Density floating problem, wood and lead

In summary, the conversation discusses a homework problem involving a block of wood and lead that must float in water with a certain percentage of its volume submerged. The conversation includes equations and attempts at solving the problem, with a focus on finding the mass of lead needed in two different scenarios (attached to the top and bottom of the wood). Eventually, the conversation arrives at the correct equations for both situations, with the key being the multiplication of density of water and gravity in the equation.
  • #1
1. Homework Statement
A block of wood has a mass of 3.47 kg and a density of 628 kg/m3. It is to be loaded with lead (1.13 × 104 kg/m3) so that it will float in water with 0.848 of its volume submerged. What mass of lead is needed if the lead is attached to (a) the top of the wood and (b) the bottom of the wood?

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alright so i have been working on this for a second and I am stuck on the second part...I found the solution to the book problem, it shows me how to do it but doesn't really flush out the steps. see below for the book problem solution.

Homework Equations


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The Attempt at a Solution


Sooo, with the above in mind, where did the (density of water)/(density of lead) come from and why is that extra M(lead) on the Fb side? I start with Fb=(Dw)(Vwood + Vlead)g, with Fb going to the above (Mwood+Mlead)g, g cancels...but I guess I am stuck there..

I tried going from that to solving for Mlead but it didn't really work out..I am checking my units and stuff and it looks ok...

in the example the 0.900 is equivalent to the 0.848, and i just treat it as a percent and multiply it in as shown. Are they using the density of water over the density of lead as a relative density? I don't know..I am sure it's just a dumb mistake..

Thanks for any help.
 
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  • #2
What is the volume of the wood? What is the weight of the wood? What volume of wood is submerged? (This is the volume of water that the wood displaces)

Let M be the mass of lead. In terms of M, what is the weight of the lead? In terms of M, hat is the total weight of wood plus lead?

In the case of lead on top of the wood, there is no lead submerged. So what is the buoyant force of the water acting on the wood? How does this compare with the weight of the wood plus lead? You now have an equation for calculating the mass of lead M.

In the case that the lead is fastened underneath the wood, the entire block of lead is submerged. So, what is the volume of wood plus lead submerged (in terms of M)? What is the buoyant force of the water acting on the submerged wood and lead? How does this compare with the weight of the wood plus lead. You now have an equation for calculating the mass of lead M for this case.

chet
 
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  • #3
Hey chet!

Yes, yes, I got all of that. I actually think I figured it out. I was thinking for (b) that the equation i needed to formulate was:
g(mlead +mwood)=(%)(D_water)(Volume_wood + Volume_lead)g

but looking closer It was:
g(mlead +mwood)=(%)(D_water)(g)(Volume_wood) + (D_water)(Volume_lead)(g)

They just wrote it weird in the example. I just didnt "see" how the density of water and gravity had been multiplied through. Knew it was a simple thing.

Thanks!
 

1. What is the density of wood and lead?

The density of wood varies depending on the type, but on average it is around 0.5 g/cm³. The density of lead is much higher at 11.3 g/cm³.

2. Why does wood float and lead sink?

Wood has a lower density than water, meaning it is less dense and therefore floats. Lead, on the other hand, has a higher density than water, causing it to sink.

3. How does the density of an object affect its ability to float?

An object with a lower density than the fluid it is in will float, while an object with a higher density will sink. This is due to the principle of buoyancy, where the upward force exerted by the fluid on the object is equal to the weight of the fluid displaced by the object.

4. What other factors can affect the ability of an object to float?

The shape and size of an object can also affect its ability to float. Objects with a larger surface area will float better than objects with a smaller surface area. The weight distribution and the presence of air pockets can also impact an object's buoyancy.

5. How can we use the concept of density to explain why some objects sink and others float?

The concept of density can help us understand why some objects sink and others float by comparing the density of the object to the density of the fluid it is in. Objects with a lower density than the fluid will float, and objects with a higher density will sink. This is why wood, which has a lower density than water, floats, while lead, with a higher density, sinks.

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