A 2.6 cm thick bar of soap is floating on

a water surface so that 2.14 cm of the bar is underwater. Bath oil (of specific gravity 0.8)is poured into the water and floats on top of the water.

What is the depth of the oil layer when the top of the soap is just level with the upper surface of the oil?

Answer in units of cm.

The soap is floating in water, you can get its weight from the volume of water displaced.

Now in the second part, the volume displaced is two weights, the sum of which equals the original weight.
The top weight is the volume of oil displaced, the lower weight the the weight of water displaced. For computing purposes, assume the area of the bar is A. (thickness*A=volume).

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To find the depth of the oil layer when the top of the soap is level with the upper surface of the oil, we need to consider the buoyancy forces acting on both the soap and the oil.

First, let's calculate the buoyant force on the soap. The buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In this case, the soap displaces water.

The density of water is 1 g/cm³. Given that 2.14 cm of the soap is submerged, the volume of water displaced by the soap can be calculated as follows:
Volume of water = 2.6 cm × 2.14 cm = 5.5644 cm³

Since the density of water is 1 g/cm³, the weight of the water displaced by the soap is equal to the buoyant force acting on the soap.

Next, let's calculate the buoyant force on the oil. The density of oil is given as 0.8 times the density of water, which is 0.8 g/cm³. To find the depth of the oil layer, we need to equate the buoyant forces on the soap and the oil.

Now, let's write an equation to represent the buoyant forces on the soap and the oil:

Weight of water displaced by the soap = Weight of oil displaced by the oil

Density of water × Volume of water = Density of oil × Volume of oil

Substituting the known values and solving for the volume of oil, we get:

1 g/cm³ × 5.5644 cm³ = 0.8 g/cm³ × Volume of oil

Simplifying, we find:

5.5644 cm³ = 0.8 × Volume of oil

Dividing both sides by 0.8:

Volume of oil = 5.5644 cm³ ÷ 0.8

Volume of oil ≈ 6.9555 cm³

Since the density of oil is given as 0.8 g/cm³, we can use the density formula to calculate the depth of the oil layer:

Density of oil = Mass of oil ÷ Volume of oil

Rearranging the formula to solve for the mass of oil:

Mass of oil = Density of oil × Volume of oil

Plugging in the values, we get:

Mass of oil = 0.8 g/cm³ × 6.9555 cm³

Mass of oil ≈ 5.5644 g

Finally, to find the depth of the oil layer, we need to consider the density of the oil and the volume occupied by this mass of oil.

Let's call the depth of the oil layer "h". The volume of the oil can be calculated as:

Volume of oil = area of the oil layer × depth of the oil layer

The area of the oil layer is the same as the area of the soap, which is given as 2.6 cm × the length of the soap.

To find the length of the soap, we can subtract the submerged length from the total length of the soap:

Length of the soap = Total length of the soap - Submerged length
= 2.6 cm - 2.14 cm
≈ 0.46 cm

Therefore, the volume of the oil can be calculated as:

Volume of oil = 2.6 cm × 0.46 cm × h

Now, we can equate the volume of the oil to the mass of the oil to determine the value of "h":

2.6 cm × 0.46 cm × h = 5.5644 g

Simplifying, we find:

1.1964 cm³ × h = 5.5644 g

Dividing both sides by 1.1964 cm³:

h ≈ 4.6451 cm

Therefore, the depth of the oil layer when the top of the soap is just level with the upper surface of the oil is approximately 4.6451 cm.