Falling Motion of a Slinky Coil

A slinky coil is a spring with a small spring constant and a natural length that is almost zero.

If you hold one end and let it hang gently, it stretches a long way because of the small spring constant. After waiting until the whole system comes to rest, release the coil, then the coil falls, but the lower end remains at rest until the upper end crushes to the bottom. If you watch only the lower end, nothing seems to happen until the upper end comes down, and it looks as if the coil were not falling.

Even though the upper end of the spring is no longer supported, the lower end “doesn't notice” and does not immediately start to fall— a scene reminiscent of the cartoon “Tom and Jerry”— which gives a curious impression.

However, this is natural if we think as follows. While the spring is hanging, move its upper end up and down quickly, then the impulse travels along the spring; the speed at which the effect propagates equals the speed of waves (longitudinal density waves) along the spring. Because a slinky coil has a small spring constant, waves propagate slowly along the sping. When you release the coil, the upper end begins to fall at the speed that is basically comparable to this wave-propagation speed, but slightly faster. Therefore, until the upper end descends, the information that the support has been removed does not reach the lower end, and the lower end stays as it is, as if nothing had happened.

This simulator numerically integrates the equations of motion to simulate the falling motion of a slinky coil initially suspended by its upper end. If you want to observe the motion in detail, you can slow the simulation down with the Simulation speed slider.

If you check the Center of mass height option in Display, a green sphere appears, showing the height of the coil's center of mass. The center of mass falls with a constant acceleration. However, you will see that the upper end of the coil initially falls much faster than the center of mass, and that it actually decelerates as time passes.

If you check Pulse propagation, a second blue coil hung at its upper end is shown. When you press Start, the central coil begins to fall, and at the same time an impulse is applied to the upper end of the right-hand coil. How this impulse propagates is visualized by the color of that coil turning yellow. The upper end of the right-hand coil remains suspended at the same position even after the impulse is applied.

By comparing the two coils, you can see that the initial falling speed of the upper end of the central coil is the same as the propagation speed of the impulse's effect through the right-hand coil (see the note for analysis). The propagation speed of the impulse, as shown in the right-hand coil, decreases as it travels downward, because the tension is smaller in the lower part and the local extension decreases there. Meanwhile, the falling speed of the upper end of the falling coil also decreases, but more gradually, so when the upper end reaches the lower end, the impulse's effect has not reached the bottom yet. In other words, the lower end does not feel that the coil is falling until the upper end arrives.

Model used for the computation:

We represent a spring of natural length zero with spring constant $K$ and mass $M$ by dividing it into $N$ segments (rendered as rings). From the equilibrium state with the upper end fixed, we release the upper end and simulate the fall.

In a real slinky coil, when the spring compresses, inelastic collisions occur among the coils and the touching parts merge, so the merged portion falls as a single body. However, it is difficult to simulate both smooth spring motion and inelastic collisions that change the velocity discontinuously in a single algorithm. Therefore, when the local extension becomes negative we exponentially increase the restoring force, and we introduce dissipation proportional to the velocity with the coefficient $\gamma$, thereby approximating the effect of discontinuous inelastic collisions.

If the dissipation parameter $\gamma$ is set to zero, the collisions between coils become elastic and the mechanical energy is conserved. At the same time, because of the large restoring force to avoid overlapping coils, the speed of longitudinal waves becomes faster, and one can observe the influence of the fall of the upper end propagating faster than the upper end itself.