Question

Given P space equals space P subscript 0 space plus space rho g h, your objective is to determine the uncertainty of P by measuring P subscript 0 comma space rho comma space g space and space h , What is the expression for capital delta P subscript g , that is, the g's contributing term for the overall uncertainty of P

Answer 1

the g's contributing term for the overall uncertainty of P is
`dP_g = [(dg)/(g)]`

Explanation:

From the question we are told that

The pressure is
`P = P_o + \rho gh`

The first step in determining the uncertainty of P in by obtaining the terms in the equation contributing to it uncertainty and to do that we take the Ln of both sides of the equation

`ln P = lnP_o + ln(\rho gh )`

=>
`ln P = lnP_o + ln \rho + ln g + ln h`

Then the next step is to differentiate both sides of the equation

`(d(ln P))/(dP) = (d(ln P_o))/(dP_o) + (d(ln \rho))/(d\rho) +(d(ln g))/(dg) + (d(ln h))/(dh)`

=>
`(dP)/(P) = (dP_o)/(P_o) + (d \rho)/(\rho) +(d g)/(g) + (d h)/(h)`

We asked to obtain the contribution of the term g to the uncertainty of P

This can deduced from the above equation as

`dP_g = [(dg)/(g)] P`