Derive Bernoulli’s Theorem

In this blog post, we will discuss the famous Bernoulli theorem and how to derive it. This theorem is a very important tool for students and is used in many different applications. We will start by discussing the basic definition of the derivative and then move on to the differentiation formulas of Bernoulli’s theorem. Stay tuned, because this is going to be a really exciting journey.

What is Bernoulli’s theorem under physics?

Bernoulli’s theorem is a concept that applies to fluid dynamics to check the pressure exerted on a liquid or gas that flows through an enclosed space. The theorem is also used in probability theory where it states that if two events are independent, then the probability of one event happening is unaffected by the other event. It can be explained well with the help of its differentiation formulas.

History of Bernoulli’s theorem

Bernoulli’s theorem was first discovered by Swiss mathematician Jakob Bernoulli in 1654. He was trying to find a way to calculate the velocity of a fluid flowing through a curved tube. His work was published posthumously in 1738.

The theorem is named after him because he was the first to discover it. The theorem is a statement about the Conservation of Energy. It states that the total energy of a system is constant. The theorem is also known as the Law of Energy Conservation. The theorem was later generalized by Euler who showed that it was true for all fluids.

What is the differentiation formula under Bernoulli’s theorem?

Here is the bernoulli equation:

p+1/2V2+gh=constant

where:

P is the pressure

is the density

V is the velocity

h  is the elevation

G is the gravitational accelarartion

Bernoulli’s theorem explains that for an inviscid flow when there is an increase in the velocity of the fluid the pressure or potential energy decreases simultaneously.

Is Bernoulli’s theorem related to the conservation of energy?

The answer is yes! Bernoulli’s theorem is a direct consequence of the conservation of energy. To see, let’s start by recalling the definition of Bernoulli’s theorem.

Bernoulli’s theorem states that if a fluid is flowing through a pipe of varying cross-sectional area, the pressure at any point in the pipe is given by:

Here is the bernoulli equation:

p+1/2V2+gh=constant

where:

P is the pressure

 is the density

V is the velocity

H is the elevation

G is the gravitational accelarartion

Now, let’s take a look at the conservation of energy. The law of conservation of energy explains that the total amount of energy in a closed system is constant. In other words, energy cannot be created nor be destroyed.

The total amount of energy in a system can be classified into two categories that include kinetic energy and potential energy. Kinetic energy defines the energy of motion whereas potential energy explains the energy that gets stored in a system due to its position or configuration.

For example, consider a ball rolling down a hill. The potential energy of the ball is converted into kinetic energy as the ball rolls downhill.

In a closed system, the total amount of energy (kinetic + potential) is always constant. This means that if the potential energy decreases, the kinetic energy must increase by the same amount.

Now, let’s apply the law of conservation of energy to Bernoulli’s theorem. We’ll start with the left-hand side of the equation:

p+1/2V2+gh=constant

where:

P is the pressure

 is the density

V is the velocity

h is the elevation

G is the gravitational acceleration

Now, let’s take a look at the right-hand side of the equation. The right-hand side is a constant, which means that it represents the total amount of energy in the system. This means that the left-hand side of the equation, p+12V2+gh=constant, must also represent the total amount of energy.

What are the assumptions made in Bernoulli’s theorem?

There are certain assumptions made including:

– that the fluid is inviscid

– that the flow is steady

– that there are no sources or sinks in the fluid

– that the fluid is incompressible

– that no external forces are acting on the fluid

– that the flow is irrotational.

Conclusion

Bernoulli’s theorem explains that for an inviscid flow when there is an increase in the velocity of the fluid the pressure or potential energy decreases simultaneously. This principle is often applied in aerodynamics and hydrodynamics. In physics classrooms across the globe, students are learning about Bernoulli’s theorem and how to apply it in problem-solving situations. This article deals with the differentiation formula, its derivative, and applications. We hope this article has given you a better understanding of what Bernoulli’s theorem is and how it can be used to solve problems. If you’re interested in learning more, we suggest checking out some of the resources we’ve linked below. And as always, feel free to reach out to us if you have any questions!