Originally posted by talzamirTaking the cross section, and doing it in 2d for simplicity, which shouldn't change the answer:
A glass is shaped like a straight cylinder with a round base, height h, radius r. If the glass is half full and tilted so that the liquid just touches the rim of the glass, how high from the bottom of the glass is the surface on the opposite side of the glass?
Originally posted by iamatigerThere's an even simpler way to reach that conclusion: a cylinder is symmetrical across its central point. Any straight plane that goes through one edge and divides it exactly in two must go through the opposite point on the opposite edge, because of that symmetry.
Taking the cross section, and doing it in 2d for simplicity, which shouldn't change the answer:
So, assuming r is not 0, the edge of the water now touches the bottom of the glass.
Originally posted by talzamirAh, but they're not. They have two flat sides each, one being the circular bottom (or top) side of the cylinder, and one being the elliptical surface of the water. There's also one curved surface; and there is no singular vertex. So they're not cones at all, they just look a bit like cones.
Sorry, late here and it's been a long day. Seems I missed a few crucial words.
If you slice the glass diagonally in two, then yes, by symmetry each half has a volume equal to one half of the full glass.
However, those halves would appear to be cones,
Originally posted by talzamirIf it were a cone, every point on its surface would be on straight line between the apex and the circumference of the base. It is fairly straightforward to prove that's not the case.
Indeed so. Thinking of the 2D cross section, it is obvious that the water needs to cover the bottom of the glass exactly, based on symmetry. The volume that it forms looks a lot like a cone - every point on the circumference of the bottom of the glass joined to the point on the rim along a straight line.. and straight the lines would be, being in the plane ...[text shortened]... is not above the middle point of the bottom of the glass, but not any other kind of cone either.