What is the Mass Density of a Floating Cylinder in Water?

[ch2kb0x]

Homework Statement

A 6cm tall cylinder floats in water with its axis perpendicular to the surface. The length of the cylinder above water is 2cm. What is the cylinders mass density?

Homework Equations

Fb-Fg = pf*vf*g = mo*g
=pf*vf*g = po*vo*g
mass density = p = m / V

*note: pf=density of fluid
vf= volume of fluid
mo = mass of object
vo = volume of object.
water density = 1000 kg/m^3

The Attempt at a Solution

I basically drew the Free body diagram which had the buoyuant force pointing up and force of object pointing down to due (mg). Help! An answer would be nice

 

[Doc Al]

Homework Equations

Fb-Fg = pf*vf*g = mo*g
=pf*vf*g = po*vo*g

So far, so good. (You set the net force = 0.) Now make use of the data given in the problem. What is vf in terms of vo?

Then you can just solve for po, which is the cylinder's density.

 

[Delphi51]

Fb-Fg = pf*vf*g = mo*g

Sorry to interrupt - just wondering if you would take a moment to explain what these symbols mean so I can follow the solution. Interesting problem!

I would have started with
Fb - Fg = ma = 0 (force of buoyancy up minus force of gravity down)
The total force must be zero since the cylinder is not accelerating.

Ah, Fg = m*g so perhaps your mo is the mass of the object.
Fb would be the mass of the water displaced by the sunken part of the object times g and related to the density of water and the volume of the sunken part.
But what are pf and vf and why isn't Fb-Fg equal to zero?

 

[Doc Al]

Sorry to interrupt - just wondering if you would take a moment to explain what these symbols mean so I can follow the solution. Interesting problem!

The OP is using:
vo = volume of object
vf = volume of fluid displaced
mo = mass of object
po = density of object (ρ)
pf = density of fluid

I would have started with
Fb - Fg = ma = 0 (force of buoyancy up minus force of gravity down)
The total force must be zero since the cylinder is not accelerating.

That's what he did, he just didn't write it all that carefully.

When he wrote: Fb-Fg = pf*vf*g = mo*g
I presumed (based on what followed) that he meant:
Fb-Fg = 0 → pf*vf*g = mo*g

(I should have pointed this out earlier.)

 

[Delphi51]

Thanks Doc Al; much appreciated. You have a sharp eye!
I've been in high school for 30 years and have been using D for density rather than rho.

 

[ch2kb0x]

Sorry, still not getting it. I know this should be a fairly easy question.

 

[Doc Al]

Answer the question I posed in post #2:

What is vf in terms of vo?

 

[badger2212]

I'm a student working on a similar problem but I'm not getting it either. I know the weight of the fluid displaced equals the weight of the object and I got the equation but how can we find this without the volume of the cylinder?

 

[Delphi51]

how can we find this without the volume of the cylinder?

Good point! I didn't even notice that problem because the volume canceled out in my calculation. The actual size or volume doesn't matter because you would get the same answer for a cylinder with a radius of 1 or one with a radius of 10. In fact, you could think of the bigger one as being made of a bunch of little ones, perhaps with a bottom area of 1 cm squared.

There is a lesson in here somewhere. Don't worry too much about things you don't know yet and don't put the numbers into the problem too early. Work with symbols only and watch for things to cancel out.