Pressure

of Pressure with Depth. Pressure Measurements. Buoyant Forces and Archimedes's

Principle. Fluid Dynamics. Bernoulli's Equation.

randomly arranged and held together by weak cohesive forces and

by forces exerted by the walls of a container. Both liquids and gases are fluids.

In our treatment of the mechanics of fluids, we do not need to learn any new physical principles to explain such effects as the buoyant force acting on a submerged object and the dynamic lift acting on an airplane wing. First, we consider the mechanics of a fluid at rest—that is, fluid statics. We then treat the mechanics of fluids in motion— that is, fluid dynamics. We can describe a fluid in motion by using a model that is based upon certain simplifying assumptions.

submerged object, the force exerted by the fluid is perpendicular

to the surface of the object. The force exerted by the fluid on the walls of the container is perpendicular to the walls at all points.

Pressure

by a fluid.
If F is the magnitude of the force

exerted on the piston and A is the surface area of the piston, then the pressure P of the fluid at the level to which the device has been submerged is defined as the ratio F/A:

(7.1)

Note that pressure is a scalar quantity because it is proportional to the magnitude of the force on the piston.

the infinitesimal force dF on an infinitesimal surface element of

area dA as

where P is the pressure at the location of the area dA. The pressure exerted by a fluid varies with depth. Therefore, to calculate the total force exerted on a flat vertical wall of a container, we must integrate Equation 7.2 over the surface area of the wall.

(7.2)

Because pressure is force per unit area, it has units of newtons per square meter (N/m2) in the SI system. Another name for the SI unit of pressure is pascal (Pa):

(7.3)

spread the downward force you exert on the snow over

a large area, reducing the pressure on the snow surface.

(darker region) in a larger volume of fluid is singled

out. The net force exerted on the parcel of fluid must be zero because it is in equilibrium.

a depth h below a point in the liquid at

which the pressure is P0 is greater by an amount ρgh.

(7.4)

is the pressure at the surface of the liquid, then

P0 is atmospheric pressure. In our calculations and working of end-of-chapter problems, we usually take atmospheric pressure to be

In view of the fact that the pressure in a fluid depends on depth and on the value of P0, any increase in pressure at the surface must be transmitted to every other point in the fluid. This concept was first recognized by the French scientist Blaise Pascal (1623–1662) and is called Pascal’s law: a change in the pressure applied to a fluid is transmitted undiminished to every point of the fluid and to the walls of the container.

increase in pressure is the same on the two sides,

a small force Fl at the left produces a much greater force F2 at the right.

increase in pressure is the same on the two sides,

a small force Fl at the left produces a much greater force F2 at the right. (b) A vehicle undergoing repair is supported by a hydraulic lift in a garage.

P - P0 is equal to ρgh. The pressure P

is called the absolute pressure, while the difference P - P0 is called the gauge pressure. For example, the pressure you measure in your bicycle tire is gauge pressure.

to push a beach ball underwater. (b) The forces on

a beach ball–sized parcel of water. The buoyant force B on a beach ball that replaces this parcel is exactly the same as the buoyant force on the parcel.

(b)

(a)

object is called a buoyant force.
Buoyant Forces and Archimedes’s Principle
The

magnitude of the buoyant force always equals the weight of the fluid displaced by the object. This statement is known as Archimedes’s principle.

on the cube of liquid are the gravitational force Fg

and the buoyant force B. Under equilibrium conditions, B = Fg .

(7.5)

on the cube of liquid are the gravitational force Fg

and the buoyant force B. Under equilibrium conditions, B = Fg .

(b)

(a)

a fluid is determined only by the densities of the

object and the fluid.

Case 1: Totally Submerged Object

fluid experiences two forces, the gravitational force Fg and the

buoyant force B. Because the object floats in equilibrium, B = Fg .

Case 2: Floating Object

of a floating object that is below the fluid surface

is equal to the ratio of the density of the object to that of the fluid.

(7.6)

Case 2: Floating Object

characterized as being one of two main types. The flow

is said to be steady, or laminar, if each particle of the fluid follows a smooth path, such that the paths of different particles never cross each other. In steady flow, the velocity of fluid particles passing any point remains constant in time.

Figure 7.10 Laminar flow around an automobile in a test wind tunnel.

smoke particles. The smoke first moves in laminar flow at

the bottom and then in turbulent flow above.

Above a certain critical speed, fluid flow becomes turbulent; turbulent flow is irregular flow characterized by small whirlpool-like regions, as shown in Figure 7.11.

Fluid Dynamics

of fluid flow to characterize the degree of internal friction

in the fluid. This internal friction, or viscous force, is associated with the resistance that two adjacent layers of fluid have to moving relative to each other. Viscosity causes part of the kinetic energy of a fluid to be converted to internal energy. This mechanism is similar to the one by which an object sliding on a rough horizontal surface loses kinetic energy.

and not fully understood, we make some simplifying assumptions in

our approach. In our model of ideal fluid flow, we make the following four assumptions:
1. The fluid is nonviscous. In a nonviscous fluid, internal friction is neglected. An object moving through the fluid experiences no viscous force.
2. The flow is steady. In steady (laminar) flow, the velocity of the fluid at each point remains constant.
3. The fluid is incompressible. The density of an incompressible fluid is constant.
4. The flow is irrotational. In irrotational flow, the fluid has no angular momentum about any point. If a small paddle wheel placed anywhere in the fluid does not rotate about the wheel’s center of mass, then the flow is irrotational.

is called a streamline. The velocity of the particle is

always tangent to the streamline, as shown in Figure 7.12. A set of streamlines like the ones shown in Figure 7.12 form a tube of flow. Note that fluid particles cannot flow into or out of the sides of this tube; if they could, then the streamlines would cross each other.

Fluid Dynamics

Figure 7.12 A particle in laminar flow follows a streamline, and at each point along its path the particle’s velocity is tangent to the streamline.

a pipe of varying cross-sectional area. The volume of fluid

flowing through area A1 in a time interval ∆t must equal the volume flowing through are A2 in the same time interval. Therefore, A1v1 = A2v2.

fluids. It states that
the product of the area and the

fluid speed at all points along a pipe is constant for an incompressible fluid.

(7.7)

constricted pipe. The volume of the shaded portion on the

left is equal to the volume of the shaded portion on the right.

fluid. It is often expressed as
(7.8)
(7.9)

on the highway at the opening of this section. As

air passes between you and the truck, it must pass through a relatively narrow channel. According to the continuity equation, the speed of the air is higher. According to the Bernoulli effect, this higher speed air exerts less pressure on your car than the slower moving air on the other side of your car. Thus, there is a net force pushing you toward the truck!

moving airplane wing. The air approaching from the right is

deflected downward by the wing. By Newton’s third law, this must coincide with an upward force on the wing from the air—lift. Because of air resistance, there is also a force opposite the velocity of the wing— drag.

of air, a spinning golf ball experiences a lifting force

that allows it to travel much farther than it would if it were not spinning.

passing over a tube dipped into a liquid causes the

liquid to rise in the tube.

who steps back and accidentally stomps on your foot with

the heel of one shoe. Would you be better off if that person were (a) a large professional basketball player wearing sneakers (b) a petite woman wearing spike-heeled shoes?

filled glass of water (ρ=1 000 kg/m3) is P. The

water is poured out and the glass is filled with ethyl alcohol ρ=806 kg/m3). The pressure at the bottom of the glass is (a) smaller than P (b) equal to P (c) larger than P (d) indeterminate.

variety of fluids. For which of the following fluids will

the column of fluid in the barometer be the highest? (a) mercury (b) water (c) ethyl alcohol (d) benzene

below the surface of a container of water. The apple

is then moved to a deeper point in the water. Compared to the force needed to hold the apple just below the surface, the force needed to hold it at a deeper point is (a) larger (b) the same (c) smaller (d) impossible to determine.

to each other at the ends of strings secured to

a table. The facing surfaces of the balloons are separated by 1–2 cm. You blow through the small space between the balloons. What happens to the balloons? (a) They move toward each other. (b) They move away from each other. (c) They are unaffected.

end-to-end to make a longer straw with no leaks. The

two straws have radii of 3 mm and 5 mm. You drink a soda through your combination straw. In which straw is the speed of the liquid the highest? (a) whichever one is nearest your mouth (b) the one of radius 3 mm (c) the one of radius 5 mm (d) Neither—the speed is the same in both straws.