11.2

Pressure

People who have fixed a flat tire know something about pressure. The final step in the job is to reinflate the tire to the proper pressure. The underinflated tire is soft because it contains an insufficient number of air molecules to push outward against the rubber and give the tire that solid feel. When air is added from a pump, the number of molecules and the collective force they exert are increased. The air molecules within a tire are free to wander throughout its entire volume, and in the course of their wandering they collide with one another and with the inner walls of the tire. The collisions with the walls allow the air to exert a force against every part of the wall surface, as Figure 11-1 shows. The pressure P exerted by a fluid is defined in Section 10.7 (Equation 10.19) as the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts:

| (11.3)

Figure 11-1 In colliding with the inner walls of the tire, the air molecules (blue dots) exert a force on every part of the wall surface. If a small cube were inserted inside the tire, the cube would experience forces (blue arrows) acting perpendicular to each of its six faces.

The SI unit for pressure is a newton/meter2 (N/m2), a combination that is referred to as a pascal (Pa). A pressure of 1 Pa is a very small amount. Many common situations involve pressures of approximately 105 Pa, an amount referred to as one bar of pressure. Alternatively, force can be measured in pounds and area in square inches, so another unit for pressure is pounds per square inch (lb/in.2), often abbreviated as “psi.”
Because of its pressure, the air in a tire applies a force to any surface with which the air is in contact. Suppose, for instance, that a small cube is inserted inside the tire. As Figure 11-1 shows, the air pressure causes a force to act perpendicularly on each face of the cube. In a similar fashion, a liquid such as water also exerts pressure. A swimmer, for example, feels the water pushing perpendicularly inward everywhere on her body, as Figure 11-2 illustrates. In general, a static fluid cannot produce a force parallel to a surface, for if it did, the surface would apply a reaction force to the fluid, consistent with Newton’s action–reaction law. In response, the fluid would flow and would not then be static.

Figure 11-2 Water applies a force perpendicular to each surface within the water, including the walls and bottom of the swimming pool, and all parts of the swimmer’s body.

While fluid pressure can generate a force, which is a vector quantity, pressure itself is not a vector. In the definition of pressure, , the symbol F refers only to the magnitude of the force, so that pressure has no directional characteristic. The force generated by the pressure of a static fluid is always perpendicular to the surface that the fluid contacts, as Example 2 illustrates.

| Example 2 | | | The Force on a Swimmer

Suppose the pressure acting on the back of a swimmer’s hand is , a realistic value near the bottom of the diving end of a pool. The surface area of the back of the hand is. (a) Determine the magnitude of the force that acts on it. (b) Discuss the direction of the force.
Reasoning From the definition of pressure in Equation 11.3, we can see that the magnitude of the force is the pressure times the area. The direction of the force is always perpendicular to the surface that the water contacts.
Solution

a. | A pressure of is . From Equation 11.3, we find

This is a rather large force, about 230 lb.
b. | In Figure 11-2, the hand (palm downward) is oriented parallel to the bottom of the pool. Since the water pushes perpendicularly against the back of the hand, the force is directed downward in the drawing. This downward-acting force is balanced by an upward-acting force on the palm, so that the hand is in equilibrium. If the hand were rotated by 90°, the directions of these forces would also be rotated by 90°, always being perpendicular to the hand.

Problem solving insight Force is a vector, but pressure is not.

A person need not be under water to experience the effects of pressure. Walking about on land, we are at the bottom of the earth’s atmosphere, which is a fluid and pushes inward on our bodies just like the water in a swimming pool. As Figure 11-3 indicates, there is enough air above the surface of the earth to create the following pressure at sea level:

Figure 11-3 Atmospheric pressure at sea level is , which is sufficient to crumple a can if the inside air is pumped out.

This amount of pressure corresponds to 14.70 lb/in.2 and is referred to as one atmosphere (atm). One atmosphere of pressure is a significant amount. Look, for instance, in Figure 11-3 at the results of pumping out the air from within a gasoline can. With no internal air to push outward, the inward push of the external air is unbalanced and is strong enough to crumple the can.
The physics of lynx paws. In contrast to the situation in Figure 11-3, reducing the pressure is sometimes beneficial. Lynx, for example, are well suited for hunting on snow because of their oversize paws (see Figure 11-4). The large paws act as snowshoes and distribute the weight over a large area. Thus, they reduce the weight per unit area, or the pressure that the cat applies to the surface, which helps to keep it from sinking into the snow.

Figure 11-4 Lynx have large paws that act as natural snowshoes. (Peter Weimann/ Animals Animals/Earth Scenes)

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.