A solid cube is floating in equilibrium in liquid mercury. The cube is made of iron of density7900 kg m–3.The cube floats with 42% of its volume above the surface of the mercury.What is the density of the mercury?A 3300 kg m–3B 4600 kg m–3C 14 000 kg m–3D 19 000 kg m–3

The density of a substance is defined as its mass per unit volume. In this case, the cube is floating in equilibrium, which means the weight of the cube is equal to the weight of the mercury displaced by the cube.

The weight of an object is given by its mass times the acceleration due to gravity. The mass of the cube is its volume times its density. Therefore, the weight of the cube is its volume times its density times the acceleration due to gravity.

The weight of the mercury displaced by the cube is the volume of the mercury displaced times its density times the acceleration due to gravity. Since the cube is floating with 42% of its volume above the surface of the mercury, it means that 58% of its volume is submerged in the mercury. Therefore, the volume of the mercury displaced is 58% of the volume of the cube.

Setting the weight of the cube equal to the weight of the mercury displaced gives:

(volume of cube * density of iron * g) = (0.58 * volume of cube * density of mercury * g)

The acceleration due to gravity (g) and the volume of the cube cancel out on both sides, leaving:

density of iron = 0.58 * density of mercury

Solving for the density of mercury gives:

density of mercury = density of iron / 0.58

Substituting the given density of iron (7900 kg/m^3) gives:

density of mercury = 7900 kg/m^3 / 0.58 = 13620.69 kg/m^3

This value is closest to option C (14 000 kg/m^3), so the density of the mercury is approximately 14 000 kg/m^3.

The principle of flotation states that a floating object displaces its own weight of the fluid in which it floats. This means that the weight of the cube is equal to the weight of the mercury displaced by the cube.

The weight of the cube (Wc) is its volume (Vc) times its density (ρc) times the acceleration due to gravity (g). So, Wc = Vc * ρc * g.

The weight of the mercury displaced (Wm) is the volume of the cube submerged (Vs) times the density of the mercury (ρm) times g. So, Wm = Vs * ρm * g.

Since the cube is floating in equilibrium, Wc = Wm.

Therefore, Vc * ρc * g = Vs * ρm * g.

We can cancel out g from both sides, and we know that Vs = 0.58 * Vc (since 42% of the cube's volume is above the surface, 58% is submerged).

So, Vc * ρc = 0.58 * Vc * ρm.

We can cancel out Vc from both sides, and rearrange to find ρm.

ρm = ρc / 0.58.

Substituting the given value for the density of iron (7900 kg/m^3), we find:

ρm = 7900 / 0.58 = 13620.69 kg/m^3.

The closest answer to this is C, 14 000 kg/m^3.