Question

To examine the speed that a wave can propagate, we will use a simple model of dominos toppling. A schematic of the system is shown below. Assume the velocity v depends on the spacing d, height h, and the gravitational acceleration constant g. Based on the data provided below, determine the two dimensionless Pi's that govern the wave speed: II1 = G(112) Total number of parameters: n = 4;(v,d, h,g) Number of primary dimensions: r = 2; (Lt) Dimensional parameters: m = 2;gh Number of dimensionless groups: n - m = 2 Dimensionless Pi 1: II = II (ghi) = gºhºus, c = 1 Dimensionless Pi 2: 12 = 12(ghd) = gdhed, f = 1

Answer 1

Formation of dimensionless a group f(v,d,h,g)=0 k=4 Let 'r' independent dimensions r = 2(g, h) Number of a terms, n=k-r = 4-2 = 2 1st a group (MLT system) T =gº hº vº Acceleration due to gravity (g), [LT-?] Height, h(m)-[L] Diameter, D(m)-[2] Velocity, v(m/s)-[LT-] [M°L°T°)=[LT-?]° [1]° [LT-'] On comparing the terms,

Lia+b+1=0 T:-2a-1=0 On solving a = 2 b= 2 1 11 gh 2nd 7 group Ty = go he di Acceleration due to gravity (g), [11²] Height, h(m) - [L] Diameter, D(m)-[2] [M°LT"]=[LT] [4] [L] On comparing the terms, L:d+e+1=0 T:-2d = 0

On solving e=-1 d h wenn Every d V=.