling apples to weighing the Earth
Lesson page: echoesof.science/school/gravity/ages-15-16/ · Pupil worksheet: echoesof.science/school/gravity/ages-15-16/print/
Learning aims
- Apply F = Gm₁m₂/r² and the inverse-square law.
- Derive g = GM/R² and explain why it is independent of the falling mass.
- Use t = √(2h/g) for free fall on different worlds.
- Measure g with a pendulum, using a linear graph (T² against L) and evaluating errors.
- See how evidence and stories differ in the history of science (Pisa, Delft, the apple).
Lesson plan
| Time | Activity |
|---|---|
| 5 min | Starter: drop two balls of different mass. Predict, observe, explain. |
| 10 min | Read and discuss the text; work through g = GM/R². |
| 25 min | Pendulum practical in pairs, 4 lengths × 3 runs. |
| 5 min | Graph T² against L, calculate g, compare across the class. |
| 5 min | Quiz, or the Supply drop game as homework. |
Materials for the experiment
- 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.
Common misconceptions
| Pupils may think… | Actually… |
|---|---|
| Heavier objects accelerate faster. | The force is larger, but a = F/m is the same. |
| There is no gravity in orbit. | g at 400 km is about 89% of the surface value; orbit is free fall. |
| The pendulum period depends on the bob's mass or the swing size. | For small angles (under about 15°) it depends only on L and g. |
| Galileo's Pisa drop is a proven fact. | It rests on one later biography; Stevin's 1586 Delft drop is documented. |
Worksheet answers
- F = G·m₁·m₂/r²: F force (N), G = 6.674 × 10⁻¹¹ N·m²/kg², m₁ and m₂ masses (kg), r distance between centres (m).
- F = GMm/R² and a = F/m, so a = GM/R², with no m in it.
- Earth: t = √(90/9.8) ≈ 3.0 s. Mars: t = √(90/3.7) ≈ 4.9 s.
- Expected 9.5–10.1 m/s². Errors: reaction time (time 20 swings, repeat), measuring L to the bob's centre, swings too large (keep under 15°), the stand moving.
Quiz answers
- The distance between two masses doubles. The gravitational force becomes… 4 times smaller. F ∝ 1/r², so (1/2)² = 1/4.[5]
- Why does a heavy ball fall with the same acceleration as a light one (no air)? The larger force is cancelled by its larger mass (a = F/m). F = GMm/R² and a = F/m, so a = GM/R²: m cancels.[5]
- At the space station's height (about 400 km), gravity is about… 90% of its surface value. NASA gives 88.8% at about 250 miles (400 km).[4]
- Who published a documented drop of two lead balls of different weights in 1586? Simon Stevin, in Delft. Stevin and De Groot dropped them from the church tower in Delft; the Pisa story rests on one later biography.[10]
- A pendulum's length is made 4 times longer. Its period becomes… 2 times longer. T ∝ √L, and √4 = 2.[7]
- An object is dropped from 20 m on the Moon (g = 1.6 m/s²). How long does it fall? 5.0 s. t = √(2h/g) = √(40/1.6) = √25 = 5.0 s.[6][2]
Going further
Use the Supply drop game: at 100 m height and 20 m/s, how far before the target must you release on each world? Check your answer in the game. Then try the Newton's Cannon game on echoesof.science.
Sources used in this lesson
- [2] Planetary Fact Sheet (Metric) (NASA Space Science Data Coordinated Archive): reliability High
- [4] What Is Microgravity? (NASA): reliability High
- [5] College Physics 2e, 6.5 Newton's Universal Law of Gravitation (OpenStax, Rice University): reliability High
- [6] University Physics Volume 1, 3.5 Free Fall (OpenStax, Rice University): reliability High
- [7] University Physics Volume 1, 15.4 Pendulums (OpenStax, Rice University): reliability High
- [8] June 1798: Cavendish weighs the world (APS News, American Physical Society): reliability High
- [9] Story of Newton's encounter with apple goes online (Phys.org (news report, 2010)): reliability Medium-high
- [10] Galileo Galilei's and Simon Stevin's experiments (John Plaice (Substack essay)): reliability Medium
- [11] The Delft Tower Experiment (Amusing Planet (popular history site, 2024)): reliability Medium
Full list with links and a note on each source's reliability: echoesof.science/school/gravity/sources/
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