An open cylinder of radius (R) is partly filled with liquid. It is rotated about its axis at (ω) rad/s without spilling. At the walls, the rise of liquid surface above the original level will be (where, g is acceleration due to gravity)
- Aω²R² / 8g
- Bω²R² / g
- Cω²R² / 4gCorrect
- Dω²R² / 2g
Solution
Concept
Rise of liquid at the wall of a rotating cylinder
Formula
Rise at wall = ω²R²/(4g)
Step-by-step solution
1. Total height of paraboloid h = ω²R²/(2g). 2. The volume of a paraboloid is half of its circumscribing cylinder, so the surface falls by h/2 at the axis and rises by h/2 at the wall. 3. Rise = h/2 = ω²R²/(4g).
Explanation
Because the volume of the paraboloid is half of the cylinder, the original level lies half way between the lowest and highest points of the free surface. So the rise at the wall is ω²R²/4g.
Shortcut
Total height = ω²R²/2g; rise at wall = half of it.
Common mistake
Giving the total height instead of the rise above the original level.
Correct answer: C) ω²R² / 4g