How to play Big and Fading Swings
- Choose a test and read the pendulum, its starting swing, and the share each swing keeps on the card.
- Big swings take longer than T₀ = 2π·√(L ÷ g). Fading swings shrink by the same factor every swing.
- Type your prediction and press Start. A gray ideal pendulum swings beside the real one.
- Watch the real pendulum fall behind the ideal one, or its swing shrink, and compare with your orange marks.
What does the pendulum formula leave out?
T = 2π·√(L ÷ g) comes from assuming the pull back towards the middle grows in step with the angle. That's true for small swings, but the real pull grows like sin θ, which falls behind at big angles, so wide swings take longer: about 1.7% longer at 30°, 7% at 60°, and 18% at 90°. A clock pendulum that swung wider would run slow, which is why clockmakers keep the swing to a few degrees. Real swings also fade, because air and friction take a little energy every time. A simple model that fits many real oscillators is that every swing keeps the same share of its size, say 90%. Then the size shrinks by multiplying, not subtracting: 20°, 18°, 16.2°, and so on, and it halves after the same number of swings whatever size it starts at, like a radioactive half-life. Because energy goes with the size squared, keeping 90% of the size means keeping only 81% of the energy. Car shock absorbers fade the bounce on purpose; a good bell or tuning fork fades as slowly as possible.
A focused learning model
Big swings are worked out exactly from θ'' = −(g ÷ L)·sin θ, with no friction, and run for three or four swings in real time. The fading swings keep 90% of their size every full swing, shrinking smoothly as θ = θ₀·0.9ⁿ·cos(2π·t ÷ T₀), and play two or three times faster. The pendulum is 1 m long and drawn to scale; the gray ideal pendulum follows the small-swing formula. Answers within 2% are right. g = 9.8 m/s².