# Lateral pressure of a liquid exerted on a plunger

• A13235378
In summary, the conversation discusses the correct expression for calculating pressure force in a rectangular plate under certain simple circumstances. The force can be calculated by taking the average of the pressure at the top and bottom of the plate and multiplying it by the width and height of the plate. This only applies when the pressure variation is linear and the width of the plate is constant. Otherwise, the general expression must be used.
A13235378
Homework Statement
Consider a liquid exerting a force on a plunger. Find that force

I saw somewhere that I don't remember anymore that you can use medium force, specifically medium pressure:

(P up + P down) / 2

I also saw the following expression:

dF = (P + ugh) dS that represents the force exerted by each minuscule piece.

I was wondering if those two expressions are right and how can I start from the second and get to the first. In short, I wanted to better understand the concept of average quantities. In speed for example, I can use ( Vinitial+ Vfinal) / 2 to find the average speed, that is, when I can do a simple arithmetic medium.
Relevant Equations
P2 = P1 + ugh , u = Especific mass.

The second expression is correct and applies in general, but it only leads to the first in certain simple circumstances, such as a rectangular plate with a horizontal base.

etotheipi
To complement what @haruspex said, orient the rectangular plate, of width ##w## and height ##h##, in the ##x##-##y## plane, and consider the fluid to occupy ##z < 0##. The pressure force will be$$F_z = \int_0^w \int_0^h P_0 - \rho g y \, dy \, dx = \int_0^w P_0 h - \frac{1}{2}\rho g h^2 \, dx = P_0 h w - \frac{1}{2}\rho g h^2 w$$The pressures you labelled would be ##P_{down} = P_0## and ##P_{up} = P_0 - \rho g h##. So looking at the average pressure, the force can be written as $$\frac{P_{up} + P_{down}}{2} homework = P_0 homework - \frac{1}{2} \rho g h^2 w$$which is the same as we obtained with the general expression. This only occurs here because the pressure variation is linear with depth and because the width of the lamina is constant. In general it will not hold, in just the same way that something like SUVAT will not hold unless certain criteria are met!

## 1. What is lateral pressure of a liquid?

The lateral pressure of a liquid refers to the force exerted by a liquid against a surface that is parallel to the direction of the liquid's flow. This force is perpendicular to the surface and is caused by the weight of the liquid and the acceleration due to gravity.

## 2. How is lateral pressure of a liquid calculated?

The lateral pressure of a liquid can be calculated using the formula P = ρgh, where P is the pressure, ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height of the liquid. This formula assumes that the surface area is large enough to be considered flat and that the liquid is incompressible.

## 3. What is the unit of measurement for lateral pressure of a liquid?

The unit of measurement for lateral pressure of a liquid is typically expressed in pascals (Pa), which is equivalent to one newton per square meter (N/m²). However, other units such as pounds per square inch (psi) or atmospheres (atm) may also be used.

## 4. How does the shape of a plunger affect the lateral pressure of a liquid?

The shape of a plunger can affect the lateral pressure of a liquid as it determines the surface area that the liquid is exerting force on. A larger surface area will result in a greater lateral pressure, while a smaller surface area will result in a lower lateral pressure.

## 5. What are some real-world applications of lateral pressure of a liquid?

The lateral pressure of a liquid has many practical applications, such as in hydraulic systems, where it is used to transfer force from one point to another. It is also important in the design of dams and other structures that must withstand the pressure of large bodies of water. Additionally, understanding lateral pressure is crucial in industries such as oil and gas, where it is used to control the flow of fluids through pipelines and other equipment.

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