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Re: Q re gyroscope-based toy


George J. Bugh  wrote in message 3963C107.B2F8FED3@flash.net">news:3963C107.B2F8FED3@flash.net...
> Is there anyone in Australia that can send me a copy of that article
> please? I can pay for slow mail. Is that something amazon.com would
> have?
>
> Tom Snyder wrote:
> >
> > There was an article about the DynaBee in The Physics Teacher:
> >
> > Physics Teach., Vol. 18, No. 2, February 1980 Pages 147 - 148
> > "The physics of the 'Dyna Bee'"
> > by J. Higbie, Department of Physics, University of Queensland,
> > Brisbane, Australia 4067

Thanks to Tom Snyder for the reference.  I went to the
library and made a copy of the article.  Here is its
text.  It also contains some diagrams, which I can't
reproduce here, though I've included their captions.

--
  Yaakov Eisenberg


----
     A new "toy" for grown-ups has recently appeared on the
market called the "Dyna-Bee."  It is about the size of a
soft ball and has a wheel inside it which can be speeded-up
by appropriate wrist-action when it is held in the hand and
given an initial spin.  It is sold as a wrist exerciser but
let's face it, it's a toy.
     The real challenge is to get it to work.  Only those
with an inborn muscular coordination can make it work right
away.  The rest of us need lots of practice.  But, like
riding a bicycle, once you get the knack, you never forget.
The interesting question is what makes it work?  It turns
out that this is a very handy device for illustrating the
principles of gyroscopic motion since precession actually
drives the internal wheel.
     The device consists of an internal weighted wheel
mounted on a small diameter rigid axle.  The axle runs in
a U-shaped groove inside the plastic housing.  The groove
extends all the way around the inside of the spherical
housing so that the axle can roll through a full circle
(Fig. 1).  There is a hole in the bottom of the spherical
housing which exposes part of the wheel's rim so that it
can be given its initial spin by flicking it with the heel
of the hand.

Fig. 1.  Schematic of the Dyna Bee.
The internal U-groove extends all the way around and
holds the axle tips.  There is also a steel ring with
two holes in it for the axle tips which slides along
the inside of the housing in contact with the groove.
This ring keeps the axle along a diameter of the groove
circle.

     Once the wheel is spinning, the housing is twisted so
that the bottom of the U-groove presses upward on one end
of the axle and the top of the groove presses down on the
opposite end.  This gives the spinning wheel a torque and
it responds by precessing the axle tips along the length of
the groove.  The axle tips move in just the right direction
so that as they roll on the side of the groove, it causes
the wheel to speed-up.  That is, the precessional motion
moves the axle tip along in the groove and the friction is
in just the right direction to cause the wheel to spin
faster (Fig. 2).

Fig. 2.  The torque applied by the sides of the groove
causes the spinning wheel to precess.

     As the axle tips move around, the forcing torque must
move around as well and this is what gives the circular
wrist action needed to make it work.  As the wheel gains
speed the precessional velocity decreases and the tangential
velocity of the axle's lateral edge in contact with the
groove increases.  These two effects merge together when the
precessional speed of the tip is just sufficient to cause
it to roll without slipping in the groove.  At this stage
a dynamical equilibrium is reached where the wheel will no
longer speed up.  It just "holds its own."  If you want the
Dyna Bee to go faster, you have to increase the precessional
speed by increasing the applied torque.
     The precessional speed of the axle tip is V = R d\theta/dt
= R\Omega, where R is the radius of the circular U-groove
(length of half the axle) and \Omega is the angular precession
rate.  The tangential speed of the axle's lateral edge is v =
r\omega, where r is the radius of the axle shaft and \omega is
the angular speed of the internal wheel.  The angular momentum
of the wheel is L = I\omega, where I is the moment of inertia
of the wheel.  The torque causes the angular momentum vector
to precess: \tau = dL/dt = L d\theta/dt = L\Omega (Fig. 2.).
The precessional speed of the axle tip is then:

         V = R\Omega = R\tau / L = R\tau / I\omega

The relative velocity between [the] edge of the axle in contact
with the groove and the groove itself is

         v_r = V - v = R\tau / I\omega - r\omega

When this becomes zero, the axle rolls in the groove without
slipping.  For a constant torque, the maximum angular speed
is then: (Fig. 3)

         \omega_0 = \sqrt{R\tau/Ir}

Fig. 3.  Graph of V and v versus \omega for constant torque

     Unfortunately this device is too small for lecture
demonstrations, but it may be possible to construct a
larger one operating on the same principle for lecture
use.  However, for small classes or laboratory groups
it should be quite effective since everyone wants to
"have a go" as we say in Australia.