Derive the equation Vi/Vtot = robject /rfluid

[Joe Ramsey]

Realy Confused?

I’m really confused. I have 2 questions and I was wondering if anybody can help me.

(1) First question: to Derive the equation Vi/Vtot = robject /rfluid .

The only other information given is a picture of an oval vertical shaped object that by looking at the picture is approximately 90% submerged in a fluid.

A: I know this problem is suppose to be worked out algebraically because no dimensions where given. I really having problem’s solving it. I know that Vi is the volume of fluid displaced by the object. I also have figured that to get the volume of the object is (4/3)(pi)abc, where a, b,c are the distances from the center to the edge on each axis. Also, I know that the density of the fluid is pgy. However I have no idea how to go about deriving this equation?



(2) Second question: What is the maximum thickness a steel sphere of radius R can have and still float on the surface of water?

Find the mass of such as sphere for
R = 2 m and R = 4 m.

This is what I have so far…….
Vol= 4/3 pi r^2

Vol= 4/3 pi 2m^2 -4/3 pi r(hollow center)^2= volume of the steel part

The density of steel: 7,800kg/m^3
Density of water: 1,000 kg/m^3
Density of air: 1.29 kg/m^3

mass of steel= 7800kg/m^3 * (4/3 pi 2m^2 -4/3 pi r(hollow center)^2)


density = mass/volume
100kg/m^3= mass of the steel/ volume

I am stumped here….I do not know if I have even gone about this problem the correct way. Thanks for your help.

 

[HallsofIvy]

(1) First question: to Derive the equation Vi/Vtot = robject /rfluid .

The only other information given is a picture of an oval vertical shaped object that by looking at the picture is approximately 90% submerged in a fluid.

You are not even told what Vtot, robject, rfluid mean?
I don't see how you can do it without that information! Since you say that you know that Vi is the volume of the object (how did you figure that out without any other information?) I might guess that Vtot is the total volume of object and liquid but I have no idea what "rfluid" and "robject" mean.

 

[AD]

The first question is a problem with Archimedes' principle. The volume of the submerged portion of an object floating in a fluid as a proportion of its total volume is equal to its density divided by the density of the fluid. (The letter 'R' is the Roman equivalent of the Greek letter rho, which is frequently used to denote density.)

How do you derive it? Well, from Archimedes' principle, you know that the bouyancy force acting on an object is equal to the weight of displaced fluid.

F_b = ρ_fluid * V_i * g

If the object is floating, this is also equal to the object's weight.

ρ_object * V_tot * g = ρ_fluid * V_i * g

The constant g cancels, so the mass of displaced fluid is equal to the mass of the object.

ρ_object * V_tot = ρ_fluid * V_i

Dividing both sides by V_tot and ρ_fluid :

V_i/V_tot = ρ_object/ρ_fluid