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

We know,

Dimension of pressure is, [P] = [ML^-1T^-2]

Dimension of density is, [d] = [ML^-3]

Dimension of height is, [h] = [L]

Dimension of ‘g’ is [g] = [LT^-2]

Now,

Let, [P] = k [d]^x [g]^y [h]^z  [k is a dimensionless constant]

=> [ML^-1T^-2] = [ML^-3]^x [LT^-2]^y [L]^z

Thus,

x = 1

-3x +y + z = -1

-2y = -2

Solving we get,

x = 1, y = 1, z = 1

Thus, P = kdgh

It is found that k = 1

Therefore, P = dgh