Concept Summary

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Topic | Discussion | Learning Aids
| 11.1 MASS DENSITY Fluids are materials that can flow, and they include gases and liquids. The mass density of a substance is its mass m divided by its volume V: |
Mass Density |

| (11.1)

| The specific gravity of a substance is its mass density divided by the density of water at 4 °C (): |
Specific gravity |

| (11.2)

| 11.2 PRESSURE The pressure P exerted by a fluid is the magnitude F of the force acting perpendicular to a surface embedded in the fluid divided by the area A over which the force acts: |
Pressure |

| (11.3)

| The SI unit for measuring pressure is the pascal (Pa); . |
Atmospheric pressure | One atmosphere of pressure is or 14.7 lb/in.2 |
| 11.3 PRESSURE AND DEPTH IN A STATIC FLUID In the presence of gravity, the upper layers of a fluid push downward on the layers beneath, with the result that fluid pressure is related to depth. In an incompressible static fluid whose density is , the relation is

| (11.4)

where P1 is the pressure at one level, P2 is the pressure at a level that is h meters deeper, and g is the magnitude of the acceleration due to gravity.
| 11.4 PRESSURE GAUGES Two basic types of pressure gauges are the mercury barometer and the open-tube manometer. |
Gauge pressure and absolute pressure | The gauge pressure is the amount by which a pressure P differs from atmospheric pressure. The absolute pressure is the actual value for P. |
Pascal’s principle | 11.5 PASCAL’S PRINCIPLE Pascal’s principle states that any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and the enclosing walls. |
Buoyant force | 11.6 ARCHIMEDES’ PRINCIPLE The buoyant force is the upward force that a fluid applies to an object that is partially or completely immersed in it. |
| Archimedes’ principle states that the magnitude of the buoyant force equals the weight of the fluid that the partially or completely immersed object displaces: |
Archimedes’ principle |

| (11.6)

Use Self-Assessment Test 11.1 to evaluate your understanding of Sections 11.1–11.6.

Steady flow | 11.7 FLUIDS IN MOTION, 11.8 THE EQUATION OF CONTINUITY In steady flow, the velocity of the fluid particles at any point is constant as time passes. |
Ideal fluid | An incompressible, nonviscous fluid is known as an ideal fluid. |
| The mass flow rate of a fluid with a density , flowing with a speed v in a pipe of cross-sectional area A, is the mass per second (kg/s) flowing past a point and is given by |
Mass flow rate |

| (11.7)

| The equation of continuity expresses the fact that mass is conserved: what flows into one end of a pipe flows out the other end, assuming there are no additional entry or exit points in between. Expressed in terms of the mass flow rate, the equation of continuity is |
Equation of continuity |

| (11.8)

where the subscripts 1 and 2 denote two points along the pipe.
Incompressible fluid | If a fluid is incompressible, the density at any two points is the same, . For an incompressible fluid, the equation of continuity becomes

| (11.9)

The product Av is known as the volume flow rate Q:
Volume flow rate |

| (11.10)

| 11.9 BERNOULLI’S EQUATION, 11.10 APPLICATIONS OF BERNOULLI’S EQUATION In the steady flow of an ideal fluid whose density is , the pressure P, the fluid speed v, and the elevation y at any two points (1 and 2) in the fluid are related by Bernoulli’s equation: |
Bernoulli’s equation |

| (11.11)

When the flow is horizontal (), Bernoulli’s equation indicates that higher fluid speeds are associated with lower fluid pressures.
| 11.11 VISCOUS FLOW The magnitude F of the tangential force required to move a fluid layer at a constant speed v, when the layer has an area A and is located a perpendicular distance y from an immobile surface, is given by |
Force needed to move a layer of viscous fluid with a constant speed |

| (11.13)

where is the coefficient of viscosity.
| A fluid whose viscosity is , flowing through a pipe of radius R and length L, has a volume flow rate Q given by |
Poiseuille’s law |

| (11.14)

where P1 and P2 are the pressures at the ends of the pipe.

Use Self-Assessment Test 11.2 to evaluate your understanding of Sections 11.7–11.11.

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