Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: Pressure Forces exerted on and by liquids Pressure as a function of depth

2. Pressure … object exerting force -> pressure N Most useful for liquids or gases

3. Definition of pressure: where |N| is the magnitude of the (normal) force exerted on one object by another and A is the area of the surface at which the two objects are in contact.

4. Pressure in liquids (gases) Normal forces on element of liquid (gas) act in all directions – may vary with position – useful to define quantity at a point - pressure

5. Pressure in liquids (and gases): • Pressure is not a vector; (does not have a direction). • Normal force (N) is between two “objects” which can be adjacent regions of water in the same container. • Pressure in liquids is associated with a geometric point, not with an object or pair of objects.

6. Which of the three force probes measures the largest force? 1. A 2. B 3. C 4. All three probes measure equal forces.

7. Show experiment: Pressure probe Vary depth, orientation -- pressure increases with depth

8. Pressure as a function of depth in a liquid The pressure in a liquid increases linearly with depth from its value P0 at the surface that is open to the atmosphere or from some other reference point. The increase in pressure for every one meter increase in depth is proportional to the density of the liquid.

9. Comment -- Gases ? • Equally true for gases but since density is so much lower – usually neglect effect in lab scale containers • Everest …

10. Derivation … Mass M=volume times density M=rAh Weight force=rAhg Pressure p=force/area P= rhg h A

11. Comparing pressures at different points The difference in pressure between two points in a liquid depends on the difference of their depths below the open surface (i.e., below the points where P = P0.)

12. Rank the pressures at the three points 1. PA < PB = PC 2. PA < PB < PC 3. PA < PC < PB 4. PC < PA < PB

13. Important point … • Pressure at a point depends only on vertical distance to free surface – one can choose any path through the liquid to get there …

14. The water level in the right-hand tube is 1. higher than on left 2. lower than on left 3. equal to that on left 4. Can’t tell.

15. explanation • Levels will be equal. • Why ? A point just below one surface can be connected to the other surface. Pressure is same so net vertical distance must be same

16. Show experiment Communicating tubes or“Pascal’s Vases”

17. “Milk seeks its own level”

18. A 100-g mass is placed on a flat piece of plastic which is then held underneath a glass tube and submerged in a large container of water. The plastic plate and the mass are observed to stay attached to the glass tube. In order to make the plate and mass fall, would you have to raise or lower the tube? 1. Raise it. 2. Lower it. 3. Either will make the plate fall. 4. Neither will make the plate fall.

19. explanation • Net force on plate = rAhg • Must balance weight force = mg • I.e rAhg=mg height of water needed proportional to mass.

20. Three containers of different shape but with bases of equal area (or equal “footprints”) are filled with water to the same height. The weight of the water is the greatest in container… 1. A 2. B 3. C 4. The weight of the water is the same in all three containers.

21. Three containers of different shape but with bases of equal area (or equal “footprints”) are filled with water to the same height. The pressure at the bottom of the container is the greatest in container… 1. A 2. B 3. C 4. The pressure at the bottom is the same in all three containers.

22. “Hydrostatic Paradox” For differently shaped containers that have the same bottom surface and are filled with the same liquid up to the same height • the weights of the contents are different • the pressures at the bottom are the same

23. Resolving the hydrostatic paradox