How Does Bath Oil Depth Affect Soap Buoyancy?

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The discussion focuses on the relationship between bath oil depth and soap buoyancy, specifically how the addition of oil affects the submerged depth of a soap bar. A 2.09 cm soap bar initially floats with 1.50 cm underwater, and when oil of specific gravity 0.639 is added, the soap rises, altering the submerged depth. Participants emphasize the importance of using Archimedes' principle to equate the buoyant forces from both water and oil to the weight of the soap. The challenge lies in mathematically expressing the changes in submerged depth and oil volume in terms of the variables involved. Understanding the relationships and dimensions of the quantities is crucial for solving the problem accurately.
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A 2.09 cm thick bar of soap is floating on a water surface so that 1.50 cm of the bar is underwater. Bath oil of specific gravity 0.639 is poured into the water and floats on top of the water. What is the depth of the oil layer when the top of the soap is just level with the upper surface of the oil?

Given the thicknesses and distances of the materials i have found that the distance between the waters surface and that of the highest point of the bar of soap is :

2.09-1.50=5.9e-3m

since the bath oil floats on the water i figured that the depth of the bath oil will be the distance of the part of soap that is petruding out of the water. However this is not the case.

I also know that the depth or volume is not the petruding distance multiled of divided by the density of the liquid.

Any help??
 
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does anyone have any suggestions for this one??
 
DDS said:
does anyone have any suggestions for this one??

Both liquids will provide a buoyant force. The net buoyant force will equal the weight of the soap, and the total volume displaced (two different liquids) will equal the volume of the soap. You can't just use the exposed height of the soap before the oil is added because adding the oil will raise the level of the soap relative to the water surface. You have to look at the system as a whole.
 
I understand your analysis its just I am still confused as to how to apply this in a matematical sense. Can you help me with that
 
DDS said:
I understand your analysis its just I am still confused as to how to apply this in a matematical sense. Can you help me with that

Think of a bar of soap as having perfectly uniform thickness of h = 2.09 cm. It does not matter what volume the soap has. Assume the soap has volume V = A*h, where A is a cross-sectional area. The volume of the water displaced intially is 1.5cm*A. Use the density of water to find the weight of water displaced. It will be in terms of A. That is also the weight of the soap.

Let x = thickness of oil layer, and y = the depth the soap sinks into the water. You can easily write y in terms of x. The total weight of liquid displaced will be the sum of two terms: the weight of water displaced, which depends on the water density and its volume plus the weight of oil, which depends on the density of oil and its volume. All volumes and weights will be proportional to A, so A will divide out in the end. The only variable in the equation will be x, and you can solve for that.

That is a verbal description of all the relationships you need. See if you can turn that into the equation you need to solve.
 
what do i assume my density of water is?
 
jsut 1000 kg/m^3
 
this is what i have come up with so far:

H=0.0209 m
Volume of H20 displaced=0.015 m
Exposed portion= 5.9e-3 m

Denstity=D=0.639

Weight of H20 Displaced:

w=(1000)(0.015)(9.81)
w=147.15 i havea feeling thas wrong because its not interms of A as you said

what i think it could be but doesn't make sense to me is that if the volume of soap is V=A*H
then u sub that V into the equation for weight to get it in terms of A. That doesn't make sense to me becuase
the relation V=A*H describes the volume of soap so why would the volume of soap go into the position of the volume of fluid in Archimedes Principle.

Nevertheless could it be:

w=(1000)(A*0.0209)(9.81)
w=205.029 A
 
am i on the right track
 
  • #10
should it be interms of A or can i use my first answer
 
  • #11
can anyone let me know if I am on the right track...anyone??
 
  • #12
DDS said:
this is what i have come up with so far:

H=0.0209 m
Volume of H20 displaced=0.015 m
Exposed portion= 5.9e-3 m

Denstity=D=0.639

Weight of H20 Displaced:

w=(1000)(0.015)(9.81)
w=147.15 i havea feeling thas wrong because its not interms of A as you said

what i think it could be but doesn't make sense to me is that if the volume of soap is V=A*H
then u sub that V into the equation for weight to get it in terms of A. That doesn't make sense to me becuase
the relation V=A*H describes the volume of soap so why would the volume of soap go into the position of the volume of fluid in Archimedes Principle.

Nevertheless could it be:

w=(1000)(A*0.0209)(9.81)
w=205.029 A

I think it will help you to keep track of dimensions of quantities. How can volume have units of meters? Volume is three dimensional, and must have dimensions of length cubed.

If the volume of the soap is V=AH, where H is .0209 m, and the soap sinks into the water a depth of .015m, then the volume of the water displaced is V_w = A(.015m). The density of water is 1000 kg/m^3 as you said, so you can use volume and density to find the weight of displaced water = weight of soap. When the oil is added, the volume of oil displaced is A times the depth of oil, and the volume of water displaced is A times the depth the soap sinks into the water. Every quantity that depends on volume in this problem will have an A in it, and the As will all divide out in the end.

In the final configuaration, the volume of fluid is the voilume of the soap because the soap floats with its upper surface aligned with the top surface of fluid. The soap is displacing its entire volume.
 
  • #13
so where did i go wrong originally and which equation do i use, the one with the A or the first one without the A
 
  • #14
DDS said:
so where did i go wrong originally and which equation do i use, the one with the A or the first one without the A

I have told you where you went wrong, and I have told you a complete outline of how to do the problem. Pay attention to the dimensions of the quantities involved and you will see that you need the area for intermediate steps to follow the logic that follows from Archimedes Principle. You need to write the equations yourself based on the information already given. It's not enough to plug numbers into equations that someone else writes for you.
 
  • #15
This is where i get to and then i get confused by your explanation:

Volume of soap:
Vs=AH
Vs=0.0209A

Volume of H20 Displaced:
Vw=0.015A

Weight of H20:
W=Dw*Vw*g
w=1000*0.015A*9.81
=147.15A

Weight of soap:
147.15A
 
  • #16
DDS said:
This is where i get to and then i get confused by your explanation:

Volume of soap:
Vs=AH
Vs=0.0209A

Volume of H20 Displaced:
Vw=0.015A

Weight of H20:
W=Dw*Vw*g
w=1000*0.015A*9.81
=147.15A

Weight of soap:
147.15A

Now add the oil until the top of the oil is even with the top of the soap. When that is done the thickness of the oil will be some value not yet introduced (except in my earlier post) Let's call it x. Then how deep does the soap sink into the water expressed in terms of x?

Now, in terms of x calculate the volume of oil displaced, and the new volume of water displaced (it is going to change when the oil is added). Use these two volumes to calculate the weight of oil displaced and the weight of water displaced, using the fact that their sum must be the weight of the floating soap. All of this will still involve A, but it will divide out in the end.
 
  • #17
Then how deep does the soap sink into the water expressed in terms of x?

thats what confuses me, i know that the soap will be added to the exposed area which is 5.9e-3 but that's all i know . How do i calculate how deep the soap sinks. Is it just equalt to the exposed area?

and when i calculate the volume of oil displaced what weight do i use, the weight of the soap?
 
  • #18
DDS said:
Then how deep does the soap sink into the water expressed in terms of x?

thats what confuses me, i know that the soap will be added to the exposed area which is 5.9e-3 but that's all i know . How do i calculate how deep the soap sinks. Is it just equalt to the exposed area?

and when i calculate the volume of oil displaced what weight do i use, the weight of the soap?

Draw yourself a picture of the soap. Its thickness is 2.09cm: given. Extend the top line in both directions. That line is the top of the oil layer. Draw a line parallel to the top and bottom of the soap, through the soap, somewhere near the middle of the soap. That line represents the bottom of the oil and the top of the water. Call the distance from the top of the soap to the oil-water line x. How far is it from this line to the bottom of the soap?
 
  • #19
5.9e-3m

because if your given the thickness to be 2.09 and the submerged distance is 1.50 then the oil occupies the remaining space which is 0.59 cm
 

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  • #20
DDS said:
5.9e-3m

because if your given the thickness to be 2.09 and the submerged distance is 1.50 then the oil occupies the remaining space which is 0.59 cm

No, it's not. I told you that when you add the oil the distance the soap sinks into the water is going to change. The buoyant force of the oil is going to raise the soap in relation to the water line. Call the distance from the top of the oil to the water line x. What is the distance from the the water line to the bottom of the soap in terms of x?
 
  • #21
is it 1.50 cm
 
  • #22
DDS said:
is it 1.50 cm

I know you are having a hard day, but I'm afraid you are not really paying attention. Read what I just asked you again and answer the question in the form I stated, in terms of x.
 
  • #23
this is how i understand it , the free space is 5.9e-3 m


As the bath oil is being added the soap will rise above the water line. I know the distance is smaller than 1.50 the only think i can think of is subtrancting 1.50-5.9e-3 getting 1.491x

my brain is fired, its just I am so determined to get my last 4 questions done but no matter how hard i try it seems like I am only getting in circles
 
  • #24
DDS said:
my brain is fired, its just I am so determined to get my last 4 questions done but no matter how hard i try it seems like I am only getting in circles

Because you are not listening, you didn't listen in the vibrator thread, too. It looks you're more interested in getting the correct answers rather than understanding the problem :frown:
 
  • #25
no no i was just doing to many problems at once i figured the vibrator out just I am stuckn on this one still
 
  • #26
As for the fluids question i have gotten this far:

I have drawn two FBD's one for the soap in water and one for the soap in water and oil:

FBD1:

Fbw=mg
147.15A=mg

FBD2:

Fbo + Fbw - mg = o
(0.639*Vo*9.81) + Fbw2 - mg =O

i know that my Fbw2 is not the same value as my first Fbw because it is under oil, I know that older dan said we should call it x and y but i am not sure on how to equate the two becuase it seems like i have too many unknows , any help?
 
  • #27
DDS said:
As for the fluids question i have gotten this far:

I have drawn two FBD's one for the soap in water and one for the soap in water and oil:

FBD1:

Fbw=mg
147.15A=mg

FBD2:

Fbo + Fbw - mg = o
(0.639*Vo*9.81) + Fbw2 - mg =O

i know that my Fbw2 is not the same value as my first Fbw because it is under oil, I know that older dan said we should call it x and y but i am not sure on how to equate the two becuase it seems like i have too many unknows , any help?

You don't need to include Fbw 1 and 2 (I assume you're thinking that these are the weight of the fluids?)

As Archemides principle says, the buoyant force equals to the weight of the fluid displaced. You already have the weight of the fluid from Fbo.

There are 2 fluids which the soap displaces and their weight must equal to the weight of the soap. Reread OldDan's post on setting up an equation to represent x and y...
 
  • #28
i have an i am still confused . I know that when the oil is added the bar of soap will rise( first unknown) secondly when the soap reaches equilibirum the volume of oil added will be some number ( another unknown)

these two unknows are throwing me off..

As for Fbw2 the force of boyancy of water in oil is different then just the FBw when no oil has been added.

How do i calcuate FBW2 you said u don't need it but i can't see how u can go about it without it
 
  • #29
DDS said:
i have an i am still confused . I know that when the oil is added the bar of soap will rise( first unknown) secondly when the soap reaches equilibirum the volume of oil added will be some number ( another unknown)

these two unknows are throwing me off..

As for Fbw2 the force of boyancy of water in oil is different then just the FBw when no oil has been added.

How do i calcuate FBW2 you said u don't need it but i can't see how u can go about it without it

Uhh... the force of buoyancy of water changes only because the force of buoyance of oil also contributes to keep the soap afloat. It's the same principle; the weight of the both fluids displaced equals to the weight of the soap. So set up an equation(s) to find the volume of oil displaced and water displaced.
 
  • #30
This is what i have:

Fbw=mg
dw*vw*g=ds*vs*g
density of soap=717.7

Fbo + Fbw2 = mg
(0.639*XA*9.81) + (100*YA*9.81) = d*v*g
cancel terms and you get
(0.639*X) + (1000*Y) = d

i know you relate them through thickness of the soap but how?
 
  • #31
DDS said:
This is what i have:

Fbw=mg
dw*vw*g=ds*vs*g
density of soap=717.7

Fbo + Fbw2 = mg
(0.639*XA*9.81) + (100*YA*9.81) = d*v*g
cancel terms and you get
(0.639*X) + (1000*Y) = d

i know you relate them through thickness of the soap but how?

The volume has not been divided out completely. Reread OldDan's previous posts more carefully, about drawing out the soap... X and Y are the heights of the soap subermerged in each liquid... just read the previous posts more carefully
 
  • #32
I know i made a mistake in righting it, its supposed to be:

(0.639*X) + (1000*Y) = ds * 0.0209


then form here is used my two equaions to find x and y but i seem to be getting the answer from which i originally thaught it would be:


i know x+y =0.0209 m

and i also know that

x=23.45 - 1564.9y
which i got from simplyfiying the very first equation in this post.


from there i found that

Y= 0.0149m
and X=5.91e-3

then wouldn't the depth of the oil layer be equal to my value from X or am i missing something??
 
  • #33
DDS said:
I know i made a mistake in righting it, its supposed to be:

(0.639*X) + (1000*Y) = ds * 0.0209


then form here is used my two equaions to find x and y but i seem to be getting the answer from which i originally thaught it would be:


i know x+y =0.0209 m

and i also know that

x=23.45 - 1564.9y
which i got from simplyfiying the very first equation in this post.


from there i found that

Y= 0.0149m
and X=5.91e-3

then wouldn't the depth of the oil layer be equal to my value from X or am i missing something??

the density of oil is not 0.639.
 
  • #34
of course its not i negelcted to convert specific gravity to density

i got the answer
 

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