The Falling Plate Paradox: Does Gravity or Perfect Vacuum Have More Force?

[loddie]

I have a theoretical question for physicists. For reference, see attached image.

Suppose you have two perfectly flat plates aligned laying adjacent to each other. The plates are parallel to gravity. (perpendicular to the ground). Between the plates are perfect seals (no leaking). The seals have zero friction, regardless of compression. The plates also have zero friction. One plate, plate A, has a hole through it leading to a chamber with perfect vacuum and is rigidly supported. The other plate, plate B, is not secured to anything. Would plate B fall?

Please support your answer with formulas, free body diagrams, etc.

Initially I believed the plate would fall. However, I am wondering:

1. Does the mass of Plate B matter?
2. Is the force of a perfect vacuum greater than the force of gravity?

Thanks for reading and sharing your knowledge!

 

[sophiecentaur]

If the conditions are really 'ideal' then the answer is different from a practical case. "Perfectly flat" and "zero friction" implies a fictitious and ideal situation so how can you have any vertical force except that of the weight of the red plate

[Mentor's note: Edited as part of some general thread cleanup]

 

[loddie]

I believe the plate would fall. My argument was in a real world situation, the force preventing the red plate from falling would only be friction. Vacuum enables friction, but vacuum would not directly prevent the red plate from falling. However, coefficient of friction greater than the force of gravity would.

I'm interested in the answer to the theoretical question as it will help my understanding of physics. I'm guessing that the forces of perfect vacuum would have no effect on the red plate as the forces are perpendicular to gravity, but it has been a long time since high school physics.

 

[loddie]

I'm guessing most people would think the plate would fall, but I could be wrong. So I created another version of the problem. See attached image which I created (not copied from a textbook or exam) where the plates are now parallel to the ground. In this case, as long as the mass is of reasonable amount, I would not expect the plate to fall. Somehow, I'm missing something as the two situation are very similar, but I'm expecting different outcomes.

 

[Nugatory]

I'Somehow, I'm missing something as the two situation are very similar, but I'm expecting different outcomes.

There is no such thing as "force of vacuum". What seems to be a force from the vacuum is the outside air, which is at atmospheric pressure $10^5 N/m^2$, pressing on one side of the plate and no pressure at all from the other side.

In the first case (vertical alignment) the plate will fall. You can see this by looking at the forces on the plate:
- Gravity is acting downwards and no force is acting upwards.
- The force from the air pressure (not "force of vacuum", as I said above) is acting to push the plate to the left, but this force is opposed by the seals resisting being compressed so there's no net sideways force.
So we end up with a net downwards force and the plate falls. The mass of the plate is irrelevant because no mattrer how small the gravitational force, there's nothing to oppose it.

In the second case (horizontal alignment) the force from the air pressure is acting upwards, opposing the force of gravity. If the gravitational force ($mg$, where $m$ is the mass of the plate and and $g$ is $10 m/sec^2$) is greater than the force from the pressure ($10^5 N/m^2$ times the area of the plate that is exposed to vacuum) the plate will fall. If not, it will stay put.

 

[loddie]

Nugatory, thank-you for your detailed explanation! The component I was missing was the zero net sideways force due to the seals. I also fell into the trap of misunderstanding vacuum. Now I'll be able to sleep.

 

[sophiecentaur]

What seems to be a force from the vacuum is the outside air, which is at atmospheric pressure 105N/m210^5 N/m^2, pressing on one side of the plate and no pressure at all from the other side.

Also, whether you haver a 'perfect' vacuum or just a 'pretty good' one, makes very little difference to the actual force acting against the plate. It's a popular misconception that a perfect vacuum has a touch of the 'infinities' about it. Not so. The only infinite thing associated with a perfect vacuum is how difficult it is to achieve - i.e. removing the last few dozen molecules would take years and years and still you couldn't be sure. Deep space, even, has about one particle in every metre cube.