Hydrostatic Law Explained

The question asks about the specific law that describes the relationship between the rate of change of pressure in the vertical direction within a fluid and the fluid's weight density at a given point. Let's examine the options provided.

• Pascal’s Law: This law states that pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid and the container walls. This relates to pressure transmission, not its change with depth or vertical direction.
• Newton’s Law: This refers to various laws formulated by Isaac Newton, such as laws of motion or law of universal gravitation. These laws are fundamental in physics but do not specifically govern the vertical pressure gradient in a fluid at rest.
• Hydrostatic Law: This law deals with fluids at rest (hydrostatics). It states that the rate of increase of pressure in the vertical direction in a fluid is equal to the weight density of the fluid at that point. Weight density is defined as the weight per unit volume, which is equal to the product of mass density ($\rho$) and acceleration due to gravity ($g$). Mathematically, the hydrostatic law can be expressed as: $$ \frac{dp}{dy} = \rho g = \gamma $$ where:
  • $p$ is the pressure
  • $y$ is the vertical distance (often measured upwards)
  • $\rho$ is the mass density of the fluid
  • $g$ is the acceleration due to gravity
  • $\gamma$ is the weight density of the fluid
  If $y$ is measured downwards, the rate of increase of pressure would be positive. This law directly describes the relationship mentioned in the question.
• Hooke’s Law: This law relates the force applied to a spring or elastic material to the resulting displacement or deformation. It is relevant to solid mechanics, not fluid pressure at rest.

Based on the analysis of each option, the law that precisely states that the rate of increase of pressure in a vertical direction is equal to the weight density of a fluid at a point is the Hydrostatic Law.