Wilberforce's Spring

Plate 5 speaks for itself: “Bob, then twist, then bob again.”

The Physics

A mass on a helical spring can bob up and down and twist. The spring weakly couples the two motions, so when their frequencies are tuned equal the energy swaps slowly between bobbing and twisting, like two coupled pendulums beating.

f_T = (1/2π)√(k/m) ≈ f_R = (1/2π)√(κ/I)

• f_T, f_R: translational and rotational frequencies, Hz
• k: spring constant, N m⁻¹
• m: bob mass, kg
• κ: torsion constant, N m rad⁻¹
• I: bob moment of inertia about the spring axis, kg m²

Tuned so f_T ≈ f_R, the energy swaps from bobbing to twisting and back. Wilberforce, 1894.

Sources

• L. R. Wilberforce, Phil. Mag. 38, 386 (1894).
• R. E. Berg, T. S. Marshall, Am. J. Phys. 59, 32 (1991).

On the poster

Plate 005, “Wilberforce's Spring”: the pixel-art plate with its slogan “Bob, then twist, then bob again.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.