Question

Which of the following formula is used to calculate the hydrostatic pressure at a given depth in a fluid?

• A. P = mv
• B. P = W/t
• C. P = pgh
• D. P = F/A

Hint:

Hydrostatic pressure:
\[ P = \rho g h \] It depends only on:
- Fluid density (\(\rho\))
- Gravity (\(g\))
- Depth (\(h\))
It is completely independent of the total volume or shape of the container.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity acting on its mass.
It increases linearly with depth because of the increasing weight of the fluid column above.
Key Formula or Approach:
The pressure \(P\) exerted by a vertical column of fluid of height \(h\) and density \(\rho\) is:
\[ P = \rho g h \] Where:
- \(\rho\) is the density of the fluid (\(\text{kg/m}^3\))
- \(g\) is the acceleration due to gravity (\(\approx 9.81 \text{ m/s}^2\))
- \(h\) is the depth of the fluid column (m)

Step 2: Detailed Explanation:
Let us derive this from the basic definition of pressure:
\[ P = \frac{\text{Force}}{\text{Area}} = \frac{\text{Weight of fluid column}}{\text{Area}} \] The mass \(m\) of the fluid column of height \(h\) and cross-sectional area \(A\) is:
\[ m = \text{Density} \times \text{Volume} = \rho \times (A \times h) \] The force (weight, \(W\)) exerted by this mass is:
\[ F = m \times g = \rho \times A \times h \times g \] Substituting this into the pressure equation:
\[ P = \frac{\rho A h g}{A} = \rho g h \] Let us review the other options:
- \(P = mv\) is the formula for linear momentum (where \(P\) is momentum).
- \(P = W/t\) is the formula for power (work done per unit time).
- \(P = F/A\) is the general definition of mechanical pressure, but it is not specific to the depth of a fluid column.
Therefore, the specific formula for hydrostatic pressure at a depth is \(P = \rho g h\) (often written as \(P = pgh\)).

Step 3: Final Answer:
The formula used to calculate hydrostatic pressure at a depth is \(P = \rho g h\).