Optional Unit VI: Fluid Mechanics
B. Pressure

Key Concepts

Pressure is the force acting perpendicular to a given surface area.

Pressure is a scalar quantity, acting in all directions.

The SI fundamental unit for pressure is N/m². The derived unit is the pascal (Pa).

Meteorologists express pressure in millibars.

1 mb = 0.10 kPa

1 Pa = 1 N/m², 1 kPa = 1 x 10³ Pa

One pascal is a relatively small amount of pressure, roughly equivalent to the pressure exerted by a five dollar bill on a level surface.

Standard atmospheric pressure is 1.013 x 10⁵ Pa Pa, or 101.3 kPa.

Atmospheric pressure decreases at higher elevations.

Standard atmospheric pressure will support a column of mercury 760 mm high. This can be expressed as a pressure of 760 mm Hg, or 760 Torr. A barometer is used to measure atmospheric pressure.

The pressure (P) exerted by an incompressible fluid is directly proportional to the depth in the fluid (h), its density (), and the gravitational field strength (g). The pressure is independent of the total volume or the shape of the container.

Absolute pressure (P_(abs)) is the pressure measured above zero pressure. Gauge pressure(P_G) is a measure of the difference between absolute pressure and atmospheric pressure (P_(atm)).

P_abs = P_G + P_atm)

Learning Outcomes

1. Define the following terms: pressure, standard atmospheric pressure, barometer, absolute pressure, gauge pressure.

2. Demonstrate an understanding of accepted usage of units for pressure.

3. Recognize other non-SI units for pressure that are in common use.

4. Recognize that the historical development of ideas in science has often led to the adoption of accepted standards and conventions. (e.g., Torricelli's mercury column used to measure atmospheric pressure.)

5. Solve problems related to pressure in fluids.

6. Distinguish between absolute pressure and gauge pressure.

Teaching Suggestions, Activities and Demonstrations

1. Design a U-tube manometer to test the pressure on an object in a liquid at different depths below the surface. Compare the results obtained with tap water to a different fluid such as salt water.

2. Perform an activity to estimate the pressure exerted by various different objects, and then measure or calculate the actual pressure to determine the reasonableness of the estimates.

3. Design an experiment to investigate the relationship between the air pressure in a bicycle tire and the contact area between the tire and the ground. (Slight variations in tread patterns are evident when rolling or standing a fully loaded tire on a sheet of white paper.)

   As an added challenge, students might want to try to investigate the effects of air pressure on the rolling resistance of an inflated bicycle tire. Part of the challenge of such an assignment is to figure out a way of setting up the experiment so that meaningful data can be obtained.

4. Drill three small holes at different heights in a can. (Holes can be punched out using a nail. Beware of any sharp protrusions that result.) Place the can under a tap and pour water into the can so that a steady state of water flow into the can and out of the can (into a sink) is reached. (This helps to show the difference between equilibrium and steady state.)

   Measure the horizontal distance that the water jets travel from the holes to the sink below. (Projectile motion! Some interesting possibilities for integration with other physics topics exist.) Try to develop a relationship between the height of the water column above the hole and the force exerted on the water as it leaves the can. This shows that pressure increases at greater depth in water.

5. Perform the necessary measurements of surface area to determine the pressure exerted by a flat- bottomed shoe and a high-heel tip on the floor. Compare this to the pressure exerted when wearing cross-country skis or snowshoes. Use these ideas to explain how skis and snowshoes work.

6. Place a one-holed stopper in a pop bottle. Run a tube through the stopper. Invite a student to take a sip from the bottle. Liquid can not be drawn up through the straw in this way, since sealing off the inside the bottle will reduce the inside pressure when one tries to draw liquid up the straw.

   A variation is to use a two-holed stopper, keeping one finger over the empty hole on the stopper while trying to draw liquid up the straw. Once the empty hole is opened, the liquid can be sucked up the straw normally.

7. Using a thick plastic tube (1 cm in diameter), secure two balloons on either end. Have one balloon partially filled, and the other nearly completely filled. Beforehand, ask the students to predict what will happen. Most of them will likely predict that air will flow from the more full balloon to the less full balloon to "equalize pressure." Actually, the smaller balloon will loose some of its air to the one that has more air in it, to minimize surface area.

   (This has some interesting aspects to it. If one balloon remains less full, students may suspect that it is because that balloon is not as elastic, perhaps because it has not been pre-stretched by filling it. If the air from the full balloon is ssqueezed gently, the empty balloon will fill, and the one that was full will remain empty. It is very difficult to form a concept of what is happening in this demonstration.)

8. If you like theatrics, then the nail-bed demonstration is one which students enjoy. Prepare a nail bed by nailing lots of nails, evenly spaced, into a board, long and wide enough so that a person can lie face-up over the entire board.

   One can lie on the nail bed safely, because the weight is distributed over the entire nail bed. The pressure on each point is fairly small. (Calculations can be done beforehand to confirm this, to relieve any anxiety over trying it.) A variation on this theme is to use a second nail bed, and sandwich the teacher in between. A concrete block can be placed on top of the upper nail bed and broken with a sledge hammer.

   (Caution: This demonstration should only be performed by the teacher under careful supervision. It is not safe if any of the nails are sticking up higher than the others, or if someone happens to slip and fall on it, bearing all their weight down on only a small surface of the body! Exercise extreme care. It may be better to have another teacher instead of a student using the sledge hammer!)

9. Connect two heavy masses to a piece of piano wire. Hang the masses over a block of ice. The wire will melt through the ice due to the increase in pressure. Once the wire passes through the ice, the refrozen line where the wire has passed can be seen.

10. Using a set of Magdeburg hemispheres, connect them to a vacuum pump. Evacuate the inside. Seal the stopcock. Once most of the air has been removed and the external pressure exceeds the internal pressure, the hemispheres will be very difficult to separate.

    (Rubber cup-type versions are also available. An inexpensive alternative which illustrates the same thing is to use two toilet plungers having the same diameter. Squeeze them together. Pulling them apart will be difficult.

    A classic demonstration involves heating an empty spirit duplicating fluid can, sealing it, and allowing air pressure to crush the can. These types of cans are no longer commonly found in schools. An alternative is to use the small cardboard juice containers that are supplied with the straw taped to the side. Heat the empty container by pouring hot water inside. Pour out the water, then seal the opening with tape.)

11. Place uninflated balloons 30 cm apart around the perimeter of a rectangular table. Invert a second table face down on the one below, so that the open ends of the balloons are all protruding out from between the tables. Invite one student to sit in the middle of the inverted table. Get as many volunteers as there are balloons. At the same time, have each volunteer blow into a balloon, to see if they can lift the inverted table with the person sitting on it.

12. Place a thick lead strip over the edge of a desk. At least 10 cm of the lead strip should protrude beyond the edge of the desk. Observe the metal after about one hour. The lead will fatigue and droop down under its own weight.