Question

the hydrostatic pressure P of a liquid column depends on the density D height of the liquid column age and acceleration due to gravity G using dimensional analysis derive formula for hydrogen pressure​

Answer 1

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JinKazama1

JinKazama1Samaritan

Final Answer: P = DHg.

Steps:

1) We know that,

dim(P) = \frac{M}{LT^{2}}

dim(D) = \frac{M}{L^{3}} \\ \\ 
dim( H) = L \\ \\ 
dim( g) = \frac{L}{T^{2}}

2) Since , Hydrostatic Pressure is dependent on above given quantities .

Therefore,

P = k D^{x}H^{y}g^{z} where k is constant.

=> dim(P) = dim(k D^{x}H^{y}g^{z}) \\ \\ 
=\ \textgreater \ \frac{M}{LT^{2}} = (\frac{M}{L^{3}} )^{x}*(L)^{y} *( \frac{L}{T^{2}} )^{z} \\ \\ =\ \textgreater \ 
 \frac{M}{LT^{2}} = \frac{M^{x}}{L^{(3x-y-z)}*T^{2z}}

=> Comparing Powers ,we get

x = 1 , 2z=2 => z= 1

=> 3x-y-z=1

=> 3*1-y-1=1

=> y= 1

3) Therefore, Formula of Hydrosatic Pressure is given by :

\boxed{H=kDHg}

By Experiments ,it is observed that value of k = 1 ;

\boxed{H=DHg}