A Carpenter's Level

You have probably seen a carpenter’s level with a glass tube with a bubble that deviates from the centre when placed on a sloping surface. The bigger the slope, the more does the bubble deviate from the mark.



It moves because, being lighter than the liquid in the tube, it rises to the surface. If the tube were straight, the bubble would move to the end of the tube, that is, to its highest point. A level like that, as it may easily be seen, would be very inconvenient. That is why the tube is usually arched.
When the level is horizontal, the bubble, situated at the highest level point of the tube, is in the centre;
if the level is sloped, the highest point is then not its centre, but some point next to it, and the bubble moves from the mark to another part of the tube.*

The problem is to determine how many millimetres the bubble will move away from the mark if the level is sloped 1/2° and the radius of the arch of the tube is 1 metre.


* It would be more correct to say that "the mark moves from the bubble", because the latter really remains in its place while the tube and the mark glide past.