Liquid Pressure. Pascal's Principle

we can use the alternative formula

for calculating the liquid pressure. This formula offers some advantages compared to the standard formula of pressure such as

1. No area is involved in the calculations. This means we don't have to remember all geometric formulae for area of figures or to find the area of irregular shapes by integration.
2. The shape of container is not important for the same reason.
3. It is not necessary to find the weight of liquid. This advantage is especially important when dealing with objects immersed in seawater as we don't need to calculate the mass of the entire seawater.
4. It is enough if we know the density of liquid and the depth of the immersed object from the surface to calculate the liquid's pressure. Hence, it is enough a small sample of liquid to calculate its density experimentally. As for the liquid's depth, we can use sensors that use sound echo to calculate it.

In this way, we obtain an important property of liquid pressure:

"Liquid's pressure depends only by depth if the density of liquid is known."

In other words, liquid pressure is the same everywhere at a certain depth, regardless its position within the container.

It must me noted that:

1. The quantity in the formula P_liquid = ρ_liquid × g × h shows the depth from the surface, not the height from the ground as in Kinematics.
2. The shape and size of the base area do not affect the magnitude of liquid pressure as long as the depth is the same. Look at the figure below.

Water pressure is otherwise known as "gauge pressure" as we often use a pressure gauge to measure it.

We can use the formula of liquid pressure to calculate the density of an unknown liquid by using a U-shaped tube in which we pour two non-mixable liquids. We measure the levels starting from the meeting position and then, use the equation

to calculate the density of an unknown liquid.

Pressure in liquids obeys to a universal rule, known as "Pascal's Principle" which says:

"In a fluid at rest in a closed container, a pressure change in one part is transmitted without loss to every portion of the fluid and to the walls of the container."

This principle is particularly useful in communicating vessels, in which we can use a small force in a small area to transmit pressure in a larger area and in this way, to increase the value of force.

The equation that represents mathematically the Pascal's Principle is