Block floating in water - Oil is added

[emenos]

Homework Statement

A block of wood floats on water. A layer of oil is now poured on top of the water to a depth that more than covers the block.If 75% of the wood is submerged in water before the oil is added, find the fraction submerged when oil with a density of 715 kg/m3 covers the block. (Do not neglect the buoyant force of air before the oil is added.)

Homework Equations

The Attempt at a Solution

 

[aniketp]

Hey emenos,
The fraction of the block floating initially is given. Get the density of the block from this assuming Dwater=1000kg/m^3. Once you get the density use archimedes principle and Newton's 2nd law to get the required fraction.

 

[Moetasim]

The fraction of the block submerged in water can be represented by the ratio of the block's volume to the total volume of the block and water. Before the oil is added, this fraction is 75%. After the oil is added, the oil will displace some of the water and increase the total volume. The fraction of the block submerged in water will decrease because the total volume has increased but the block's volume remains the same.

To find the new fraction submerged, we can use the equation:

f = Vb / (Vb + Vw + Vo)

Where:
f = fraction submerged
Vb = volume of the block
Vw = volume of the water
Vo = volume of the oil

We know that the volume of the block and water remain the same before and after the oil is added, so we can set Vb + Vw as a constant. This means that the fraction submerged will only depend on the volume of the oil added.

Let's assume the original volume of the block and water is 1 unit. This means that before the oil is added, the volume of the block is 0.75 units and the volume of the water is 0.25 units.

After the oil is added, the total volume will increase by the volume of the oil, which we can represent as Vo. The volume of the water will decrease by the same amount, since the oil displaces the water. This means that the new volume of the water is 0.25 - Vo.

Substituting these values into the equation, we get:

f = 0.75 / (0.75 + 0.25 - Vo)

We also know that the density of the oil is 715 kg/m3, which means that the volume of the oil can be represented by its mass divided by its density, or Vo = m / 715.

Substituting this into the equation, we get:

f = 0.75 / (0.75 + 0.25 - m / 715)

We can solve for the fraction submerged by setting m = 0, which represents the case where no oil is added. In this case, the fraction submerged is 0.75, which matches our initial condition.

Setting m = 715, which represents the case where the entire block is covered in oil, we get:

f = 0.75 / (0.75 + 0.25 - 715 /