To determine the submerged depth of the cube of steel floating in mercury, we can use Archimedes' principle, which states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
The buoyant force (F_b) can be calculated as the difference between the weight of the fluid displaced and the weight of the object itself.
Let's denote: V_cube as the volume of the steel cube, ρ_steel as the density of steel (given as 7.8 g/cm³ or 7800 kg/m³), ρ_mercury as the density of mercury (given as 13.6 g/cm³ or 13600 kg/m³), g as the acceleration due to gravity (approximately 9.8 m/s²).
The volume of the steel cube can be calculated as: V_cube = (0.3 m)^3 = 0.027 m³
The weight of the fluid displaced (W_fluid) can be calculated as: W_fluid = ρ_fluid * V_cube * g = ρ_mercury * V_cube * g
The weight of the steel cube (W_steel) can be calculated as: W_steel = ρ_steel * V_cube * g
The buoyant force (F_b) is the difference between these two weights: F_b = W_fluid - W_steel = ρ_mercury * V_cube * g - ρ_steel * V_cube * g = (ρ_mercury - ρ_steel) * V_cube * g
Since the cube is floating, the buoyant force is equal to the weight of the cube: F_b = W_cube = ρ_steel * V_cube * g
Setting these two equal, we can solve for the submerged depth (h) of the cube:
(ρ_mercury - ρ_steel) * V_cube * g = ρ_steel * V_cube * g
(ρ_mercury - ρ_steel) * h = ρ_steel * h
Simplifying the equation, we find:
(ρ_mercury - ρ_steel) * h = ρ_steel * h
ρ_mercury * h - ρ_steel * h = ρ_steel * h
ρ_mercury * h = 2 * ρ_steel * h
h (ρ_mercury - 2 * ρ_steel) = 0
Since the height (h) cannot be zero, we have:
ρ_mercury - 2 * ρ_steel = 0
ρ_mercury = 2 * ρ_steel
Now, we can substitute the given values:
13.6 g/cm³ * 1000 cm³/kg = 2 * 7.8 g/cm³ * 1000 cm³/kg
13600 kg/m³ = 2 * 7800 kg/m³
13600 kg/m³ = 15600 kg/m³
Since the equation is not satisfied, we can conclude that the cube of steel will not float in mercury.