The slinky starts out under tension, like an extended spring. The top is being pulled down toward the center and the bottom is being pulled up toward the center.
To give the slinky its other interesting properties, the spring-rate has been matched to gravitational forces.
As a result, the bottom is pulled up by the tension in the 'spring' with about the same force as the gravitational effect, so it barely moves, and that is true elsewhere in the slinky, until the coil above it has collapsed.
At the start, the second coil is pulled up by the pull from the top coil, but the top coil is pulled down by both the tension and gravitation, and falls. As it falls, the next coil down loses the pull from the top coil and starts to fall under gravity no longer balanced by the upward pull from the top coil, and so on down the slinky.
If the spring constant of the slinky were higher (or the weight of the coils lower), the bottom coil would start to rise instead of staying put -- but it wouldn't be a slinky, able to do those weird tricks, then.
Well, sort of ...
First, an anecdote from my college days while taking elementary physics.
I was fortunate enough to have "tutors" in the form of mechanical engineers while I was working and attending college. Almost every day included a science lesson of some sort.
As an interesting project, I was encouraged to create a device which consisted of a single spring, similar to the slinky and a cylindrical mass attached to the bottom of the spring with the top of the spring attached to a rigid support.
The spring constants, the weight of the cylindrical mass and the moment of inertia about its axis were chosen to accomplish a system of two linked oscillators.
One oscillator stored energy alternately in the extension or compression of the spring and in the vertical motion of the mass. In an idealized spring-mass system, the spring constant and the weight allow one to calculate the frequency of oscillation.
The "twist" in this project was to select the mass such that the torsional spring constant and the moment of inertia of the mass about its vertical axis would form a torsional spring-mass system. In this system, the energy would alternately be stored in the twisting of the spring and in the angular momentum of the mass as it rotated about its vertical axis.
By a suitable choice of diameter for the cylindrical mass, it was possible to create a system of two linked oscillators, the natural frequencies of which differed by a small amount.
The result could be observed by pulling the mass downward and releasing it to impart the initial energy to the system. After several vertical oscillations, the magnitude of the vertical motion would decrease, eventually reaching zero. During the decrease in magnitude of the vertical oscillations, the spring was delivering energy into the torsional spring-mass system. The mass would begin rotating first one way about the vertical axis and then reverse to the opposite direction.
The secret to the effect was the fact that a helical spring must "unwind" a little bit as it is stretched. Each coil, being further from its neighbors when the spring is stretched, must cover a greater distance. The stretching of the spring creates a twisting force. This force is reversed when the spring is unstretched. Each stretching of the spring in the spring-mass system I described transfers a bit of the energy which was in the vertically translating mass into rotation of the mass about the vertical axis.
The above description explains why we see the twisting of the slinky in the video as the compression of the slinky changes.
The other key observation I would make concerns the fact that the slinky has no "mass" attached to the bottom of it. In a classical spring-mass system, the spring is typically mass-less. The spring-constant describes the extension or compression of the mass-less spring when acted upon by an external force, typically a mass acting under the effects of gravity.
In the slinky video, the spring is not mass-less. The extension of the slinky is entirely due to ITS OWN mass. The result is that the upper coils of the slinky are far apart due to the fact that these coils are supporting the entire slinky. To a first-approximation, the coils at the center of the slinky should be half as far apart, since those coils are supporting only the half of the slinky that hangs below them. The coils at the bottom are supporting hardly any mass at all.
Suspending the slinky as it is in the video results in a non-uniform extension of the spring, with the extension starting at zero at the bottom of the slinky and rising linearly to a value at the top which can be predicted by knowing the spring constant of the slinky.
Now imagine for a moment that we could somehow take this slinky, in its non-uniformly extended state, and place it in a space without a gravitational field. Recall that the extension of the spring at the top was exactly that which was needed to counteract the affect of gravity on the bottom of the spring. Without the gravitational field, that spring force would be that which would accelerate the bottom of the spring at 32 feet per second per second toward the center of mass of the spring.
The top of the spring would accelerate toward the center of mass of the spring at a greater rate, that which would result in the center of mass remaining still, since the spring is not being acted upon externally.
When the slinky is released as in the video, the bottom of the spring is accelerating toward the center of mass of the spring at 32 feet per second per second. At the same time, the center of mass of the spring is accelerating toward the center of the earth at 32 feet per second per second. The two accelerations cancel exactly at the bottom of the spring, causing the bottom to be motionless in the non-accelerating frame of reference which is the earth. The bottom of the spring IS accelerating in the frame of reference of the moving center of mass of the slinky.
If you had different tensions/thickness of steel at the oppsite ends of the slinky, I wonder if that would produce humorous results upon release?
W. G. Unruh's succinct paper on the matter here.
When a Slinky is dropped, the bottom of the Slinky remains motionless as the top collapses towards it, making it appear to the observer as though the Slinky is levitating. By considering the Slinky as a tightly wound, pretensioned spring, the static equilibrium of a hanging Slinky was solved for using Hooke's law (Equation 1). This result was used to measure the spring constant of an actual metal Slinky. The motion of the Slinky after it is released at time t=0 was then solved for to derive an expression for the time over which the bottom of the Slinky remains motionless and the Slinky appears to levitate (Equation 7). This expression gave a value of t = 0.29 ± 0.05 seconds for the Slinky used in the experiments, which matches up very well with the experimentally measured value of t = 0.4 ± 0.1 seconds