# Archimede's Principle problem

## Homework Statement

A hollow cubical box is 0.30 meters on an edge. This box is floating in a lake with 1/3 of its height beneath the surface. The walls of the box have a negligible thickness. Water is poured into the box. What is the depth of the water in the box at the instant the box begins to sink?

The answer from the book that i got the problem from is 0.2 meters.

## Homework Equations

buoyancy force = (density of the water)(volume of displaced water)(gravity)

## The Attempt at a Solution

First I did the buoyancy force, which i think is: (1000 kg/m^3)(1/3)(0.3)^3 m^3)(9.8 m/s^2) = 88.2 N.

But what I don't get is, if the walls of the box have negligible thickness, wouldnt that mean the mass of the box is basically zero? So how can the box even have 1/3 of its volume in the water when it has no weight?

Also if the buoyancy force is 88.2 N just holding the box in equilibirium, wouldn't that mean the box weighs 88.2 N...

Any help at all would be really appreciated..!

Last edited:

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There do seem to be a few logical problems with this question (it seems like they are treating the box to have zero mass), but I think they just want you to calculate the height of water necessary to overcome an 88.2N buoyant force. Doing so gives:

(88.2N)/(9.8m/s^2)(1000kg/m^3) = V = .009 m^3

h = (.009m^3)^1/3 = .208m

Dick
Homework Helper
You don't have to calculate anything. If the box is massless, it can't sink if it's not full of water, can it? Even if it is full of water, will it sink if it's massless? I have some problems with this question as well. But at least you can avoid the roundoff error by answering that you simply need to fill the box with water before it can even begin to 'sink'.

You don't have to calculate anything. If the box is massless, it can't sink if it's not full of water, can it? Even if it is full of water, will it sink if it's massless? I have some problems with this question as well. But at least you can avoid the roundoff error by answering that you simply need to fill the box with water before it can even begin to 'sink'.
my initial reaction was the same as yours, however i do believe that the intent of the question is to consider that the box does have mass.

As the OP stated, this box mass would be 88.2N (without checking your maths)

Dick
Homework Helper
Rrrright. That was stupid of me. Think I should take a nap. Thanks! The box doesn't have zero mass, it just doesn't weigh much.

thanks for the help.

There do seem to be a few logical problems with this question (it seems like they are treating the box to have zero mass), but I think they just want you to calculate the height of water necessary to overcome an 88.2N buoyant force. Doing so gives:

(88.2N)/(9.8m/s^2)(1000kg/m^3) = V = .009 m^3

h = (.009m^3)^1/3 = .208m

doesnt that imply the volume of water is cubical?

Dick
Homework Helper
thanks for the help.

There do seem to be a few logical problems with this question (it seems like they are treating the box to have zero mass), but I think they just want you to calculate the height of water necessary to overcome an 88.2N buoyant force. Doing so gives:

(88.2N)/(9.8m/s^2)(1000kg/m^3) = V = .009 m^3

h = (.009m^3)^1/3 = .208m

doesnt that imply the volume of water is cubical?
Yes, it does. And no, it's not cubical. You know the box is floating with 0.1m of it's height submerged when empty. If you pour 1cm of water in, how much does the box sink? Try and answer that without calculating anything. Just think about it. Then answer the original question.