## Classical Mechanics: Lecture 27 Notes

**Overview**

Today is a 49 minute lecture on hydrostatics and fluid mechanics.

**Details**

Ignoring gravity, a compartment filled with fluid. It has area A at the top, a force F is applied.

Pressure = Force/Area in N/m^2 or Pascals

The pressure is everywhere the same. This is Pascal’s principle. The pressure is transmitted undiminished everywhere; it has no direction, it is a scalar. The force is everywhere perpendicular to the wall. If not, it would begin to move.

dF/dA = P

Hydraulic jacks use this principle. A force on one area F1. The pressure will be F1/A1.

F1/A1 = F2/A2

If A2/A1 = 100, then the forces can be different. There is no violation of conservation of energy. The volume of fluid is the same and does not change.

W = Fd

F1d1 = F2d2

Work in = Work out

Gravity has an effect. At the bottom of the oceans, pressure is high. For a slab on the ocean floor:

Mass = Area x Density

F1 up -F2 down – mg = 0

– density x gravity = dPressure/dheight

This is hydrostatic pressure, due to gravity. Most liquids are practically incompressible. You cannot change the density of a liquid.

Demonstration: A paint can filled with water and another with air. The air acts like a cushion. The force is far higher when hit with the water case.

If a paint can shot with a bullet has water in it, there is a high force and the small area of the bullet. The paint can with air will not explode due to the cushion effect. Clearly he could not bring his pellet gun to class in the year 2012.

Liquids are completely incompressible.

Pascal’s Law

P2 – P1 = dP = – d g (y2-y1)

Fill a vessel with water. The pressure on the bottom is the same for all points.

Weight = Area x height x density x gravity

This is a force. We live at the bottom of an ocean of air. Liquids are linear because the density is the same.

1 kg of force/cm^2

1 Newton/cm^2

10^5 Pascals

…is what the air weights from the sea level to the top of the atmosphere. This is also called barometric pressure. A hand has an area of 150 cm^2.

To measure atmospheric pressure, immerse a hose in a liquid. Lift it out and observe the liquid’s level. Pull out enough and it will empty. Measure the length of hose you can pull out. You must put your finger on top.

P1 – P2 = d g h

P2 = 0

P1 = dgh

Know g, h, and d and you’ve got pressure. Mercury has a density of 13.5 x 10^3 kg/m^3 with h of .76 meters.

P1 = 1.03 x 10^5 Pascals

When there is 760 mm of Hg, it is one atmosphere. High pressure is good weather, low pressure is bad weather. Pascal is reputed to have done this with wine.

Every 10 meters down in the ocean, pressure increases 10 times. Submarines can go 900 meters down, 90 atmospheres. The inside of the vessel is 1 atm, the outside is much higher, there is an overpressure out there.

The next demonstration is pumping the air out of a paint can. The pressure inside is smaller than outside.

If scuba diving with a 30 meter long tube, you cannot breathe with the tube if 30 meters down.

A manometer is a tube with liquid inside, open at both ends. One level falls, one level rises. The height difference is h.

P1 – P2 = d h g

where P2 is 1 atm.

You can snorkel at maybe 1 meter. Don’t make the liquid oscillate, push hard. You can also measure underpressure by sucking. It’s about the same. Underwater, its easy to let air out. To expand lungs, to raise your chest is hard – this determines the depth you can snorkel.

Could you steal beer from a 1st story person if you were at the 2nd floor? He gets up on the top of the stage and does this. The solution has something to do with the sealed top.

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