A 2.0-cm thich bar of soap is floating on a water surface so that 1.5 cm of the bar is under water. Bath oil of specific gravity 0.60 is poured into the water and floats on top of it. What is the depth of the oil layer when the top of the soap is just level with the upper surface of the oil?

Hints: You are given the soap's thickness, but not its other two dimensions, lenght and width... call them L & W

Consider, first, the soap floating in the two liquids. Draw a new diagram, create a new buoyant force equation, and solve for the unknow (soap's density)

Then, consider the soap floating in the two liquids. Draw a new diagram, create a new buoyant force equation, and solve for the unknown (oil's thickness)

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

First, let's consider the soap floating in the two liquids. We are given that 1.5 cm of the soap is under water, so the remaining 0.5 cm is above the water surface. We are also given the thickness of the soap as 2.0 cm.

To find the density of the soap, we can use the buoyant force equation:

Buoyant force = weight of the displaced fluid

The buoyant force is equal to the weight of the water displaced by the submerged portion of the soap:

Buoyant force = density of water * volume of water displaced

The volume of water displaced can be calculated as the product of the cross-sectional area of the soap (L * W) and the depth submerged in water (1.5 cm).

The weight of the displaced water is given by the product of the density of the water (1000 kg/m^3) and the volume of water displaced.

Therefore, we can write the buoyant force equation as:

density of water * L * W * 1.5 cm = density of soap * L * W * 1.5 cm

We can cancel out the cross-sectional area and the depth submerged to simplify the equation:

density of water = density of soap

Now that we have the density of the soap, we can move on to the next step.

Next, let's consider the soap floating in the two liquids, with the oil layer on top. We want to find the depth of the oil layer.

Again, we can use the buoyant force equation:

Buoyant force = weight of the displaced fluid

The buoyant force is equal to the weight of the oil displaced by the submerged portion of the soap:

Buoyant force = density of oil * volume of oil displaced

The volume of oil displaced can be calculated as the product of the cross-sectional area of the soap (L * W) and the depth submerged in oil (unknown).

The weight of the displaced oil is given by the product of the density of the oil (0.60 * density of water) and the volume of oil displaced.

Therefore, we can write the buoyant force equation as:

density of oil * L * W * unknown = density of water * L * W * 1.5 cm

We can cancel out the cross-sectional area to simplify the equation:

density of oil * unknown = density of water * 1.5 cm

Now, we know the density of the oil (0.60 * density of water), so we can solve for the unknown (the depth of the oil layer) by rearranging the equation:

unknown = (density of water * 1.5 cm) / (0.60 * density of water)

We can cancel out the density of water to simplify the equation:

unknown = 1.5 cm / 0.60

Simplifying further, we find:

unknown = 2.5 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 2.5 cm.