# Finding volume of a submerged object

## Homework Statement

There is a block of wood floating on the surface of a body of water, with a ball attached to the bottom of the block by a string. I am asked to find the volume of the ball given the tension in the string. We also know the volume of the wood block from an earlier problem if applicable (I don't think it is needed for this problem but I may be wrong).

## Homework Equations

B = mg = ρfluidgVobject
∑F = ma
ρV = m

## The Attempt at a Solution

I started out with a fbd and summed the forces on the ball:
∑F = T + B - mballg = 0.
Substituted in the buoyancy equation:
∑F = T + ρwatergVball - mballg = 0.
Then rearranging and substituting in ρV = m for the ball:
T + ρwatergVball = ρballVg.
Finally rearranging for the volume of the ball:
Vball = T / ((ρball - ρwater)g).
Alas, this did not result in the correct answer and I am not sure where I went wrong. I think it is somewhere with my forces, buoyancy has always confused me and its exact function as a force. Any help appreciated!

gneill
Mentor
Hi quaticle, Welcome to Physics Forums.

What parameters are known about the ball?

What parameters are known about the ball?
I am only given the density of the ball, which is iron (7.87e3 kg/m3).

SteamKing
Staff Emeritus
Homework Helper

## Homework Statement

There is a block of wood floating on the surface of a body of water, with a ball attached to the bottom of the block by a string. I am asked to find the volume of the ball given the tension in the string. We also know the volume of the wood block from an earlier problem if applicable (I don't think it is needed for this problem but I may be wrong).

## Homework Equations

B = mg = ρfluidgVobject
∑F = ma
ρV = m

## The Attempt at a Solution

I started out with a fbd and summed the forces on the ball:
∑F = T + B - mballg = 0.
Substituted in the buoyancy equation:
∑F = T + ρwatergVball - mballg = 0.
Then rearranging and substituting in ρV = m for the ball:
T + ρwatergVball = ρballVg.
Finally rearranging for the volume of the ball:
Vball = T / ((ρball - ρwater)g).
Alas, this did not result in the correct answer and I am not sure where I went wrong. I think it is somewhere with my forces, buoyancy has always confused me and its exact function as a force. Any help appreciated!

If the ball were suspended by a string in air, what would the tension be on the string?

What would the tension be on the string when the ball is submerged?

It's helpful to draw a free body diagram in these situations.

You say you did not get the correct answer. Were you given the correct answer?

If the ball were suspended by a string in air, what would the tension be on the string?

What would the tension be on the string when the ball is submerged?

It's helpful to draw a free body diagram in these situations.

You say you did not get the correct answer. Were you given the correct answer?

So the tension in the string if in air would simply be the mass of the ball * gravity. The tension when submerged is given to me (0.8N). I am given the correct answer for the volume of the ball as 12cm3. Using my final equality my answer is something on the order of 10-5...

Going off what you are saying I am contriving this:
The buoyant force would be the Tair - Tsubmerged , correct? And knowing B = mfluid*Vdisplaced*g I can then divide by ro*g to solve for the volume of the ball? But since I am only given the ball's density ( that of iron) I can say that Tair = ρironV ?

So the tension in the string if in air would simply be the mass of the ball * gravity. The tension when submerged is given to me (0.8N). I am given the correct answer for the volume of the ball as 12cm3. Using my final equality my answer is something on the order of 10-5...

Going off what you are saying I am contriving this:
The buoyant force would be the Tair - Tsubmerged , correct? And knowing B = mfluid*Vdisplaced*g I can then divide by ro*g to solve for the volume of the ball? But since I am only given the ball's density ( that of iron) I can say that Tair = ρironV ?
. When I rearrange terms to solve for the volume though I end up with the same equality I had in the original post.

SteamKing
Staff Emeritus
Homework Helper
So the tension in the string if in air would simply be the mass of the ball * gravity. The tension when submerged is given to me (0.8N). I am given the correct answer for the volume of the ball as 12cm3. Using my final equality my answer is something on the order of 10-5...

Going off what you are saying I am contriving this:
The buoyant force would be the Tair - Tsubmerged , correct? And knowing B = mfluid*Vdisplaced*g I can then divide by ro*g to solve for the volume of the ball? But since I am only given the ball's density ( that of iron) I can say that Tair = ρironV ?

Remember, tension is a force, while ρ is the mass density of the iron ball.

Typically, mass density is given in kg/m3. How many cm3 are in 1 m3?

Oh I completely forgot the g term, and of course the conversion! Thank you for the help, once I take those two into consideration I do obtain the desired answer. Just to reiterate and reinforce my understanding... here the buoyant force is the difference in the tensions? i.e the amount force alleviated from the string when submerged?

SteamKing
Staff Emeritus