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Let's say that if I put a milk canister on a scale it shows 10kg. If I fill it with Helium it must necessarily be less than this, right?

a milk canister

Can someone explain this (probably trivial) thing to me? :)

What do we weigh anyway? The canister with all the air above it? And what does all that have to do with how submarines work?

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The idea is that helium weighs less than normal air.

To consider a more extreme example, imagine two situations where the milk canister is underwater. In the first situation it is filled with air; in the second, lead.

In the case where the milk canister is filled with air, it may even float. It would still want to go down because of gravity, but the surrounding water would want to go down even more, and so it would push the milk cansister up to make space. The phenomenon of a surrounding fluid pushing an object up is called buoyant force. While the usual gravitational force points down and has strength $Mg$ (where $M$ is the mass of the object and $g$ is gravitational acceleration), the buoyant force points up and has strength $M_d g$, where $M_d$ is the mass of the displaced fluid. Thus if the mass of the object is the same as the mass of the water it displaces, then there won't be a net force up or down.

Now imagine the case where the milk canister is filled with lead. Then it will certainly want to go down, because it has more mass than the water it displaces. The net force on it would still be less than its weight outside of water. The amount it would be less by is the buoyant force, $M_d g$.

Before we consider your question, let's examine one more situation, which is the milk canister in a vacuum chamber. This is the easiest situation of all because there is no buoyant force, so the net force on the milk canister is just the force of gravity $Mg$. So in this case, the canister filled with helium will weigh less than the canister filled with air simply because the helium has less mass.

Now if we go back to your situation (say by letting air into the vacuum chamber) there will be a non-zero buoyant force, so the net force on each canister will decrease, but since both canisters displace the same amount of air, the net force will decrease by the same amount for both the air-filled and the helium-filled. Thus the helium-filled one will still be lighter.

Now what does a scale weigh? When you put an object on a scale, the object sits there stationary. This means there is no net force on the object. The three forces acting on the object are gravity, buoyant force, and the normal force from the scale. Thus the normal force must exactly cancel the vector sum of gravity and the buoyant force. Thus the scale is exactly measuring the $Mg-M_dg$ force we have been talking about all along.

You can say that a scale weighs all the air above the canister, but you don't gain anything. To see why, consider the case where you put nothing on the scale. Then the scale reads zero. One way of thinking about this is that you have nothing on the scale so you get zero. This is the simple way of thinking about it. The other way of thinking about it is that you air weighing the column of air above the scale, and you get zero because the mass of the air above the scale is the same as the mass of air it is displacing.

Submarines float because they have a mass equal to the mass of the water they displace, and so they have no net force. Actually there is an additional subtlety. Not only does a submarine float, but it floats at a particular depth in the water. This is because as you go farther underwater, the pressure increases, which squeezes the water and causes its density to increase. Thus as you go deeper and deeper, and mantain constant volume, you displace more and more water. Thus if you have no net force at some depth and you go up, you will displace less water, giving you a net force down, which will cause you to go back down. Similarly if you go down, you will displace more water, and get a net force up and you will go back up. Thus your depth will tend to stay the same.

This also means that you can control your depth by making small changes to your total mass. In a submarine these small changes are made by either letting water into the submarine or expelling water out of the submarine.

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  • $\begingroup$ Thanks, great writeup! I wish I could vote you up, but I don't have enough reputation points to do that. $\endgroup$ – user65380 Nov 27 '14 at 12:46
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The answers so far are OK, but at the risk of being downvoted for humor, let me suggest reading This What-if Comic

If you start with an evacuated canister, things will be different from starting with a canister full of atmosphere at STP.

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  • $\begingroup$ pretty cool, have to re-read it later I think. :) $\endgroup$ – user65380 Nov 27 '14 at 12:45
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The difference will be : $$\Delta M=V*(\rho_{He}-\rho_{air})$$ where $\rho$ stands for the volumetric mass density and $V$ the volume inside the canister. Therefore $V*\rho_{air}$ is the mass of air inside the canister. $$M_{displayed}=V_{inside}*\rho_{gaz}+V_{canistersteel}*\rho_{steel}-V_{tot}*\rho_{air}$$

The scale measures the difference between the weight of the canister and the gaz inside and the weight of the same volume of air (archimedes).

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