Pressure Increase in a Liquid

Pressure in a fluid at rest changes with depth. This phenomenon is fundamental in fluid mechanics, particularly in hydrostatics. When considering a vertical column of liquid, the pressure at any point is due to the weight of the liquid column above it, plus any pressure applied to the surface.

Rate of Pressure Increase with Depth

Let's consider the pressure change as we move vertically downwards in a liquid. Imagine a small cylindrical element of liquid with cross-sectional area \(A\) and height \(dh\). If we move downwards by a small distance \(dh\), the increase in pressure, \(dP\), is due to the weight of this small liquid element.

The weight of the liquid element is given by:

\( \text{Weight} = (\text{mass}) \times (\text{acceleration due to gravity}) \)

Mass can be expressed as density (\(\rho\)) times volume. The volume of the cylindrical element is \(A \times dh\). So, the mass is \( \rho \times A \times dh \).

Weight \( = \rho \times A \times dh \times g \)

This weight acts on the area \(A\). The pressure increase \(dP\) is the force (weight) per unit area:

\( dP = \frac{\text{Weight}}{\text{Area}} = \frac{\rho \times A \times dh \times g}{A} \)

Simplifying this, we get:

\( dP = \rho \times g \times dh \)

The rate of increase of pressure in the vertical downward direction is \( \frac{dP}{dh} \).

\( \frac{dP}{dh} = \rho \times g \)

Connecting Rate of Pressure Increase to Liquid Properties

Now let's look at the options provided:

• Density (\(\rho\)): Density is mass per unit volume. It is part of the formula \( \frac{dP}{dh} = \rho \times g \), but the rate of pressure increase is equal to \( \rho \times g \), not just \( \rho \).
• Specific gravity: Specific gravity is the ratio of the density of a substance to the density of a reference substance (usually water at a specific temperature). It is a ratio and dimensionless, not equal to the rate of pressure increase.
• Specific volume: Specific volume is the volume per unit mass, which is the reciprocal of density (\(1/\rho\)). This is not equal to \( \rho \times g \).
• Specific weight (\(\gamma\)): Specific weight is defined as the weight per unit volume. Mathematically, specific weight (\(\gamma\)) is given by \( \gamma = \rho \times g \), where \( \rho \) is the density and \(g\) is the acceleration due to gravity.

Comparing the derived rate of pressure increase \( \frac{dP}{dh} = \rho \times g \) with the definitions of the properties, we see that the rate of increase of pressure in a vertical downward direction is equal to the specific weight (\(\gamma\)) of the liquid.

\( \frac{dP}{dh} = \gamma \)

Therefore, the rate of increase of pressure in a vertical downward direction is equal to the specific weight of the liquid.