Question

Determine (a) gauge pressure and (b) the absolute pressure of water at a depth of 9 m from the surface and the atmospheric pressure is $10N/{m^2}$. ($g = 10m/{s^2}$)

Answer 1

Hint: The gauge pressure is equal to the product of the density of water, the acceleration due to gravity and the depth below the surface of the water. The absolute pressure is equal to the sum of the gauge pressure and the atmospheric pressure.
Formula used:
The gauge pressure of fluid is given in terms of the density of the fluid and depth below the surface by the following expression:
$P = \rho gh$
The absolute pressure is defined in terms of the gauge pressure and the atmospheric pressure by the following expression:
${P_{absolute}} = {P_{atm}} + {P_{gauge}}$

Complete answer:
We are required to find out the gauge pressure and absolute pressure of water. The depth at which we need to find these values is given as
$h = 9m$
Taking $g = 10m/{s^2}$, the gauge pressure of water at this depth can be calculated in the following way by taking the product of the density of water, the acceleration due to gravity and the depth below the surface of the water. Therefore, we have
${P_{gauge}} = \rho gh = 1000 \times 10 \times 9 = 90000N/{m^2}$
Here we have taken the density of water in newtons per metre square which is given as $1000N/{m^2}$.
Now we need to find out the absolute pressure of water at the given depth. This pressure is equal to the sum of the gauge pressure of water and the atmospheric pressure on the water surface. We are given the value of atmospheric pressure as
${P_{atm}} = 10N/{m^2}$
Therefore, we can calculate the absolute pressure of water in the following way.
${P_{absolute}} = {P_{atm}} + {P_{gauge}} = 90000 + 10 = 90010N/{m^2}$
Thus, we have the gauge pressure of water as $90000N/{m^2}$ and the absolute pressure of water is $90010N/{m^2}$.

Note:
The gauge pressure can be understood as the pressure exerted by the water relative to the atmospheric pressure above the surface of the water. If this pressure is greater than the atmospheric pressure then it is taken as positive while if the gauge pressure is lower than the atmospheric pressure then it is taken as negative.