ONE FORMULA, ONE ANSWER
Rotating Liquid Free-Surface Height Difference Calculator
Calculate the signed free-surface height difference between two radial positions when liquid rotates as a rigid body with its container. Choose input units and calculate one value within the stated conditions.
Inputs
Calculations run in this browser. Scientific notation such as 1.2e-3 is supported. Do not group thousands.
Result
Second minus first surface height Δz
Enter values and select Calculate.
Δz = z(r₂)−z(r₁) = ω²(r₂²−r₁²)/(2g)
Assumptions
Required: steady rigid-body rotation about a vertical axis, uniform gravity and constant surface pressure, with liquid present at both radii and no overflow or dry bottom. The result is z(r₂)−z(r₁), negative when the second radius is smaller. Excluded: determining absolute depth or liquid volume.
Displayed values are floating-point approximations. Design safety factors and unprovided error contributions are not calculated.
Formula references and credits
NaNa Lab created the code, descriptions, icons and preview images. Text, figures and code from the linked references have not been reproduced.
Physical constants: NIST / CODATA 2022. The values of μ₀ and ε₀ are measured recommendations, not exact defining constants. Values and precision notes are in the source record.
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