Physics

The hydrostatic pressure 'P' of a liquid column depends upon the density 'd' , height 'h' of liquid column and also an acceleration 'g; due to gravity. Using dimensional analysis, derive formula for pressure P.


SOLUTION

It is given that the hydrostatic pressure is directly proportional to the following terms

  $$ P=K{{d}^{x}}{{h}^{y}}{{g}^{z}}............(1) $$

 $$ K=cons\tan t $$

Writing dimensional formula of pressure, density, height and acceleration due to gravity as

  $$ \left[ \frac{M}{L{{T}^{2}}} \right]={{\left[ \frac{M}{{{L}^{3}}} \right]}^{x}}{{\left[ L \right]}^{y}}{{\left[ \frac{L}{{{T}^{2}}} \right]}^{z}} $$

 $$  $$

On comparing there powers, we get

  $$ x=1 $$

 $$ z=1 $$

 $$ \And \,\,y=\,1 $$

Equation (1) becomes

$$P=dgh$$

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