Kevin S. Huang

2016: Problem 2

Topic: Fluids
Concepts: Buoyant force, Floating

Solution:

Before additional oil is added, we balance forces on the cylinder. The buoyant forces from the oil and water must equal the weight:

W=F_{B,\text{oil}}+F_{B,\text{water}}

\rho_{\text{obj}}Vg=\rho_{\text{wat}}\left(\frac{V}{3}\right)g+\rho_{\text{oil}}\left(\frac{V}{3}\right)g

Thus, the density of the object is

\rho_{\text{obj}}=\frac{1}{3}(\rho_{\text{wat}}+\rho_{\text{oil}})

Once the cylinder is only in oil, recall the ratio of densities yields the fraction of the object submerged in the fluid:

f_{\text{sub}}=\frac{\rho_{\text{obj}}}{\rho_{\text{oil}}}=\frac{1}{3}\left(\frac{\rho_{\text{wat}}}{\rho_{\text{oil}}}+1\right)=\frac{1}{3}\left(\frac{1000}{800}+1\right)=\frac{3}{4}

so the answer is B.