Fluid Mechanics - Bouyancy Question

[wesDOT]

Homework Statement

A block of wood floats on water. The density of the
wood is .634g/cm^3. Its mass is 243 g. Find the minimum mass (g) of lead we
must attach to the block so that it will sink, along with the lead. The density of lead is
11.3 g/cm^3. Caution: you must take into account the buoyant force on the lead, as well
as on the wood.

Homework Equations

B=pgV=Mg(weight of fluid displaced)

The Attempt at a Solution

I know that I must apply bouyancy principles to Newton's Second Law but I am not sure exactly how to set up my equations. How should I attack this problem?

 

[diazona]

You might consider drawing a free-body diagram first, to show the forces acting on the wood and lead. Or at least making a list of the forces involved. (You can consider the wood and lead combined to be a single free body - don't worry about the force of the lead pushing on the wood or vice-versa) The buoyant force ($$B = \rho g V$$) will be one force, but what else is there?

Once you've done that, you can add up all the forces to get the total force. Remember that in order for the wood and lead to sink, the total force should be downward rather than upward.

 

[nlightner]

To solve this problem, we need to consider the forces acting on the block of wood and the lead. The buoyant force, which is equal to the weight of the fluid displaced, acts in the opposite direction of gravity. Therefore, for the wood to float, the buoyant force must be equal to the weight of the wood.

We can use the given density and mass of the wood to calculate its volume using the formula V=M/density. This will give us a volume of 383.5 cm^3.

Next, we need to find the minimum mass of lead that must be attached to the block in order for it to sink. This means that the total weight of the block and the lead must be greater than the buoyant force acting on the block.

We can calculate the buoyant force on the block using the density of water (1 g/cm^3) and the volume of the block (383.5 cm^3). This gives us a buoyant force of 383.5 g.

Now, we can set up an equation to find the minimum mass of lead needed:

M(block + lead) = M(block) + M(lead) > B

Substituting in the values we know, we get:

243 g + M(lead) > 383.5 g

Solving for M(lead), we get a minimum mass of lead of 140.5 g. This means that we must attach at least 140.5 g of lead to the block in order for it to sink.

However, we must also take into account the buoyant force acting on the lead itself. This means that the actual minimum mass of lead needed may be slightly higher. To account for this, we can repeat the calculation using the density of lead (11.3 g/cm^3) and the volume of the lead needed (140.5 g/11.3 g/cm^3 = 12.4 cm^3). This gives us a buoyant force of 12.4 g.

Therefore, the final answer is that we must attach at least 140.5 g of lead to the block in order for it to sink, but the actual minimum mass may be slightly higher due to the buoyant force acting on the lead itself.