Question

A 1.2 kg iron block lies at the bottom of a water tank. Your task is to attach a string to the iron, and attach a piece of Styrofoam to the other end of the string, so that the arrangement floats near the surface of the tank. Find the smallest mass of Styrofoam that must be attached (so the arrangement barely floats). Use the following densities: water 1000 kg/m3 ; iron 7900 kg/m3 ; and Styrofoam 45 kg/m3.

Answer 1

Gravitational acceleration = g = 9.81 m/s²

Density of water = ₀ = 1000 kg/m³

Density of iron = ₁ = 7900 kg/m³

Density of styrofoam = ₂ = 45 kg/m³

Mass of the iron block = M₁ = 1.2 kg

Volume of the iron block = V₁

M₁ = ₁V₁

Buoyancy force on the iron block = F₁ = ₀V₁g

Mass of the piece of styrofoam = M₂

Volume of the piece of styrofoam = V₂

M₂ = ₂V₂

Buoyancy force on the piece of styrofoam = F₂ = ₀V₂g

The weight of the iron block and the piece of styrofoam is supported by the total buoyancy force on them.

M₁g + M₂g = F₁ + F₂

M₁g + M₂g = ₀V₁g + ₀V₂g

M₁ + M₂ = ₀V₁ + ₀V₂

M₂ = 0.0494 kg

Smallest mass of styrofoam to be attached to the iron block so that the arrangement barely floats = 0.0494 kg