Spinning Eggs and Communication Satellites

If I spin a marble egg horizontally on a work top why after a time delay will it right itself and end upright thin point up , fat point down .. Until losing speed ?
- Rob Edwards (age 45)
Witney,Oxfordshire,England

Dear Rob,  What a wonderful question!  

First the egg problem.  You have to be sure to conserve both energy and angular momentum.  Call the angular momentum about the egg's initial spinning axis L₁ = I₁ ω₁ and that along the egg's final spinning axis L₂ =  I₂ ω₂ .  The I's  are moments of inertial about their respective axes.  Conservation of angular momentutm gives you I₁ ω₁ = I₂ ω₂  where ω's are the angular velocities in radians per second.   Now for energy.  In the initial position you have E₁= mgH₁+(I₁ω₁²)/2 where H₁ is the height of the center of mass above the table.  In the final position you have the a similar equation for E₂.   Without going into the messy algebra you will find that energetically the egg could stand up IF: 1,  The moment of inertia  I₁  is smaller than I₂  AND: 2,  The center of mass of the egg whilst on end is not too high AND:  3, The egg is initially spinning fast enough. 

I went to our refrigerator and much to the chagrin of my wife started spinning hard boiled eggs on the kitchen counter.   Sure enough, it works.    Now, why should the egg decide to stand up on end?

The answer has to do with stability of spinning motion of a three dimensional object with no applied torques.  This is a problem of great interest to space craft engineers.   For example you want an observing or communication satelite to maintain its orientation as it orbits the earth.   It turns out that if you have a 3-D object with moments of inertia I₁, I₂ and I₃,  with I₁ < I₂ < I₃,  and no applied torques, then the spinning object will be stable along the I₁ and I₃ axes but it will be unstable along the I₂ axis.   You can demonstrate this yourself with a tennis racquet or ping-pong paddle.   Both are stable when spinning along the axis of the handle as well as along the axis perpendicular to the face.   But try spinning it along an axis similar to flipping eggs in a skillet and you will find it extremely difficult to make a nice 360° flip.    Such are the mysteries of 3-D motion.

LeeH

(published on 09/18/2014)