How Does a Spinning Doll Demonstrate Physics Principles?

[GENIERE]

To amuse my 1.5 yr old granddaughter I tied a 5-foot string to her doll and suspended it from the end of one blade in a ceiling fan. As you would expect, the doll spun in a circle maybe a foot or so lower than the blade.

After about 10 minutes, the doll slowly lowered to a point dead center in the rotational axis. The string was pulled inward! That means the doll was lifted a few inches. How is that explained?

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[Q_Goest]

That's pretty cool! I'll have to try that one myself. My daughter's 10 and she'll appreciate that! I wonder if it requires a certain weight of doll?

It sounds like the doll is going to a node where it's stable. You can see that same affect by holding a long rope and if you spin it just right, instead of making it fly around in a circle, you can get the center of the rope to stand still, and you'll get two hoops going, like a sine wave moving through 360 degrees (imagine graphing a sine wave from 0 to 360 degrees on an X-Y graph with the X being degrees then the line goes from 0 to 1 at 90, back to 0 at 180, down to -1 at 270 and back to 0 at 360 ). I suspect it has something to do with fan also, I wonder if it will happen with any weight of doll. The fan has a beam which attaches it to the ceiling, which is also flexing. I suspect if it didn't flex, it wouldn't do what you saw. Notice also, the centering of the doll balances the entire fan/doll system. Without the doll in that position, the fan is unbalanced. But to explain in words, why the system would tend to move towards that balanced condition is a bit beyond me. Must have something to do with String Theory! lol

 

[pervect]

The doll is going to seek the state of lowest total energy compatible with the constraints it is under.

Lifting the doll up requires some energy, but apparently it requires less energy with your particular geometry than it would to make it move in a circle along with the fan.

 

[GENIERE]

The fan is suspended by a ball and socket, so it can move freely in all directions. The fan actually tilted to the doll side as it rotated. The doll was about 8 inches long with a plastic head and a stuffed body, maybe half a pound in weight.

 

[pervect]

I had some time and I wrote down the Lagrangian for this, assuming that the fan blade continues to move in a circle. (This may be a false assumption).

The result is that the doll cannot lie on the center axis with the above assumption.

If we view the doll from a co-rotating frame from a viewpoint above the fan blade, we have the following diagram

                |y  
                |
   fan blade    | 
--------------->|------->x
       R

The equations of motion in terms of the generalized coordintes x and y are then

$$x'' = w y' + w^2(R+x) - \frac{gx}{\sqrt{D^2-x^2-y^2}}$$
$$y'' = -wx' - \frac{gy}{\sqrt{D^2-x^2-y^2}}$$

here g is the acceleration of gravity, and D is the constant length of the string.

The height of the doll is $$D - \sqrt{D^2-x^2-y^2}$$

One can see this without writing down the Lagrangian, the generalized forces on the doll in the rotating frame are due to centrifugal force, coriolis force, and the potential due to the height of the doll.

Unfortunately this means that the axis of rotation is not a stable point.
If x', x'', y', and y'' are all zero one has the relation

y=0
w^2(R+x) = gx/sqrt(D^2-x^2)

In short there is no way for the doll to be at x=-R though it is possible for the doll to be at a x < -R, which puts it on the other side of the axis of rotation. Here the centrifugal force pushes the doll in the -x direction, and it's balanced by a potential gradient force in the +x direction

 

[GENIERE]

...x < -R, which puts it on the other side of the axis of rotation. Here the centrifugal force pushes the doll in the -x direction, and it's balanced by a potential gradient force in the +x direction

Very nice - I did say dead center, but it probably was not.

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