The Sandglass

For a sandglass timer one could simply have a straight glass or plastic tube with equally spaced markings and then the whole tube would be inverted to start the time measurement. Why do ruled sandglasses have a tapered “hourglass” shape instead?

Answer

The hourglass shape ensures that the time scale on the glass is uniform, with equal distances between scale divisions corresponding to equal time intervals. If the sandglass didn’t taper, the top of the sand column would descend at increasing speed. We can mathematically determine the proper shape. The speed, V, at which the sand is escaping from the opening is given approximately by Torricelli’s formula V ~ , where g is the local acceleration of gravity and y is the height of the sand column in the upper glass. Let A = πr² be the circular cross-sectional area of the upper glass at the top of the sand column and v the speed at which this top is falling, then Av = aV, where a is the area of the opening, because the sand is approximately incompressible. Substitution produces y ~ cr⁴, with constant c = π²v²/(2ga²). A plot of y versus r will produce the familiar hourglass shape.

Sandglasses without time interval markings have been used since before the fourteenth century to time speeches at town meetings and other events. When the sand had run its course, the speaker’s time was done. They are still used today in some board games and as kitchen timers. Sandglasses with ruled markings haven’t been so popular, being replaced early on by mechanical watches and clocks.