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Re: [Phys-L] fidget spinner: more data, more analysis, timestamps, force-law plot



Regarding the difference in the deceleration vs speed profiles among the different spinner cases JD discusses below:

For example, if you think the frictional force is some function
of velocity, you can write
dv/dt = F(v)
and you can visualize the force law by plotting dv/dt as a
function of v.

It is well worth doing this for the old and new spinner data.
For the data BC got from the web, the plot is here:
https://www.av8n.com/physics/img48/fidget-spinner-force-law.png
The red line is a fitted parabola, mostly drag proportional to
v^2 but with a small linear component that is noticeable at
small v.

For the data that BC took yesterday, the situation is markedly
different:
https://www.av8n.com/physics/img48/bc-spinner-force-law.png

For one thing, we notice that the speeds are an order of
magnitude smaller. It appears that the v^2 term (presumably
aerodynamic drag) is not noticeable here. There is a small
linear term, plus a constant term. The constant term goes
away at the very smallest speeds, as it must, since the net
force must go to zero at zero speed.

I do not have a microscopic physical explanation for the
constant term. It will require more thought than I can give
it at the moment.

It seems to me that the shape of the spinner probably has the biggest effect on whether the deceleration force is dominated by a linear or quadratic term in speed (assuming comparable balance and friction in the bearing, etc). I would expect a spinner that pushes a lot of air out of the way as it spins (because of its lobed shape) would have its deceleration dominated by the quadratic drag. OTOH I would expect a spinner that is rotationally symmetric about its spin axis would be dominated by a linear term until significantly higher speeds simply because the rotation of an axially symmetric object doesn't have to push any air mass out of the way, and its drag is caused by the shearing of the air in a region around its outside in an axially symmetric way, so that the quadratic term is essentially only being generated by the intrinsic nonlinearity of the NS equations for high shear rates & Reynolds numbers caused by the strong shear field gradient pattern. To leading order the shear force is linear in the shear rate and hence the velocity. The nonlinearity occurs to higher order as it is generated internally by the fluid shear rather than the inertial reaction of continuously displacing an intervening mass of air to make room for the whirling lobes.

As far as the apparent constant term mentioned above goes, I suspect that there is some sort of kinetic dry friction coefficient effect in the bearing contacts that may explain (or at least model) it.

Both of these hypotheses are testable. BC's photos show he has both an axially symmetric and a lobed spinner whose spin-down data can be contrasted. Also, if a spinner shows a constant drag force term perhaps a tiny drop of very light lubricating oil on the bearings may make that contribution greatly reduced, if it is indeed caused by a dry kinetic friction mechanism.

David Bowman