Fluid mechanics - ice melting into water

[JRPK8@mizzou.]

Homework Statement

A pure ice cube sits in water, floating. Prove that the water level will not change after the ice cube melts (meaning, the volume displaced by the cube will equal the volume of water added once the cube is melted).

density of water = 1000 kg / m^3
density of ice = 920 kg / m^3

Homework Equations

D = m/V F(net) = mg - pVg

The Attempt at a Solution

F(net on the cube) = mg - pVg D = m / V
mg = pVg D * V = m(water afterwards)
m = p (V) (920/1000) * V = m(water after melting)
m = (920/1000) V .92 * V = mass (water after melting)
m (submerged cube) = .92 V


Thus, m(submerged cube) = m(water after melting)

 

[LowlyPion]

Maybe just cite the conservation of matter?

Since it was all water in the first place and nothing was added or removed ...

 

[thomate1]

This shows that the mass of the ice cube when submerged in water is equal to the mass of the water after the ice cube has melted. Since the density of ice is lower than the density of water, the ice cube will displace a larger volume of water when it is submerged. However, when the ice cube melts, it will occupy the same amount of space as the water it originally displaced. Therefore, the water level will not change because the volume of water added from the melted ice cube is equal to the volume of water displaced by the ice cube. This is known as Archimedes' principle, which states that the buoyant force acting on an object in a fluid is equal to the weight of the fluid displaced by the object. In this case, the buoyant force on the ice cube and the weight of the water it displaces are equal, resulting in no change in water level. This principle is an important concept in fluid mechanics and is applicable in many real-world scenarios, such as ship buoyancy and hot air balloons.