Show that liquid in a container exerts pressure equal to . What is the effect of depth on liquid pressure?

The pressure exerted by a liquid in a container depends on both the depth of the liquid and the density of the liquid. Let me explain how this works and show the relationship step-by-step.

Liquid Pressure Formula:

The pressure exerted by a liquid at a given depth can be calculated using the following formula:

P=ρgh

Where:

• $P$ is the pressure at depth $h$,
• $\rho$ is the density of the liquid,
• $g$ is the acceleration due to gravity (approximately 9.8 m/s²),
• $h$ is the depth of the liquid from the surface.

Derivation and Explanation:

The liquid exerts pressure on the walls of the container and on any object submerged in it. This pressure comes from the weight of the liquid above the point where the pressure is being measured.

1. Consider a small section of the liquid:

   • Imagine a small horizontal column of liquid at a depth $h$, with a small cross-sectional area $A$.
   • The weight of this small column of liquid can be expressed as: Weight=ρ⋅A⋅h⋅g where $\rho$ is the density of the liquid, $A$ is the cross-sectional area of the column, $h$ is the depth, and $g$ is the acceleration due to gravity.

2. Pressure is defined as force per unit area:

   P=Force/Area​

   So, the pressure at a depth $h$ is:

   P=ρ⋅A⋅h⋅g​

   Simplifying:

   P=ρgh

   This equation shows that the pressure in a liquid at a depth $h$ depends on the density of the liquid, the acceleration due to gravity, and the depth itself.

Effect of Depth on Liquid Pressure:

From the formula P=ρgh, it’s clear that:

1. Pressure increases with depth:

   • As the depth $h$ increases, the pressure $P$ increases linearly. This is because the deeper you go, the more liquid there is above you, which results in more weight and thus more pressure.

   Example:

   • If you dive deeper into a swimming pool, you’ll feel more pressure on your ears, as the pressure increases with depth.

2. Pressure is independent of the shape of the container:

   • The liquid pressure at a given depth is independent of the container’s shape. Whether the container is wide or narrow, the pressure at depth $h$ will be the same, as long as the liquid is the same.

3. Pressure is proportional to liquid density:

   • The greater the density $\rho$ of the liquid, the higher the pressure for a given depth. For example, at the same depth, the pressure in mercury (with a much higher density than water) would be much greater than in water.