Hydrostatics - Benzene poured into uniform U shaped tube with water

[ichivictus]

Homework Statement

A uniform glass tube is bent into a U shape. Water is poured into the tube till it stands 10cm in each tube.

Benzene (sp gr = 0.879) is then slowly added to the tube on the left side until the water rises 4cm on the right.

What length is the column of benzene when the situation is reached? (Water and benzene don't mix).

Homework Equations

(pgh)₁ = (pgh)₂

The Attempt at a Solution

I know that sp gr is the ratio between the density of benzene and water (p1/p2). Using the relevant equation I have to solve for h1 (1 being benzene).

h1 = h2/0.879

The book gives the answer and says h2 is 8cm. I am having trouble finding out how it equals 8cm.

You'd think that it would be 14cm since it was initially at 10cm and the right side rises 4cm. This problem is making zero sense.

Any info would be helpful. I'm just starting on fluids in this class. Thanks!

 

[Tanya Sharma]

What is the length of water column measured from base on the left and on the right side ?

 

[ichivictus]

You mean the length from side to side? It doesn't give any info. It has a small picture and details how benzene on the left is longer than water on the right, which makes sense. The length from side to side is just the bottom and is full of water before and after benzene is in the tube.

I mean it makes sense that you will need more benzene to push the water around due to its density, but coming up with 8cm, I guess there is a formula or something I haven't learned yet...

 

[lurflurf]

If the water level on the left goes up 4cm, what can you say about the water level on the right? What can you say about the sum of the left and right water levels? How is the benzene level related to the two water levels?

 

[ichivictus]

If water on left rises 4cm then the water level on the right lowers 4cm right? The sum of both sides will remain 20cm (until benzene is added).

Benzene, when added on the left side, adds weight to the water acting like a piston of a sort pushing on it and forces it onto the other side. Of course it will take more benzene than water to push it around so you will see more benzene on the left column than water on the right. (in other words: hb>hw)

 

[Tanya Sharma]

You mean the length from side to side? It doesn't give any info.

I mean the length of water column in the left side and in the right side .Initially the lengths of water column are 10 cm each on both the sides.Now if the water level rises by 4 cm in the right side then it goes down by 4 cm in the left side .So now what are the resulting lengths of water column in left and right side ?

 

[ichivictus]

left side: 10-4= 6cm
right side: 10+4 = 14cm

I'm probably missing something very obvious...

I mean I know the difference between the right side and left side is 14-6= 8cm, but that just means there is 8cm more water on the right side than left side. Is that all I have to use for the height of water? Oh hey it's beginning to make sense now I think.

 

[Tanya Sharma]

left side: 10-4= 6cm
right side: 10+4 = 14cm

Good...

What is the pressure at the interface between benzene and water on the left side ?
What is the pressure at a distance 6 cm from the base (bottommost point) on the right side ?

Just write your answers symbolically .Do not put any numerical values.

 

[ichivictus]

Pressure? Pressure is force/area or energy/volume. Energy in this case is just potential since it's in equilibrium.

mgh/v = pgh (p being density)

Ah I see now. The pressure on the left at the interface will equal the pressure on the right at 6cm from the base.

Which then I can say is p1gh1 = p2gh2

I know p1/p2 is .879, h2 = 8cm, and the g's will cancel each other out.

h1 = h2/0.879 like I wrote in first message. 8cm/0.879 = 9.1cm

Thanks for helping me understand this :)