Lesson 7: Stepping through time

"""A ball under gravity in a ring with a gap that spins; the bounces are elastic and without friction.
Run: python spinning-ring.py   (needs: pip install numpy matplotlib)
"""
import numpy as np
import matplotlib.pyplot as plt

plt.style.use("dark_background")

R, ball, g = 1.0, 0.05, 9.81  # ring radius (m), ball radius, gravity
gap, spin = np.radians(18), 2 * np.pi / 4  # an 18 degree gap, one turn every 4 seconds
p, v = np.array([0.3, 0.2]), np.array([1.8, 0.7])
dt, t, track = 0.0005, 0.0, []

while t < 60:
    v[1] -= g * dt
    p = p + v * dt
    t += dt
    track.append(p.copy())
    r = np.linalg.norm(p)
    if r > R - ball:  # the ball touches the wall, or is at the gap
        angle = np.arctan2(p[1], p[0])
        centre = -np.pi / 2 + spin * t  # where the gap is now
        off = abs((angle - centre + np.pi) % (2 * np.pi) - np.pi)
        if off < gap / 2 - ball / R:  # all of the ball fits through: out
            break
        normal = p / r
        v = v - 2 * np.dot(v, normal) * normal  # the bounce: the spin plays no part in it
        p = normal * (R - ball)

track = np.array(track)
print(f"escaped after {t:.1f} s" if t < 60 else "still inside after 60 s")
plt.figure(figsize=(6, 6))
plt.plot(track[:, 0], track[:, 1], color="#5eead4", linewidth=0.4)
plt.gca().add_patch(plt.Circle((0, 0), R, fill=False, color="white"))
plt.axis("equal")
plt.axis("off")
plt.show()

# Try: spin = 0 and centre = -np.pi / 2 (a still gap at the bottom): the bounces stay the same

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. v[1] -= g * dt

   Gravity: every step, the vertical speed drops by g·dt.

2. p = p + v * dt

   The ball moves by its velocity times the time step. This simplest way to step is called Euler’s method.

3. if r > R - ball:

   The ball’s edge reaches the wall when its centre is one ball radius away from the ring.

4. off = abs((angle - centre + np.pi) % (2 * np.pi) - np.pi)

   The angle between the ball and the middle of the gap, wrapped into −π … π, so that 359° counts as 1° away.

5. v = v - 2 * np.dot(v, normal) * normal

   The reflection: remove the part of the velocity along the normal twice. The spin of the ring does not enter it.

6. p = normal * (R - ball)

   Put the ball back on the wall, so that a step can never leave it outside.

Try it yourself

1. Check the energy, ½·|v|² + g·y per unit mass: print it at the start and at the end. Does it stay the same?

   Hint

   np.dot(v, v) is |v|².

   Answer

   Nearly: it drops by about 2 % over the run, from 3.83 to 3.75. The simple steps and the push back onto the wall lose a little.

2. Halve dt. What happens to the energy and to the moment the ball escapes?

   Hint

   dt = 0.00025.

   Answer

   The energy now drops by about 1 % instead of 2 %; the ball still escapes after about 3.9 s.

3. Why does the bounce line not use spin at all?

   Hint

   Which way does a point of the wall move?

   Answer

   Without friction the wall can only push along the normal, and a spinning wall moves along the tangent, at right angles to it.

Quiz