Question

A block of birch wood floats in oil with \(90.0 \%\) of its volume submerged. What is the density of the oil? The density of the birch is $0.67 \mathrm{g} / \mathrm{cm}^{3} .$

Short answer

Answer: The density of the oil is 0.603 g/cm³.

Step-by-step solution

Understand the principle of buoyancy

According to Archimedes' principle of buoyancy, the buoyant force acting on a submerged object is equal to the weight of the fluid displaced by the object. In this case, it must be equal to the weight of the block of wood.

Set up equation relating submerged volume, densities of wood, and oil

Let's denote the density of oil as \(\rho_o\), the density of the birch as \(\rho_b\), and the submerged volume ratio as \(r\). We have: \(r \times \rho_b = \rho_o\) We are given that the density of the birch, \(\rho_b = 0.67 \mathrm{g} / \mathrm{cm}^{3}\), and that \(90 \%\) of its volume is submerged, so \(r = 0.90\).

Solve for the density of oil

Plug the given values into the equation \(r \times \rho_b = \rho_o\): \((0.90) \times (0.67 \mathrm{g} / \mathrm{cm}^{3}) = \rho_o\) Now, calculate the density of oil: \(\rho_o = 0.603 \mathrm{g} / \mathrm{cm}^{3}\) So, the density of the oil is \(0.603 \mathrm{g} / \mathrm{cm}^{3}\).