Lesson 5: Formulas on arrays

"""Pendulum wave: 24 pendulums making 51, 52, ... 74 swings in 60 seconds.
Run: python pendulum-wave.py   (needs: pip install numpy matplotlib)
"""
import numpy as np
import matplotlib.pyplot as plt

plt.style.use("dark_background")

g, cycle = 9.81, 60.0  # gravity in m/s^2, seconds until they are back in step
n = np.arange(51, 75)  # swings per cycle, one more for each pendulum
period = cycle / n
length = g * (period / (2 * np.pi)) ** 2  # small swings: T = 2 pi sqrt(L / g)
print("lengths in cm:", np.round(length * 100, 1))

fig, axes = plt.subplots(1, 4, figsize=(12, 4), sharey=True)
for ax, t in zip(axes, [0, 15, 20, 30]):
    swing = np.cos(2 * np.pi * t / period)  # where each bob is, from -1 to 1
    ax.scatter(swing, np.arange(len(n)), c=swing, cmap="cool", s=40)
    ax.set_title(f"t = {t} s")
    ax.set_xlim(-1.2, 1.2)
    ax.axis("off")
plt.show()

# Try: t = 60 s (or 75, 80, 90): the same pictures again, as every count is whole

No Python on this computer? The notebook runs the same script, the exercises and the answers in your browser, on Google Colab.

Line by line

1. n = np.arange(51, 75)

   The 24 counts of swings per minute: 51, 52, … 74.

2. period = cycle / n

   Each pendulum’s period: 60 seconds divided by its count.

3. length = g * (period / (2 * np.pi)) ** 2

   The small-swing pendulum formula T = 2π√(L / g), solved for the length L.

4. fig, axes = plt.subplots(1, 4, figsize=(12, 4), sharey=True)

   One figure with four plots side by side; the loop fills one plot per moment.

5. swing = np.cos(2 * np.pi * t / period)

   Where each bob is at the time t, from −1 to 1: after t / period swings, the cosine of that many full turns.

6. ax.set_title(f"t = {t} s")

   An f-string puts the value of t into the title.

Try it yourself

1. How long are the longest and the shortest pendulum?

   Hint

   The script prints the lengths.

   Answer

   34.4 cm for 51 swings a minute, 16.3 cm for 74.

2. Add the moment t = 12 s. How many groups does the row split into?

   Hint

   You need a fifth plot: plt.subplots(1, 5) and [0, 12, 15, 20, 30].

   Answer

   Five: at 12 s, a fifth of the minute, neighbours are a fifth of a swing apart.

3. Make it 30 pendulums.

   Hint

   Change the end of np.arange.

   Answer

   n = np.arange(51, 81) gives 30 counts, from 51 to 80.

Quiz