Question

The force exerted on a bridge pier in a river is to be tested in a 1:10 scale model using water as the working fluid. In the prototype the depth of water is 2.0 m, the velocity of flow is 1.5 m/s and the width of the river is 20 m. Of the hydrodynamic force on the model bridge pier is 5 N, what would it be on the prototype? (hint: pressure ratio is equal to the length-scale ratio) (5 points; Ans: 5000 N)

Answer 1

Answer:

Force on the prototype is 5000 N

Solution:

As per the question:

Depth of water, x = 2.0 m

Flow velocity, v' = 1.5 m/s

Width of the river, w = 20 m

Force on the bridge pier model, F' = 5 N

Pressure, Ratio = Ratio of scale length

Scale = 1:10

Now,

$\frac{P'}{P} = \frac{x}{w} = \frac{2.0}{20}$

where

P' = pressure on model

P = pressure on prototype

$\frac{\frac{F'}{A'}}{\frac{F}{A}} = \frac{1}{10}$

where

F' = Force on model

F = Force on prototype

A' = Area of model

A = Area of prototype

Now:

$\frac{F'}{F}.\frac{A}{A'} = \frac{1}{10}$

$\frac{5}{F}.\frac{1}{\frac{1}{10}}.\frac{1}{\frac{1}{10}} = \frac{1}{10}$

F = 5000 N