Intuition problem about an accelerating barometer

[Kaushik]

Consider a barometer kept in an elevator. When the elevator accelerates upwards, there is a pseudo force along with gravitational force acting on the liquid in the barometer. Due to which pressure on the liquid should increase, right? If pressure on the liquid increases then to balance this, the water level should rise and not decrease. Isn't?
But my book states that the water level decreases. I am not able to understand why that is the case.

When the elevator accelerates downwards the pseudo force is upwards and hence the air pressure inside the elevator decreases and the level of liquid should also drop. What is wrong in my intuition here?

 

[A.T.]

there is a pseudo force along with gravitational force acting on the liquid in the barometer. Due to which air pressure should increase, right?

A force acting on the liquid in the barometer is increasing the air pressure?

 

[Kaushik]

A force acting on the liquid in the barometer is increasing the air pressure?

Oops! I meant 'increase the pressure on the liquid'. As the pressure on the interface increases the pressure of the point in the same level (under the inverted tube) also increases. Due to this increase, won't the water level rise?

 

[Kaushik]

Is it done like this:
Let the external pressure be $P_{ext}$ before the elevator accelerates and $x_0$ be the new height.
Then when it starts to accelerate,
$P_{ext} + ρax_0 = ρgx_0 \tag1$
But,
$P_{ext} = ρgh \tag2$
Comparing the above,
$gh = x_0(g-a) \tag3$
$\implies x_0 > h \tag4$Or is it done like this:
$P_{ext} = ρ(g_{eff})x_0 = ρ(g+a)x_0 \tag5$
But,
$P_{ext} = ρgh \tag6$
Comparing the above $2$,
$gh = (g+a)x_0 \tag7$
$\implies x_0 < h \tag8$

Which one is correct and why?

 

[A.T.]

Oops! I meant 'increase the pressure on the liquid'.

The pseudo force is acting on the entire volume of the liquid, not at some interface as pressure.

 

[Kaushik]

The pseudo force is acting on the entire volume of the liquid, not at some interface as pressure.

If it is acting on the entire volume, then won't the air pressure increase?

 

[A.T.]

If it is acting on the entire volume, then won't the air pressure increase?

If the elevator is airtight then the air pressure at the bottom will increase. But the pseudo force is acting on both: the liquid and the air.

 

[Kaushik]

If the elevator is airtight then the air pressure at the bottom will increase. But its acting on both: the liquid and the air.

So the air pressure at the bottom will increase because more molecules are pressing against it as compared to other heights. So then the position of the barometer in the elevator also matters?

 

[A.T.]

So then the position of the barometer in the elevator also matters?

If the elevator is airtight it has some effect. But I'm not sure if your book wants you to consider that, or just the change in effective gravity.

 

[Kaushik]

If the elevator is airtight it has some effect. But I'm not sure if your book wants you to consider that, or just the change in effective gravity.

My book never mentioned about the difference in air pressure with height in an accelerating elevator. As of now, I think we can neglect its effect.

 

[A.T.]

$gh = (g+a)x_0 \tag7$
$\implies x_0 > h \tag8$

Are you sure?

 

[jbriggs444]

My book never mentioned about the difference in air pressure with height in an accelerating elevator. As of now, I think we can neglect its effect.

Yes. This effect is typically neglected. Similarly, we normally neglect the pressure difference between the front and rear of an accelerating car. Except when carrying helium balloons to a birthday party.

And yes, position matters. The pressure at the top of the sealed elevator will decrease when the elevator accelerates upward. The pressure at the bottom will increase.

 

[Kaushik]

Are you sure?

According to my perception(as of now), this seems right!
$\frac{g+a}{g} > 1 \implies x_0 >h$
If this is wrong, where am I going wrong and why?

 

[Kaushik]

Is it done like this:
Let the external pressure be $P_{ext}$ before the elevator accelerates and $x_0$ be the new height.
Then when it starts to accelerate,
$P_{ext} + ρax_0 = ρgx_0 \tag1$
But,
$P_{ext} = ρgh \tag2$
Comparing the above,
$gh = x_0(g-a) \tag3$
$\implies x_0 < h \tag4$Or is it done like this:
$P_{ext} = ρ(g_{eff})x_0 = ρ(g+a)x_0 \tag5$
But,
$P_{ext} = ρgh \tag6$
Comparing the above $2$,
$gh = (g+a)x_0 \tag7$
$\implies x_0 > h \tag8$

Which one is correct and why?

I know that both are completely opposite. It would be nice if you point out the error. Thanks!

 

[A.T.]

According to my perception...

How about algebra? Show your steps from (7) to (8).

 

[Kaushik]

Oops!
$(4)$ was actually $x_0 > h$ and $(8)$ is $x_0 < h$. Is it correct now?

 

[A.T.]

$(8)$ is $x_0 < h$. Is it correct now?

Yes.

 

[Kaushik]

Yes.

Ok so now, which one is correct? 1st attempt or the 2nd attempt?

Is it done like this:
Let the external pressure be $P_{ext}$ before the elevator accelerates and $x_0$ be the new height.
Then when it starts to accelerate,
$P_{ext} + ρax_0 = ρgx_0 \tag1$
But,
$P_{ext} = ρgh \tag2$
Comparing the above,
$gh = x_0(g-a) \tag3$
$\implies x_0 < h \tag4$Or is it done like this:
$P_{ext} = ρ(g_{eff})x_0 = ρ(g+a)x_0 \tag5$
But,
$P_{ext} = ρgh \tag6$
Comparing the above $2$,
$gh = (g+a)x_0 \tag7$
$\implies x_0 > h \tag8$

Which one is correct and why?

EDIT: I feel like the second attempt is wrong because the two $P_{ext}$ i equated are actually not equal. Am I correct?

 

[A.T.]

Ok so now, which one is correct? 1st attempt or the 2nd attempt?

2nd

 

[Kaushik]

2nd

EDIT: I feel like the second attempt is wrong because the two PextPextP_{ext} i equated are actually not equal. Am I correct?

What about this?

The external pressure before acceleration is actually lesser than the external pressure after acceleration. So how can I equate both of them?

 

[jbriggs444]

Is it done like this:
Let the external pressure be $P_{ext}$ before the elevator accelerates and $x_0$ be the new height.
Then when it starts to accelerate,
$P_{ext} + ρax_0 = ρgx_0 \tag1$

You will need to justify this starting point better. In words, what are you equating to what? And if $x_0$ is the new height, what was the old height?

 

[A.T.]

The external pressure before acceleration is actually lesser than the external pressure after acceleration. So how can I equate both of them?

See post #9.

 

[Kaushik]

You will need to justify this starting point better. In words, what are you equating to what? And if $x_0$ is the new height, what was the old height?

The following is what I am thinking:
$h$ is the old height.
$P_{ext} = ρgh$ (before accelerating)
Now, as it starts to accelerate the pressure increases.
It increases by $ρax_0$ due to upward acceleration.
New external pressure is $P_{ext} + ρax_0$
This new external pressure should be equal to the weight of the liquid column above it.
$P_{ext} + ρax_0 = ρgx_0$

Where am I going wrong?

 

[jbriggs444]

The following is what I am thinking:
$h$ is the old height.
$P_{ext} = ρgh$ (before accelerating)
Now, as it starts to accelerate the pressure increases.

So you are contemplating a column of uniform density air above the elevator which is h units in height. Say about ten miles high.

And you imagine that the elevator is sealed to the sides of the [implausibly tall] shaft like a piston in a cylinder.

And you imagine that the column of air is sufficiently incompressible that it rises up the shaft along with the elevator as soon as the acceleration begins.

Is that what you are imagining?

 

[A.T.]

Where am I going wrong?

Hard to say. Confusing the surrounding air with the liquid? Failing to decide if the book wants you to consider the change in air pressure, and provides enough information to actually do it?

 

[Kaushik]

Hard to say. Confusing the surrounding air with the liquid? Failing to decide if the book wants you to consider the change in air pressure, and provides enough information to actually do it?

I think I am getting confused in forming the equations. Like the 'cause and effect' is where I'm getting confused!

 

[jbriggs444]

I think I am getting confused in forming the equations. Like the 'cause and effect' is where I'm getting confused!

The intent in the book is that the elevator is vented to the shaft and that the shaft is ventilated to the outside. The pressure inside the elevator is always going to be the same as the pressure outside the elevator and is not affected by the elevator's acceleration.

You are not given the elevator's speed or direction of motion. There is no intent that any Bernoulli, air-ram or altitude effects be accounted for.

 

[sophiecentaur]

So you are contemplating a column of uniform density air above the elevator which is h units in height. Say about ten miles high.

The air pressure against the surface of the mercury would only be the same as the pressure from the mercury column if you were to accelerate a total column of air up to space at the same rate as the elevator. Alternatively you could add mass to the centre of the Earth so the local g would be the same for air and mercury. The 'weights of air' and mercury would track.

 

[A.T.]

I think I am getting confused in forming the equations. Like the 'cause and effect' is where I'm getting confused!

'Cause and effect' is irrelevant for the equations.

 

[sophiecentaur]

'Cause and effect' is irrelevant for the equations.

Yeah. That problem is common until you find that the approach that ignores cause and effect actually works. In another context, amplifier feedback is mind blowing until you realize that the steady state can be described and solved very easily.