Two bulbs connected by a tube with two mercury threads

[coconut62]

Homework Statement

Please refer to the image attached.

Homework Equations

P=hρg

The Attempt at a Solution

My workings:

16000=h1ρg
8000= h2ρg

Since ρ and g are constant, therefore h1:h2 must be 2:1.

But the answer is 18:12. Why am I wrong?

 

[nasu]

There is pressure in the gas between the two mercury columns.
The equilibrium equations should include this too.

 

[coconut62]

I don't understand. If

16000= h1pg + P and
8000=h2pg + P

wouldn't it cancel out?

 

[nasu]

Not when you take the ratio of the two heights. The ratio h1/h2 is 2:1 only if p=0.

You can calculate the difference though, h2-h1, and then compare with the answers.

 

[haruspex]

Not when you take the ratio of the two heights. The ratio h1/h2 is 2:1 only if p=0.

... And only one of the offered answers is consistent with p > 0. That is all you are asked to do. There is not enough information to deduce the exact ratio.

 

[nasu]

However there is enough information to deduce the difference, h1-h2. And only one answer is compatible with that difference.

 

[coconut62]

Okay, I try and see:

16000= h1pg + P
8000=h2pg + P

(is P the same for both sides?)

8000=(h1-h2)pg

h1-h2 = 0.060m= 6.0cm

Nasu, in your post in #4, do you actually mean that,
if P=0, then I can just divide one equation by another.
and if P=/=0, then I can only solve the simultaneous equations by subtraction?

So it's just a matter of solving the two equations. I kept dividing one equation by another that's why I kept getting 2:1. Lol

 

[haruspex]

and if P=/=0, then I can only solve the simultaneous equations by subtraction?

The ratio is h1 : h2 = 16000-p : 8000-p. p > 0. You cannot determine the ratio exactly, but only one of the offered ratios is consistent with these facts. That is why the question is worded this way. It does not ask you which of the offered answers is the ratio, it asks which could be the ratio.

 

[nasu]

Actually it does not say or ask anything about a ratio.
They give you the density of the mercury so you can calculate the actual value of the difference. (6 cm).

But even so, it seems that none of the pairs will work.
Taking the last one, h1=18cm and h2=12cm, for example.
In order to have the 18 cm column in equilibrium, and considering that 16000 Pa is about 12 cm Hg, the pressure p should be negative. Unless I made an error of calculation.

For the first pair, in order to have the 4 cm column in equilibrium with the 12 cm from the bulb, p should be 8 cm Hg.
But in order to have the 2 cm in equilibrium with the 6 cm from the other bulb, p should be 4 cm Hg.

Something is fishy.

 

[haruspex]

Actually it does not say or ask anything about a ratio. All discussion about ratio was just a wrong start. There is not point to insist along that path.

They give you the density of the mercury so you can calculate the actual value of the difference. (6 cm).

You are right, you can calculate the difference and that does answer the question. But it is still the case that there is not enough information to determine the two heights - it is merely a matter of saying which pair of heights is feasible, and that can be resolved by either method.

 

[coconut62]

News:

My teacher said she gave the wrong answer. The answer should be 4:2.

What do you guys think? Lol