How do I prove That Toricelli's equation, given here is correct?

  • Thread starter Aliskahir
  • Start date
In summary, the student is looking for an equation that relates V1 and V2 in terms of the areas. By substituting V2 in for V1 in Bernoulli's equation, they find that V1 = V2 * A2 / A1. If they use P1 + 1/2 D V1^2 = P2 + 1/2 D (V1*A1 / A2) ^2, they find that V1 = Sqrt 2G(Y2-Y1) / (1-A1^2/A2^2).
  • #1
Aliskahir
6
0

Homework Statement



The problem is shown here, Problem # 48
Problem-1.jpg




Homework Equations



I know that V1 = Sqrt of 2*G (h)

But the A1^2 / A2^2 part is unknown to me, and I've spent hours trying to find an explanation, but now I'm here...

Can anyone point me in the right direction, please?
 
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  • #2
Go back to Bernoulli's Principle. Find a relationship between V2 and V1 in terms of the areas. Substitute for V2 in Bernoulli.
 
  • #3
In Bernoulli's equation I found that

V1 = V2 * A2 / A1
 
Last edited:
  • #4
Aliskahir said:
In Bernoulli's equation I found that

V1 = V2 * A2 / A1

Solve that for V2 in terms of V1 (since you're looking to replace V2 in Bernoulli's equation.

Can you write the equation for Bernoulli's principle for this case?
 
  • #5
(Using D for Density)

If I replace V2 With the equation I get

1/2 DV1^2 + DGY1 = 1/2D (A1*V1/A2)^2 + DGY2

By the way, I appreciate your help, I'm lost mostly because I missed this particular day of class, and I'm trying to understand a friend's notes and the book. My test tomorrow is going to kill me =[
 
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  • #6
Aliskahir said:
(Using D for Density)

If I replace V2 With the equation I get

P1 + 1/2 D V1^2 = P2 + 1/2 D (V1*A1 / A2) ^2

----hold on making revisions---

By the way, I appreciate your help, I'm lost mostly because I missed this particular day of class, and I'm trying to understand a friend's notes and the book. My test tomorrow is going to kill me =[

You're missing the potential energy term (D*g*h) on the left hand side. Remember, there's a height difference between the two locations. P1 and P2 are the ambient air pressure, which can be assumed equal at both locations, so they cancel. Solve for V1.
 
  • #7
1/2 DV1^2 + DGY1 = 1/2D (A1*V1/A2)^2 + DGY2

1/2 V1^2 (1-A1^2/A2^2) = G (Y2 - Y1)

V1 = Sqrt 2G(Y2-Y1) / (1-A1^2/A2^2)
 
  • #8
Yeah, I added it up top
 
  • #9
Is it... correct? or closer to being correct? :confused:
 
  • #10
Aliskahir said:
Is it... correct? or closer to being correct? :confused:

Can't you compare it with what is given in the problem statement?
 

FAQ: How do I prove That Toricelli's equation, given here is correct?

How do I understand Toricelli's equation?

Toricelli's equation is a mathematical expression that relates the height and velocity of a liquid in a container. It states that the velocity of a liquid flowing out of an opening in a container is equal to the square root of twice the gravitational acceleration (g) times the height of the liquid above the opening (h). This equation is derived from the principle of conservation of energy and is commonly used in fluid mechanics.

What is the significance of Toricelli's equation?

Toricelli's equation is important in understanding the behavior of fluids in containers, such as tanks, pipes, and dams. It is also used in various applications, such as designing water systems and predicting the flow rate of liquids. Additionally, it is a fundamental concept in fluid dynamics and is the basis for other equations, such as Bernoulli's equation.

How was Toricelli's equation derived?

Toricelli's equation was derived by Italian mathematician and physicist Evangelista Torricelli in the 17th century. He used the principle of conservation of energy to analyze the flow of water through a small opening in a container. By equating the potential energy of the water at the surface to the kinetic energy of the water flowing out of the opening, he arrived at the equation v = √(2gh), where v is the velocity, g is the gravitational acceleration, and h is the height of the liquid above the opening.

Can Toricelli's equation be applied to any type of liquid?

Toricelli's equation is applicable to any type of liquid, as long as the fluid is incompressible and the flow is steady and inviscid (no friction). This includes liquids such as water, oil, and even air. However, the equation may not be accurate for highly viscous fluids, such as honey, as they do not follow the assumptions of the equation.

How do I prove that Toricelli's equation is correct?

There are multiple ways to prove the correctness of Toricelli's equation. One way is to use experimental data and compare the predicted velocity using the equation with the actual measured velocity. Another way is to use mathematical proofs, such as using the laws of motion and the continuity equation. Toricelli's equation has been validated through numerous experiments and is widely accepted as an accurate representation of the relationship between liquid height and velocity.

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