Sol. (T ~ i.e., T ∝ d –5/2 )

The sand flows through the aperature almost uniformly, and therefore the total operating time T of the sand-glass is proportional to the volume H 3 of the sand. (As our aim is to obtain only a rough estimate, the difference between the volumes of a cone and a cube is ignored). The time T may also depend on the gravitational acceleration g, the diameter d of the aperture and density ρ of the sand, and so T ≈ H 3 × f (g,d, ρ ).

As T is a time and only g contains a time dimension, the function f has to be proportional to the reciprocal of the square root of g. Similar reasoning shows that T cannot depend on ρ , but is proportional to d –5/2 ; in summary, T ≈ H 3 / . The coefficient of proportionality is a dimensionless number, and since it does not depend on anything, can be assumed to be of order 1 (through such assumptions are notoriously dangerous in some branches of physics !). Consider some realistic data. If, for example, H is a few centimeters and d is around a millimeter, T is a few minutes, which is indeed the sort of time for which an egg should be boiled.