Serway & Jewett — Fluid Mechanics: Problem 41

The spirit-in-glass thermometer, invented in Florence, Italy, around 1654, consists of a tube of liquid (the spirit) containing a number of submerged glass spheres with slightly different masses (Fig. P14.41). At sufficiently low temperatures, all the spheres float, but as the temperature rises, the spheres sink one after another. The device is a crude but interesting tool for measuring temperature. Suppose the tube is filled with ethyl alcohol, whose density is \(0.78945 \text{ g/cm}^3\) at \(20.0^\circ\text{C}\) and decreases to \(0.78097 \text{ g/cm}^3\) at \(30.0^\circ\text{C}\). (a) Assuming that one of the spheres has a radius of \(1.000 \text{ cm}\) and is in equilibrium halfway up the tube at \(20.0^\circ\text{C}\), determine its mass. (b) When the temperature increases to \(30.0^\circ\text{C}\), what mass must a second sphere of the same radius have to be in equilibrium at the halfway point? (c) At \(30.0^\circ\text{C}\), the first sphere has fallen to the bottom of the tube. What upward force does the bottom of the tube exert on this sphere?