Fluids

[Bumbl3b33]

---

Members don't see this ad.

This is from TBR section 7, "fluids and solids", passage 4:

The question asks

"If salt is added to water, the water density increases. The force needed to submerge a floating, completely incompressible object will:
a. increase as the density increases and increase slightly as the submersion depth increases
b. increase as the density increases and decrease slightly as the submersion depth increases"

My question is...why is it "A"? Shouldn't the force needed to submerge decrease with depth? Less water pushing up, and more downward pressure from fluids above and atmosphere.

Thank you in advance!

Bumblebee.

Edit:

Another question: I see the math for it in the book, but it doesn't quite make sense to me... why do things of same densities have the same acceleration in fluid if the sizes are different? The Fb equation has the variable "volume of displaced fluid" in it...if something is bigger, won't it increase the volume of fluid displaced, thereby increasing buoyant force/acceleration for large volume objects?

 

[ilovemcat]

This is from TBR section 7, "fluids and solids", passage 4:

The question asks

"If salt is added to water, the water density increases. The force needed to submerge a floating, completely incompressible object will:
a. increase as the density increases and increase slightly as the submersion depth increases
b. increase as the density increases and decrease slightly as the submersion depth increases"

My question is...why is it "A"? Shouldn't the force needed to submerge decrease with depth? Less water pushing up, and more downward pressure from fluids above and atmosphere.

Thank you in advance!

Bumblebee.

Edit:

Another question: I see the math for it in the book, but it doesn't quite make sense to me... why do things of same densities have the same acceleration in fluid if the sizes are different? The Fb equation has the variable "volume of displaced fluid" in it...if something is bigger, won't it increase the volume of fluid displaced, thereby increasing buoyant force/acceleration for large volume objects?

They mention the object is floating. Automatically, we know the weight of the object = bouyant force.

With this in mind, we can deduce the fact that:
% Submerged = density object / density of fluid

Increasing the density of the fluid, decreases the % submerged (ie. less volume submerged under water, more volume above surface)

So, for Bouyant Force we have: pVg

where p = density of fluid
where v = volume submerged
where g = gravity

We're told (p) density of fluid increases and we can figure out that (v) volume submerged decreases. The changes (increase density / decrease volume) result in an equal Bouyant Force to the original conditions (with no salt added).

Now imagine we have two seperate containers - one with water and one with salt water. If you take the extreme scenario where the object is completely submerged and compare that to each container, you have:

Fbouyant (water) = pwater (total volume) g [smaller density = smaller bouyant force]
Fbouyant (salt water) = psalt (total volume) g [larger density = higher bouyant force]

As you can see, as the object becomes more and more submerged in each container - the Bouyant Force (in Salt) increases more than the Bouyant Force (in Water alone).

As a side note, the reason why the use the word "slightly" increases is because the density of salt water is only barely larger than water alone (something like 1.027 kg/m^3).

 

[ilovemcat]

Another question: I see the math for it in the book, but it doesn't quite make sense to me... why do things of same densities have the same acceleration in fluid if the sizes are different? The Fb equation has the variable "volume of displaced fluid" in it...if something is bigger, won't it increase the volume of fluid displaced, thereby increasing buoyant force/acceleration for large volume objects?

I'm not really sure I understand your question. The only way an object could accelerate is if the weight of the object is greater than the Bouyant Force OR if the Bouyant Force is greater than the weight of the object. In the above scenario where I considered the object to be completely submerged, I assumed that some outside force (like force from your hand), was providing an additional force to submerge the object further than it would naturally (float). In this scenario I considered that the FB = Downward Force + Weight of Object (and acceleration = 0).

The only way two objects with the same density but different volumes could accelerate at the same rate is if the net force of both objects is the same. This would mean that the bouyant force would be unequal in both cases.

So for the object's weight we'd have:

Object 1 (smaller volume) = pVg (here "p" is density of the object and "v" is volume of object)
Object 2 (larger volume) = pVg (same as above; since we have a larger "v" the weight of this object is greater).

In order for the net force to be equal for both objects:

Fnet = Object 1 - Bouyant Force
Fnet = Object 2 - Greater Bouyant Force

This makes sense because Bouyant Force = pVg (where "p" is density of fluid and "v" is volume submerged) - a greater volume results in a greater Bouyant Force.

So let's assume both objects are heavier than the liquid and are accelerating downwards.
Comparing the ratio of weight object / weight of displaced fluid we see:

pVg(object) / pVg(fluid) = p(object)/p(fluid) because volume & gravity = same and cancel out

If you have the same density for both objects and both objects are in the same fluid, then ratio for both objects will be equal (regardless of the volume size, so long as the density of both objects are greater than the density of water. Therefore, if the ratios are equal that means they have equal net forces ==> equal acceleration

This is actually a useful fact to know:
density of object > density of fluid = sink (a not 0)
density of object = density of fluid = perfectly floating just BELOW the water surface (a=0)
density of object < density of fluid = the object will float (a=0)

 

[Bumbl3b33]

They mention the object is floating. Automatically, we know the weight of the object = bouyant force.

With this in mind, we can deduce the fact that:
% Submerged = density object / density of fluid

Increasing the density of the fluid, decreases the % submerged (ie. less volume submerged under water, more volume above surface)

So, for Bouyant Force we have: pVg

where p = density of fluid
where v = volume submerged
where g = gravity

We're told (p) density of fluid increases and we can figure out that (v) volume submerged decreases. The changes (increase density / decrease volume) result in an equal Bouyant Force to the original conditions (with no salt added).

Now imagine we have two seperate containers - one with water and one with salt water. If you take the extreme scenario where the object is completely submerged and compare that to each container, you have:

Fbouyant (water) = pwater (total volume) g [smaller density = smaller bouyant force]
Fbouyant (salt water) = psalt (total volume) g [larger density = higher bouyant force]

As you can see, as the object becomes more and more submerged in each container - the Bouyant Force (in Salt) increases more than the Bouyant Force (in Water alone).

As a side note, the reason why the use the word "slightly" increases is because the density of salt water is only barely larger than water alone (something like 1.027 kg/m^3).

Hmm....maybe I am misunderstanding what you are saying, but this doesn't answer why the force needed wouldn't decrease with depth due to the combination of the weight of the water above it and the atmospheric pressure pushing it down. I believe there was a passage about that in the chemistry section, and they concurred that the lower something went, the less force it would take to push the object down.

 

[ilovemcat]

Hmm....maybe I am misunderstanding what you are saying, but this doesn't answer why the force needed wouldn't decrease with depth due to the combination of the weight of the water above it and the atmospheric pressure pushing it down. I believe there was a passage about that in the chemistry section, and they concurred that the lower something went, the less force it would take to push the object down.

Technically, part of what you said is true. As an object descends deeper into the fluid, the volume of the fluid and the object would decrease (because of the increased pressure at lower levels). This in turn would increase the density of the fluid and the object.

But if we look at the equation for Buoyant Force: p(fluid)V(object)g
You realize there's in an issue - namely, which changes at a faster rate: the increasing density of the fluid or the decreasing volume of the object. You'd have to compare the Bulk Moduli of both in order to establish if the Buoyant Force would increase or decrease with depth.

To simplify things, for the MCAT we assume that the density of the fluid is incompressible and that the Buoyant Force remains the same at all depths (when totally submerged).

 

[Bumbl3b33]

Technically, part of what you said is true. As an object descends deeper into the fluid, the volume of the fluid and the object would decrease (because of the increased pressure at lower levels). This in turn would increase the density of the fluid and the object.

But if we look at the equation for Buoyant Force: p(fluid)V(object)g
You realize there's in an issue - namely, which changes at a faster rate: the increasing density of the fluid or the decreasing volume of the object. You'd have to compare the Bulk Moduli of both in order to establish if the Buoyant Force would increase or decrease with depth.

To simplify things, for the MCAT we assume that the density of the fluid is incompressible and that the Buoyant Force remains the same at all depths (when totally submerged).

Ah, okay.

From the bottom of my heart, thank you so much! You were so very helpful 🙂