Ball on a rotating turntable     Hugh Hunt

The dynamics of a ball on a rotating table have been understood for decades.  An elegant and compact vectorial treatment can be found in Milne, E. A.”VECTORIAL MECHANICS”.  The analyses given here (top right and bottom left) are based on this vectorial approach. 


The analysis given above (top left) for a ball on a horizontal rotating table is purely Cartesian and is easier for those less skilled in vector manipulation to follow.  It makes use of     where QP is the total moment of external forces about a moving point P, hP is the moment of momentum about the moving point P,  is the velocity of point P and p is the linear momentum of the ball.  A feature of a rolling ball is that the centre of mass G is always directly above the contact point P so that   where  is the velocity of the centre of mass.  So  since .  Therefore, since QP = 0 (all forces pass through P) we have   or hP is constant ie moment of momentum is conserved about the moving point P.   This is beautiful.