Tools › Units › Calculators › Formula with Units › Example
Buckingham Pi groups: pendulum period, drag on a body and pump power
| Tool | Formula with Units |
| Source | Buckingham, E. (1914). "On Physically Similar Systems; Illustrations of the Use of Dimensional Equations." Physical Review, Vol. 4, No. 4, pp. 345–376; Sonin, A. A. (2001). The Physical Basis of Dimensional Analysis, 2nd ed. MIT Department of Mechanical Engineering, Cambridge, MA. |
| Level | Undergraduate |
Problem
For a pendulum (T, L, g), a body in a fluid stream (F, ρ, v, D, μ) and a pump (P, ρ, N, D), list the variables with their units and find the rank of the dimension matrix, the number of independent dimensionless groups and the groups themselves.
Inputs to Enter
- Pendulum: T [s], L [m], g [m/s^2]; Pi analysis checked
- Drag: F [N], mu [Pa*s], rho [kg/m^3], v [m/s], D [m]
- Pump: P [hp], rho [lb/ft^3], N [rpm], D [in] (the 'Pump power group' example)
Answers
- Pendulum: rank 2, one group Π = T² g/L
- Drag: rank 3, two groups with ρ, v, D repeating: Π1 = F/(ρ v² D²), Π2 = μ/(ρ v D) = 1/Re
- Pump: rank 3 (the radian is dimensionless), one group Π = P/(ρ N³ D⁵)
- A list of only F and L gives k = 0; σ, E and ε give rank 1, k = 2
Notes
Turn on the Pi analysis box; the pump case loads from the Example select, the others are typed into the variable rows (the formula box is ignored by the Pi analysis). Not the default case.