Classical Mechanics

Dimensional Analysis

Concept Quizzes

Buckingham Pi Theorem (Dimensional Analysis)

         

Using the Buckingham π\pi theorem, find the formula for the radius of a black hole in term of the black hole’s mass m,m, the gravitational constant G, G, and the speed of light c c

As shown in the above figure, a liquid with density ρ \rho and viscosity μ \mu flows through a pipe with diameter d. d. In a section of the pipe with length L, L, the liquid flows with a speed of v. v. Using the Buckingham π\pi theorem, determine the pressure difference ΔP=P1P2 \Delta P = P_1 - P_2 in terms of the fluid properties d,L,ρ,μ d, L, \rho, \mu and v. v .

Details

  • the function ff is unitless, so that it outputs a pure number.

The sand timer in the above figure has a due time of T. T. The radius of the hole is r, r, the initial height of the sand is H, H, and the density of the sand is ρ. \rho. Using the Buckingham π\pi theorem, determine T T in terms of the properties r,H,ρ, r, H, \rho, and the gravitational constant G. G .

As shown in the above figure, a liquid with density ρ \rho flows through a pipe with diameter d. d. In a section of the pipe with length L, L, the liquid flows with a speed of v1 v_1 at the center and v2 v_2 at the edge of the flow. The pressures are P1 P_1 at the center and P2 P_2 at the edge. Using the Buckingham π\pi theorem, express the viscosity μ \mu of the fluid in terms of the liquid properties d,L,ρ,Δv=v1v2 d, L, \rho, \Delta v = v_1 - v_2 and ΔP=P1P2. \Delta P = P_1- P_2 .

The drag force F F depends on four quantities: two parameters of the cone which are the speed of the cone v v and the size of cone r, r, and two parameters of the air which are the density of the air ρ \rho and the viscosity of the air μ. \mu. Find the independent dimensionless groups that can be produced with F,v,r,ρ F, v, r, \rho and μ. \mu .

×

Problem Loading...

Note Loading...

Set Loading...