Question

If the gauge pressure at a point well inside a liquid of density \(\rho\) in a tank is \(p\), then the depth of the point from the surface of the liquid is (atmospheric pressure is P)

• A. \(\frac{P - p}{\rho g}\)
• B. \(\frac{P + p}{\rho g}\)
• C. \(\frac{P}{\rho g}\)
• D. \(\frac{p}{\rho g}\)
• E. \(\frac{p^2}{\rho g}\)

Hint:

"Gauge pressure" specifically ignores atmospheric pressure. Whenever you see "gauge pressure" in a fluid problem, simply use the formula \(p = h \rho g\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The total pressure (\(P_{total}\)) at a depth \(h\) in a liquid is the sum of atmospheric pressure (\(P\)) and the pressure due to the liquid column (\(h \rho g\)).
Gauge pressure is defined as the difference between absolute (total) pressure and atmospheric pressure.

Step 3: Detailed Explanation:
The formula for total pressure at depth \(h\) is:
\[ P_{total} = P + h \rho g \]
By definition, Gauge Pressure (\(p\)) is:
\[ p = P_{total} - P \]
\[ p = (P + h \rho g) - P = h \rho g \]
To find the depth \(h\):
\[ h = \frac{p}{\rho g} \]

Step 4: Final Answer:
The depth is \(\frac{p}{\rho g}\).