1. Air Pocket 8th problem

2. A vertical air jet from a straw produces a cavity on a water surface. What parameters determine the volume and the depth of the cavity? The Problem

3. The Apparatus We used a square-shaped aquarium and a compressor to reproduce the phenomenon. We measured the depth and the width at the surface We changed the velocity of the air jet and the height of the straw Two pipes with different cross-sections were used

4. Speed of Air Jet Neglecting the difference between the heights of the two points: Bernoulli’s equation is:

5. The Cavity We approximated the shape of the cavity with a paraboloid A paraboloid is a parabola rotated about the y axis We measured cavity width, depth, as the function of pipe height and air speed

6. Cavity Parameters as Functions of Pipe Height

7. Cavity Parameters as Functions of Air Speed

8. Cavities with different pipe cross-sections

9. Volume of the Cavity A paraboloid is a parabola rotated about the y axis. Its volume equals its inverse function’s volume rotated about the x axis We need to calculate the volume of a solid of revolution Inverse function of the parabola

10. Volume of the Cavity

11. Volume of the cavity Parameter ‘A’ contains the rate of the width and height of the cavity, in the following way:

12. Volume of the cavity The volume of a solid of revolution: The volume of the rotated inverse parabola: This is the volume of the original paraboloid, too

13. Measured Volumes

14. Work Needed for the Formation of the Cavity We calculated this work in two ways First method: Using that, where F is the buoyancy and Using that, where F is the buoyancy and Like dipping the paraboloid from the surface to h 0 depth continually: Like dipping the paraboloid from the surface to h 0 depth continually:

15. Work Needed for the Formation of the Cavity We get Substituting back :

16. Another Method for Determining the Work When the cavity is created, the mass centre of the water which filled the cavity will move to the surface of the water, which means that its potential energy will rise: The needed work equals this change of energy:

17. The problem is then, to determine Δh Assuming that ρ is constant, Δh is the difference of the h 0 (height of the water surface) and the h m mass centre of the paraboloid: h m can be calculated from the geometry of the paraboloid Another Method for Determining the Work

18. We calculated the centre of mass of the paraboloid by this formula: We got: Another Method for Determining the Work

19. Substituting back to the equation with the work: Another Method for Determining the Work

20. Calcuating Efficiency Efficiency is the rate of input work and and useful work Input work is the mechanical energy change of air slowing down at the surface Useful work is the work done while creating the cavity

21. Calculating Efficiency Efficiency is the rate of the two: Air fill the cavity continously; its mass is

22. Efficiencies Substituting:

23. Volumes and efficiencies of different cross-sections

24. Conclusions

25. The measurements Cavity width, depth As the function of: Speed of the air jet Speed of the air jet Height of the pipe Height of the pipe We measured two types of pipem, with different cross-sections We measured two types of pipem, with different cross-sections