Calculating the Length of a Water Column in a Beaker

[patric44]

Homework Statement

a Basin contains water , a beaker is put upside down to a depth of 3m inside it , if the volume of the beaker
is 250 cm^3 . and its C.S area = 200 cm2 calculate the length of the water column which rises inside the baker , supposed that their is no air leakage from the beaker ?

Homework Equations

the formula of bolye's law : P1V1 = P2V2

The Attempt at a Solution

when i tried to solve it i used that approch :
its in the attachment .( which turns to be more complex that the solution manual )

the solution manual shows that the answer is 0.28 cm the same solution the i got . put the book didn't use
the same approch it states that : the air pressure inside the beaker after being put inside the water
Pg = Pa + ρgh (so his solution becomes significanly easier ) , he didn't take into his account the pressure due to the rising water as i did :
Pg + ρg(Lw ) = Pa + ρgh →→ Pg = Pa + ρgh - ρg(Lw ) .

so my question is : is the solution manual right ? or he had to take into his account the pressure of the rising water ? because i feel that its some how less accurate !

 

[gleem]

The book approach is approximate. Your way is more accurate and applicable to a very long beaker at depths comparable to the length. The column of water in the beaker is 1/1000 of the column of water that is compressing the air so the difference between the two approaches is small.

 

[kuruman]

The book approach is approximate. Your way is more accurate and applicable to a very long beaker at depths comparable to the length. The column of water in the beaker is 1/1000 of the column of water that is compressing the air so the difference between the two approaches is small.

There are indications pointing to the use of the approximation. The question states, "... a beaker is put upside down to a depth of 3m ..." without specifying to what point along the length of the beaker the depth is measured. This justifies the assumption that the pressure should be assumed constant along the length of the water column inside. The approximation is further validated as the "beaker" is said to have a diameter-to-height ratio of about 13, bringing it closer to being a Petri dish than a beaker, which makes the height of the column inside a small fraction of the beaker's length.

 

[patric44]

but it seems very silly from the book that he neglects the pressure of the water column that it asks to calculate !?

There are indications pointing to the use of the approximation. The question states, "... a beaker is put upside down to a depth of 3m ..." without specifying to what point along the length of the beaker the depth is measured. This justifies the assumption that the pressure should be assumed constant along the length of the water column inside. The approximation is further validated as the "beaker" is said to have a diameter-to-height ratio of about 13, bringing it closer to being a Petri dish than a beaker, which makes the height of the column inside a small fraction of the beaker's length.

i don't think that the approximation method was that obvious as you mentioned , it states that the beaker is put upside down inside the water path and the very first thing that came to the mind is to calculate the pressure from the opening of the beaker ! ( why would i neglect the rising water column that he asks to calculate )
i appreciate that the solution becomes significantly easier with the approximation , but i see that the approximation contradicted what he was asking in the first place .
( because the approximation states that there is no water column , so why you are asking for it ? )

 

[kuruman]

You are putting words in my post that simply aren't there. I did not say that the approximation is obvious;

There are indications pointing to the use of the approximation.

Secondly, the approximation does not state that there is no column;

This justifies the assumption that the pressure should be assumed constant along the length of the water column inside.

I am not belittling your work; you just did more work that could have been avoided had you made a drawing to scale. The drawing shown below is to scale except for the height of the beaker (width of the line) which is 4 times fatter than it ought to be. My screen does not have enough pixels to draw it to scale. The height of the water column inside is even thinner than that.