Empty Conical Tank: Use Torricelli's Principle to Find Time

In summary, using Torricelli's Principle and the equation for the volume of a cone, we can calculate the time it takes for a conical tank to empty when its initial depth is 2m and it has an outlet of cross sectional area 1.0cm². The time it takes is 6.18 seconds.
  • #1
Automaton2000
2
0

Homework Statement



Use Torricelli's principle to find the time it takes to empty a conical tank of circular cross section standing on its apex whose angle is 45° and has an outlet of cross sectional area 1.0cm². The tank is initially full of water and at time t = 0 the outlet is opened and the water flows out. The initial depth of the water in the tank is 2m.

Homework Equations



Torricelli's principle: √2gh
Volume of a Cone: V= (1/3) π r² h

The Attempt at a Solution



tank.jpg


where A_b is the area of the tank and A_o is the area of the hole
Is this the right conclusion and then substitute my given values into that equation to find the time when?
 
Physics news on Phys.org
  • #2
No, that is not the right conclusion. You need to use Torricelli's Principle to solve this problem. Using Torricelli's Principle, we can find the velocity of the water at any point in time. We know that the initial depth of the water in the tank is 2m, so the initial velocity of the water will be given by: v_o = √2gh = √2 x 9.8 x 2 = 28.3 m/s Now, we can use the equation for the volume of a cone to calculate the amount of water that has been discharged from the tank after a certain amount of time. The volume of the cone is given by: V = (1/3)πr²h Where r is the radius of the tank and h is the height of the water in the tank. Since the tank has a circular cross section, the area of the tank is given by A_b = πr² and the area of the outlet is given by A_o = 1 cm². We can calculate the rate of discharge from the tank by using the equation: Q = A_bv_o - A_ov Where v is the velocity of the water at any point in time. We can then use the equation for the volume of the cone and the rate of discharge to calculate the amount of time it takes for the tank to empty. The time it takes to empty the tank is given by: t = V/(A_bv_o - A_ov) Substituting in the values given, we get: t = (1/3)πr²h/(πr²v_o - A_ov) t = (1/3)h/(v_o - A_ov/πr²) t = (2/3)/(28.3 - 1/π x (1 cm²) / (π x (1m)²)) t = 6.18 seconds
 

1. What is Torricelli's Principle?

Torricelli's Principle states that the speed of a fluid flowing out of a hole in a container is equal to the speed that an object would have if it fell from the same height as the surface of the fluid to the hole.

2. How is Torricelli's Principle applied to an empty conical tank?

In an empty conical tank, the fluid is flowing out of a hole at the bottom of the tank. Using Torricelli's Principle, we can calculate the time it takes for the tank to empty by equating the speed of the fluid at the hole to the speed of an object falling from the top of the tank to the bottom.

3. What information is needed to use Torricelli's Principle to find the time for an empty conical tank?

To use Torricelli's Principle, we need to know the height of the tank, the radius of the hole at the bottom, and the acceleration due to gravity.

4. How does the shape of the tank affect the application of Torricelli's Principle?

The shape of the tank does not affect the application of Torricelli's Principle. As long as the tank has a hole at the bottom and the fluid is flowing out of the hole, we can use Torricelli's Principle to find the time it takes for the tank to empty.

5. Are there any limitations to using Torricelli's Principle for an empty conical tank?

Yes, there are a few limitations to using Torricelli's Principle for an empty conical tank. The fluid must be flowing out of a hole at the bottom of the tank and there should be no other forces acting on the fluid. Additionally, this principle assumes that the fluid is non-viscous and incompressible.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
Replies
3
Views
3K
  • Calculus and Beyond Homework Help
Replies
16
Views
3K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
3K
Back
Top