Object suspended over an empty bucket

[Tlaloc86]

I was recently tutoring a first year student, and a question of her assignment was as follows:

Suppose that that you have a bucket of water over a scale. If you then partially submerge an iron rod in the water, while holding the rod so that it does not touches the bucket, will the reading on the scale change? The answer is that yes, the reading n the scale will increase. One possible explanation of this is that the water exerts a buoyant up force over the rod, and therefore the rod will exert a force down over the water, and this force will be registered in the scale.

I was then thinking what would happen in the same scenario, but this time the bucket is empty. After all, air is also a fluid, so it also exert a buoyant force on the rod. Will this also register on the scale? Of course if it does the increase will much smaller than in the previous case since air has much less density than water, but I'm curious if the effect will be there at all. The reason I think there may not be any increase here is that in the previous case we had two phases: water and air, whereas in this case we only have one phase: air, so the two scenarios are not completely analogous. Or maybe the water is "resting" over the bucket, while the air is not...

Anyway, just a rather silly question I cannot wrap my head around :) Thanks!

 

[A.T.]

The reason I think there may not be any increase here is that in the previous case we had two phases: water and air, whereas in this case we only have one phase: air, so the two scenarios are not completely analogous.

Right, the bucket is supporting the water, while it's not supporting the air. In fact, the air is supporting the bucket, if we want to consider the volume of the bucket walls.

 

[jbriggs444]

I was then thinking what would happen in the same scenario, but this time the bucket is empty. After all, air is also a fluid, so it also exert a buoyant force on the rod. Will this also register on the scale?

It's a trick question -- the scenarios are different because you set them up differently.

The bucket is half full of water. It is completely full of air. That makes all the difference. Try the thought experiment with a bucket that is completely full of water before you dip the iron rod in. Does the scale reading change now?

 

[Dale]

If you then partially submerge an iron rod in the water, while holding the rod so that it does not touches the bucket, will the reading on the scale change? The answer is that yes, the reading n the scale will increase.

In order to answer your second question, it helps to answer the first question a little more fully. So, for the first scenario you have correctly indicated that the reading will increase. How much will it increase? Assume the density of the water is $\rho_0$, the density of the rod is $\rho_1$, the volume of the water is $V_0$, and the volume of the submerged portion of the rod is $V_1$, the total mass of the rod is $M$ and the mass of the bucket is $m$, and assume that the experiment is carried out in vacuum

 

[Tlaloc86]

It's a trick question -- the scenarios are different because you set them up differently.

The bucket is half full of water. It is completely full of air. That makes all the difference. Try the thought experiment with a bucket that is completely full of water before you dip the iron rod in. Does the scale reading change now?

Thank you, I think this solves it. If the bucket is full of water there is no change of the scale (You have a reaction to the buoyant force pointing down, but that same weight of water leaves the bucket when you introduce the rod, so no net change in the balance). Thanks!