A block of wood has a mass of 3.6kg and specific gravity of 0.6. It is to be loaded with lead of density of 11300 kg per cubic metre so that it floats in water with 0.90 of its volume immersed. What mass of lead is needed?

(a) if the lead is on top of the wood?
(b) if the lead is attached below the wood?

a. if it is on top, then the water displaced is

densitywater*volume*.9, which must equal at the mass of lead+masswood.

masslead=-masswood+densitywater*volume*.9
but mass wood=volume*densitywood
masslead=masswood+densitywater*masswood* .9/densitywood
solve for masslead.

To find the mass of lead needed in each case, we need to consider the buoyant force acting on the wood and lead, and ensure that the buoyant force is equal to the gravitational force acting on them.

Let's start with part (a), where the lead is on top of the wood.

In this case, the buoyant force acting on the block of wood and lead should be equal to the gravitational force acting on them. The buoyant force is equal to the weight of the water displaced by the combined volume of wood and lead.

1. Calculate the volume of the wood:
The specific gravity of the wood is 0.6, which means it is 0.6 times as dense as water. Density is mass divided by volume, so the volume of the wood can be calculated as:
Volume of wood = Mass of wood / Density of water
Volume of wood = 3.6 kg / (1000 kg/m^3) (since the density of water is 1000 kg/m^3)
Volume of wood = 0.0036 m^3

2. Calculate the total volume of the wood and lead combination:
The volume of the water displaced is given as 0.90 of the total volume of the block, so:
Total volume of the block = Volume of wood / (1 - 0.90)
Total volume of the block = 0.0036 m^3 / 0.10
Total volume of the block = 0.036 m^3

3. Find the volume of the lead:
Since the lead is on top of the wood, the volume of lead will be equal to the total volume of the block, minus the volume of the wood:
Volume of lead = Total volume of the block - Volume of wood
Volume of lead = 0.036 m^3 - 0.0036 m^3
Volume of lead = 0.0324 m^3

4. Calculate the mass of lead needed:
The buoyant force on the combined block is equal to the gravitational force acting on it. The buoyant force can be calculated as the weight of the water displaced by the lead, which is equal to:
Buoyant force = Volume of lead * Density of water * Acceleration due to gravity
Buoyant force = 0.0324 m^3 * 1000 kg/m^3 * 9.8 m/s^2 (taking the density of water as 1000 kg/m^3 and the acceleration due to gravity as 9.8 m/s^2)

The buoyant force should be equal to the gravitational force, which can be calculated as:
Gravitational force = Mass of wood + Mass of lead * Acceleration due to gravity
Since the buoyant force and the gravitational force are equal, we can set the equations equal to each other and solve for the mass of lead:
Volume of lead * Density of water * Acceleration due to gravity = Mass of wood + Mass of lead * Acceleration due to gravity

From this equation, we can solve for the mass of lead:
0.0324 m^3 * 1000 kg/m^3 * 9.8 m/s^2 = 3.6 kg + Mass of lead * 9.8 m/s^2

Solving for Mass of lead:
Mass of lead * 9.8 m/s^2 = (0.0324 m^3 * 1000 kg/m^3 * 9.8 m/s^2) - 3.6 kg
Mass of lead = [(0.0324 m^3 * 1000 kg/m^3 * 9.8 m/s^2) - 3.6 kg] / 9.8 m/s^2

Calculating the value gives us the mass of lead needed when it is on top of the wood.

Now let's move on to part (b), where the lead is attached below the wood.

In this case, the wood provides some buoyancy which is added to the buoyancy provided by the lead. We need to adjust the calculations accordingly.

5. Calculate the mass of lead needed when it is attached below the wood:
The buoyant force is defined as the weight of the water displaced by the total volume of the block, which now includes the volume of the wood and the volume of the lead. Let's calculate the volume of the lead first using the same steps as before.

The volume of lead = Total volume of the block - Volume of wood
Volume of lead = 0.036 m^3 - 0.0036 m^3
Volume of lead = 0.0324 m^3

The buoyant force is now given by:
Buoyant force = (Volume of wood + Volume of lead) * Density of water * Acceleration due to gravity

Setting the buoyant force equal to the gravitational force:

(Volume of wood + Volume of lead) * Density of water * Acceleration due to gravity = Mass of wood + Mass of lead * Acceleration due to gravity

From this equation, we can solve for the mass of lead in the same way as before.

Using the calculated value from step 4, we can find the final result for the mass of lead needed when it is attached below the wood.

So, in summary:
(a) Calculate the mass of lead using the equation: (0.0324 m^3 * 1000 kg/m^3 * 9.8 m/s^2) - 3.6 kg

(b) Calculate the mass of lead using the equation: [(Volume of wood + Volume of lead) * Density of water * Acceleration due to gravity - 3.6 kg] / 9.8 m/s^2