Kaplan Full length 5 Q. 3

[Axis8Mundi]

Okay, so the following question is just not registering with me.

3. A second object besides the balloon is suspended motionless when released at depth d1. What can be concluded?

A. Its mass is equal to that of the balloon.
B. Its volume is equal to that of the balloon.
C. Its density is equal to that of the balloon.
D. The buoyant force on it is equal to the buoyant
force on the balloon.

Answer C is given as correct. Kaplan's explanation seems to be incorrect though, as it states that the bouyant force on the Second object is given as ρoVog, where where mo is the object’s mass and ρo is the object’s density. Is it not true that the density in the buoyancy force equation is that of the fluid displaced and not the object itself. How then, can this be correct?

The kaplan explanation is as follows:


The buoyant force on the balloon is given by F = ρwVbg, where ρw is the density of water, Vb is the volume of the object, and g is the acceleration due to gravity. Therefore, the buoyant force on the second object can be expressed as F = ρwVog, where Vo is the volume of the object. Moreover, the weight of the object is given by W = mog = ρoVog, where mo is the object’s mass and ρo is the object’s density. The object is motionless when the buoyant force exactly equals the weight; mathematically expressed, ρwVog = ρoVog. By canceling Vog on both sides, we obtain ρw = ρo. In other words, to be in equilib- rium, the object must have the same density as water. Since the balloon and the second object are both in equilibrium at depth d1, their densities must both be equal to ρw. Their densities are thus equal to each other, and choice C is correct.

We can only determine the density, which is an object’s mass to volume ratio, from the information given. We cannot infer anything about the mass or volume separately, so choices A and B are wrong.

The buoyant force is dependent on an object’s volume and no other characteristic of the object. Since the relative volumes of the balloon and the second object are unknown, the rela- tive buoyant forces on the balloon and the second object cannot be determined, and choice D is incorrect.

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[ashtonjam]

The values V(liquid displaced) in buoyant force and V(object) in the gravitational force can be the same value, I'm pretty sure. After all, if the object is completely submerged, it will displace a volume of water equal to its own volume. Probably a poor choice of words on Kaplan's part, but in this case no harm is done.

 

[Yorick]

It's correct because the object is suspended motionless, so the only necessary condition for that is the buoyant force equals the weight of the object, so Fb= Wo or pwVog = poVog. Draw a free body diagram, the forces acting on the object and balloon are the respective weights down and the buoyant force up.

If the object was floating, this relation would be incorrect and we could only say Fb = pwVog because the buoyancy force is now greater than the weight of the object. What you're saying is true, but because the sum of the forces on the object is zero, the density of the object, water, and balloon must be equal by the given equations.