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using the dimensional analysis derive the formula of pressure
Asked by kartikchauhan2425 | 13 Jan, 2024, 05:37: PM

Let us derive a relation for absolute pressure P at a point A in liquid

P = Po + p

where Po is atmospheric pressure and p is hydrostatic pressure at the point A

Hydrostatic pressure p depends on three parameters namely

(i) height h of liquid column above point A

(ii) density ρ of liquid

(iii) acceleration due to gravity g

Hence we write the relation for hydrostatic pressure as

p = k ρa gb hc ..........................(1)

where k, a, b and c are constant numbers to be determined .

LHS of eqn.(1) is pressure :-  Force/ area = ( mass × acceleration ) / area

Dimension of LHS =  ( M × L T-2 ) / L2 = M L-1 T-2

Dimension of density ρ = mass / volume = M L-3

Dimension of acceleration due to gravity g = L T-2

Dimension of height h = L

Hence if we equate the dimension on both sides of eqn.(1), then we get

M L-1 T-2 = [ M L-3  ]a  [  L T-2 ]b  [ L ]c

M L-1 T-2 = Ma L-3a+b+c T-2b

By equating powers , we get

a = 1

-3a+b+c = -1

-2b = -2

By solving above expressions , we get , a= b = c = 1

Hence from eqn.(1), we get hydrostatic pressure p as

p =  k × ρ × g × h

From experiments it can be determined that k = 1

Hence , we get ,    p = ρ × g × h

Absolute pressure P at a point in liquid is

P = Po + ( ρ × g × h )

Answered by Thiyagarajan K | 14 Jan, 2024, 10:11: AM
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