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Foreword: I have been trying to solve this problem for several days: [Does diffusion cause the bottle to move to the left?] But the answers presented there are contradictory. Some say the bottle will not move. Some say the bottle will move. But I could not find a definite reason. I thought of a way to do this in practice. I think I was a bit successful there. But I had this problem which I am asking your help with. Though I can honestly say that this is neither a duplicate of the previous question nor a direct answer to that question.

A box-like container is placed on a frictionless horizontal plane. Suppose there are gaseous positive ions inside the container. They are evenly distributed throughout the container. Therefore, their center of mass is in the middle of the system (note that the center of mass of the container is also in the middle). Now the positive ions are attracted to the left side by some negatively charged object outside the container. The positive ions then concentrate to the left. Apparently then there is extra pressure on the left wall of the container. Therefore the container with positive gaseous ions accelerates to the left. At this point, the center of mass of the system weighs slightly to the left (because there is an external force caused by the negatively charged object). Now the negatively charged object is removed at once. Then the net external force on the system is zero. Therefore it moves uniformly with the obtained velocity. But here the positive ions must now slowly and evenly propagate in the container again. Then the center of mass returns to the center of the bowl. But since the center of mass of the system travels in a certain trajectory at a uniform velocity, the center of mass of the container (excluding ions) must displace to the left as the center of mass of the gaseous ions shifts to the right. That is, from the outside, the container appears to accelerate slightly to the left. But what creates that force over the container here to accelerate it?

The previous question discusses the necessity of the centre of mass being fixed, but no one is pointing out what force is responsible. That is what I am asking here.

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ACB
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2 Answers2

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Unless if the ions have a temperature of 0, they have random thermal velocities. So some randomly move left and some randomly move right. As long as the center of mass of the ions is on the left side of the box, the ions moving left collide with the box-wall sooner, imparting their momentum in a shorter time. $F = \frac{dp}{dt}$ will be larger on the left side than the right side, since dp is the same but dt is smaller.

Edit:

Let's say the negatively charged object is gone and the fields are gone. A moment afterwards, the ions will still be on the left side of the box. The force happens during the process of spreading throughout the box, so that's why I begin the explanation before they have finished spreading but after the negatively charged object is removed. Certainly, after they finish spreading, there will be no force, but by that time the CoM of everything will also have finished adjusting.

ACB
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Alwin
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You are right to say that the box and its contents will slide to the left when the attractive external force is applied, and will continue to slide at a constant speed when the force is switched off. However, immediately after the force is switched off there will be a transient effect when the positive ions are randomly redistributed throughout the box. The centre of mass of the box and its contents will shift to the right relative to the centre of mass of the box alone. However, since there is no external force involved, the combined centre of mass will continue to move to the left at a constant speed- the box itself will move relative to it.

The situation is analogous to a raft bearing some passengers floating down stream with a constant speed. If the passengers move from near the front of the raft to nearer the back, they will shift the centre of gravity of the combined raft/passenger system relative to the centre of mass of the raft alone. The effect is that the raft will be momentarily accelerated as the passengers move backwards, then decelerated as they stop moving. Throughout the movement, the centre of mass of the raft and passengers combined will continue to move at a constant speed with the stream.

The force that causes the raft to accelerate momentarily is the force applied by the passengers to move themselves along it. In the case of the box, the force is due to the impact of the positive ions- for a transient period after the external force is switched off, the positive ions will continue to impact one side of the box more than the other until they gradually and randomly redistribute themselves into a state of equilibrium in which they have the same effect on both sides of the box.

Professor Sushing
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