determine the submerged depth of a cube of steel 0.3m on each side floating in mercury given the specific gravities of steel and mercury as 7.8 and 13.6 respectively.

Principle of floatation:

A floating object displaces its own weight of the fluid in which it floats.

Volume of steel
=Vs

Mass of steel
Ms
= ρs*Vs

Volume of mercury displaced, Vm = Ms / ρm

Percentage of volume displaced
=Vm / Vs

If the steel block floats with bottom face parallel to surface of mercury, submerged depth equals height of block multiplied by percentage of volume displaced.
It is slightly less than two-thirds of the height of the block.

1.7199m

0.0155

.172

Sorry, I'm not sure what those numbers refer to. Can you please provide more context or information so I can assist you better?

To determine the submerged depth of a cube of steel floating in mercury, you can use Archimedes' principle. Archimedes' principle states that the buoyant force experienced by an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

First, let's calculate the weight of the steel cube:

1. Determine the volume of the steel cube:
The volume of a cube is given by side^3. In this case, the side length of the cube is 0.3m. So the volume of the steel cube is 0.3^3 = 0.027 m^3.

2. Calculate the weight of the steel cube:
The weight of an object is calculated using the formula weight = mass × acceleration due to gravity. However, since we are given the specific gravity of steel, we can use it to find the ratio of the density of steel to the density of water, which is equal to the ratio of their specific gravities.

The density of water is 1000 kg/m^3.

Density of steel = Specific gravity of steel × Density of water
= 7.8 × 1000 kg/m^3
= 7800 kg/m^3

Now, calculate the weight of the steel cube using the formula:
Weight of steel cube = Density of steel × Volume of steel cube × acceleration due to gravity
= 7800 kg/m^3 × 0.027 m^3 × 9.8 m/s^2

Now, let's calculate the buoyant force experienced by the steel cube when submerged in mercury:

The buoyant force is equal to the weight of the fluid displaced by the cube. Since mercury is the fluid in this case, we need to calculate the weight of the mercury displaced.

1. Calculate the density of mercury using its specific gravity:
Density of mercury = Specific gravity of mercury × Density of water
= 13.6 × 1000 kg/m^3
= 13600 kg/m^3

2. Calculate the volume of the steel cube submerged in mercury:
The volume of the cube submerged in mercury is equal to the volume of the water displaced by the cube, i.e., its submerged volume.

Volume submerged in mercury = Volume of the steel cube × Specific gravity of steel / Specific gravity of mercury
= 0.027 m^3 × 7.8 / 13.6

Finally, we can calculate the submerged depth of the cube in mercury:

Submerged depth = Volume submerged in mercury / Area of the cube face

The area of the cube face is (0.3 m)^2 = 0.09 m^2.

Substituting the values, we can calculate the submerged depth.