Why does chalk dust stick to a chalkboard?

Friction causes the chalk to stay on the chalkboard. While the chalkboard appears smooth, under a microscope its surface is rough. Chalk is a much weaker material than the chalk board. When it is forced across the chalk board, small parts of chalk ('dust') are broken and remain trapped by friction in the surface asperities of the chalk board. The rougher the chalkboard, the harder the chalk board is to clean because chalk gets trapped 'deeper' in the chalkboard.

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