Finding volume of a submerged object

[quaticle]

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 = ρ_fluidgV_object
∑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 - m_ballg = 0.
Substituted in the buoyancy equation:
∑F = T + ρ_watergV_ball - m_ballg = 0.
Then rearranging and substituting in ρV = m for the ball:
T + ρ_watergV_ball = ρ_ballVg.
Finally rearranging for the volume of the ball:
V_ball = T / ((ρ_ball - ρ_water)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]

Hi quaticle, Welcome to Physics Forums.

What parameters are known about the ball?

 

[quaticle]

What parameters are known about the ball?

I am only given the density of the ball, which is iron (7.87e3 kg/m³).

 

[SteamKing]

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 = ρ_fluidgV_object
∑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 - m_ballg = 0.
Substituted in the buoyancy equation:
∑F = T + ρ_watergV_ball - m_ballg = 0.
Then rearranging and substituting in ρV = m for the ball:
T + ρ_watergV_ball = ρ_ballVg.
Finally rearranging for the volume of the ball:
V_ball = T / ((ρ_ball - ρ_water)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?

 

[quaticle]

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 T_air - T_submerged , correct? And knowing B = m_fluid*V_displaced*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 T_air = ρ_ironV ?

 

[quaticle]

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 T_air - T_submerged , correct? And knowing B = m_fluid*V_displaced*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 T_air = ρ_ironV ?

. When I rearrange terms to solve for the volume though I end up with the same equality I had in the original post.

 

[SteamKing]

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 T_air - T_submerged , correct? And knowing B = m_fluid*V_displaced*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 T_air = ρ_ironV ?

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?

 

[quaticle]

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]

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?

The buoyant force is simply the weight of the water displaced by the iron ball. The tension in the string is the difference between the weight of the iron ball in air and its weight in water.