Are evenly spaced specific-gravity markings on the cylinder of a hydrometer equal distances apart? In other words, is the depth d to which the cylinder is submerged linearly related to the density ρ of the fluid? To answer this question, assume that the cylinder has radius r and mass m. Find an expression for d in terms of ρ, r, and m, and see if d is a linear function of ρ.

A hydrometer is an instrument for measuring the specific gravity of a liquid. For example, vintners use a hydrometer to determine the density changes as wine is fermented, and producers of maple sugar and maple syrup use the hydrometer to find how much sugar is in the collected sap. Markings along a stem are calibrated to indicate the specific gravity for the level at which the hydrometer floats in a liquid. The weighted base ensures that the hydrometer floats vertically.