Answer to the Question 04/01
BALL IN A BOXThe 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?
(1/03) Armin Rahmani (5/9/02) (e-mail armin_rahmani@hotmail.come) supplied a partial solution. We also got many qualitative solutions of the problem. However, the closest to a complete quantitative solution came (25/11/04) Eduardo Aoun Tannuri from University of Sao Paulo, Brazil (e-mail eduat@usp.br) that supplied a rather detailed answer that can be found in PDF file.
Answer: The the rear lid will not be reached. However, the solution of the problem is much more interesting that the final answer, because one needs to consider many interesting effects. Not all effects have been considered and we are waiting for additional emails on the subject.
The solution:
On the most elementary level the solution appears to be very simple. If we for a moment assume that as a result of acceleration the fluid density has settled into a linear profile, with the original density maintained in the center of the cylinder, then the difference between the apparent "weight" ma of the ball and the Archimedes force will create a force that increases linearly in distance from the center. Thus in the absence of friction we have a "harmonic oscillator". If the ball start at the front lid, then the amplitude of its oscillations will be half-length of the cylinder, and therefore it will reach the rear lid after half-period of its oscillation. If some friction (fluid viscosity) is introduced, the ball will not reach the rear lid, and its oscillations around the center will be damped. Eventually, it will settle at the center of the cylinder. However, the problem is slightly more difficult: it is even not trivial to identify all kinds of forces that act on the ball.
The solution of Tannuri addresses most of the aspects of the problem: when the ball is moving it is experiencing the "inertia" force (due to acceleration of the reference frame - the box), the force due to acceleration of displaced fluid (this would be present even in the absence of viscosity), the Archimedes force, and small viscous forces.
One point still needs to be clarified: How does the liquid go from the initial uniform density state to the final state where the density is a linear function of the position? It is clear that a wave will be created in the beginning of acceleration. That wave will be damped by viscosity, How long will it take? Is that time negligible compared to time scale over which the ball performs its motion?
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