Discussion of the Question 04/01

BALL IN A BOX

The question was:

A hollow cylinder, with its both ends closed, is filled with a fluid and is at rest in the space. Inside the cylinder there is a small hard ball with a density equal to the density of the fluid. The ball is initially at rest and is close to the center of one of the lids (let's call it a "front lid"). The cylinder suddenly gains an acceleration a and then moves with that constant acceleration (the motion is non-relativistic although the magnitude of the acceleration can be large). The direction of the acceleration is along the axis of the cylinder pointing from the "rear lid" to the "front lid". If viewed from the reference frame of the cylinder at the very first moment after the acceleration appears the ball will, of course, start gaining speed in the direction toward the "rear lid". The question is: will the ball hit the rear lid?


(17/8/01) Y. Kantor: We are getting many replies in which people attempt to deduce the position of the sphere from the assumption that the density of the liquid does not change under acceleration and consequently the sphere will always be at a neutral buoyancy. Clearly, such assumption of absolutely incompressible fluid only leads to a conclusion of completely "indifferent" equilibrium, and cannot serve as a guide of the final position of the sphere. If, however, the fluid is even slightly compressible, and the initial temperatures and pressure were such that the acceleration will not cause a phase separation (into, say, liquid and gaseous states) we may assume that the density will be lower at the front lid and higher at the back lid. If the changes in density are not large (that depends on the compressibility and the acceleration), then we can assume that we will have an almost linear profile of density along the axis of the cylinder. This should lead us to the conclusion regarding the final position of the sphere. However, one needs to do more careful dynamical analysis to decide what will happen before the sphere settles into its final position.
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