In a case where you vacuum your items in those airless bags(Space bags), do they weigh more ? Or is it just a feeling because of increase in density due to decrease in volume ?
[Physics] Does object sealed in a airless place weigh more
weight
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You send the box and liquid towards a barrier equipped with a gauge to measure force. The setup looks like:
When the box hits the barrier it stops, but the liquid inside it keeps moving. A short time later the liquid hits the side of the box and it too stops moving. So when you record the force at the barrier as a function of time you will get two peaks, first as the box hits the barrier and stops, then a short time later a second peak as the water hits the end of the box and stops.
If you integrate the force time curve you will get the impulse during the collision, and this is equal to the change of momentum. Since momentum is $mv$, and you know the velocity $v$, you can calculate the mass. The two peaks will give you the mass of the box and the mass of the liquid.
Needless to say, in real life you will get only approximate results. The peak for the box should be clear, however the viscosity of the liquid will mean there is a force exerted on the barrier while the liquid is moving and before it hits the end of the box. Also the liquid will splash, so the impulse you measure will be too high. However the method should give you an approximate result.
Analyzing the acceleration of the center of mass of the system might be the easiest way to go since we could avoid worrying about internal interactions.
Let's use Newton's second law: $\sum F=N-Mg=Ma_\text{cm}$, where $M$ is the total mass of the hourglass enclosure and sand, $N$ is what you read on the scale (normal force), and $a_\text{cm}$ is the center of mass acceleration. I have written the forces such that upward is positive
The center of mass of the enclosure+sand moves downward during process, but what matters is the acceleration. If the acceleration is upward, $N>Mg$. If it is downward, $N<Mg$. Zero acceleration means $N=Mg$. Thus, if we figure out the direction of the acceleration, we know how the scale reading compares to the gravitational force $Mg$.
The sand that is still in the top and already in the bottom, as well as the enclosure, undergoes no acceleration. Thus, the direction of $a_\text{cm}$ is the same as the direction of $a_\text{falling sand}$ . Let's just focus on a bit of sand as it begins to fall (initial) and then comes to rest at the bottom (final). $v_\text{i, falling}=v_\text{f, falling}=0$, so $a_\text{avg, falling}=0$. Thus, the (average) acceleration of the entire system is zero. The scale reads the weight of the system.
The paragraph above assumed the steady state condition that the OP sought. During this process, the center of mass apparently moves downward at constant velocity. But during the initial "flip" of the hour glass, as well as the final bit where the last grains are dropping, the acceleration must be non-zero to "start" and "stop" this center of mass movement.
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Best Answer
For any every day life purpose, they weight the same. But there is a difference:
Weight vs mass
Weight and mass are different things. The mass is a measurement of the amount of matter (alternatively, the opposition of a body to change trajectory), whereas the weight is the strength with the Earth pulls objects towards it. The difference is subtle: an astronaut in orbit has mass, but no weight; a scale measures weight, but it transforms it to mass.
Now think of a helium balloon as it goes up. Why does it go up, if it has some mass, and thus the Earth pulls it downwards? Because its weight is less than the air around, so the Earth is pulling from it more, and so it has preference.
Or, in other words, Archimedes principle holds. If you measure it, the balloon has negative weight.
Feathers and lead
If you weight a ton (mass) of feathers and a ton of lead, you will get that the lead is slightly heavier: the air around the feathers is pushing it upwards more than the air around the lead, because feathers take much more space, and thus, more air helps with buoyancy.
A bag full of air
Now we get to your question. We have two rigid identical boxes, one empty, and other full of air. The second one will weight a bit more, and also have more mass. The effect of the air is the same for both, as they have the same volume.
Now, take two soft plastic bag and fill one with air. Again, the mass of the one with air will be greater, but the weight will be the same. Why? Because the one that is empty has also a smaller volume.
Do a thought experiment: take the rigid box full with air and put it on a scale. Carefully open the lid. Nothing has changed, the weight is the same. In the soft bags, the extra mass added by the bag gets exactly cancelled out by the buoyancy (by Archimedes' principle) provided by the extra volume.
Perception
All this efects are small. For volumens of a liter, we are talking about differences of 1g. As you righfully note, the density increases, and we tend to feel it as heavier.