Question

$P_a$ is atmospheric pressure and P is absolute pressure at depth h. The gauge pressure at depth h is:

• A. $P+P_{a}$
• B. $\frac{P-P_{a}}{2}$
• C. $2P-P_{a}$
• D. $\frac{P+P_{a}}{2}$
• E. $P-P_{a}$

Hint:

Gauge pressure is the pressure measured relative to atmospheric pressure.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This question asks for the definition of gauge pressure in relation to absolute pressure and atmospheric pressure. These are fundamental concepts in fluid statics.
Step 2: Key Formula or Approach:
The three pressures are related by a simple formula: - Absolute Pressure (P): The total pressure at a point, including the pressure from the atmosphere above. - Atmospheric Pressure (\( P_a \)): The pressure exerted by the weight of the atmosphere. - Gauge Pressure (\( P_g \)): The pressure relative to the local atmospheric pressure. It is the pressure that most gauges are designed to measure. It can be positive (above atmospheric) or negative (below atmospheric, a vacuum). The relationship is: \[ P = P_a + P_g \] or \[ \text{Absolute Pressure} = \text{Atmospheric Pressure} + \text{Gauge Pressure} \] Step 3: Detailed Explanation:
We are asked to find the gauge pressure, \( P_g \). We can rearrange the formula from Step 2 to solve for \( P_g \). \[ P_g = P - P_a \] The gauge pressure at a depth h in a fluid of density \( \rho \) is also given by \( P_g = \rho g h \). This is the excess pressure over the atmospheric pressure.
Step 4: Final Answer:
The gauge pressure at the depth h is \( P - P_a \).