An open cylindrical can of radius $10\ \text{cm}$ and height $30\ \text{cm}$ holds water to a depth of $20\ \text{cm}$. It is spun about its vertical axis, faster and faster, very slowly, so that the water always turns with the can, as shown in the figure. Water spills over the rim once the spin is fast enough. Take $g = 10\ \text{m/s}^2$. What fraction of the water has spilled by the time the centre of the bottom of the can just becomes dry?

Fig. 1: An open cylindrical can of radius 10 cm and height 30 cm holds water to a depth of 20 cm with a flat surface. A dashed vertical axis runs through its centre, with a curved arrow showing that the can is spun about it.