Chapter 10
The Giancoli onLine Tutor

Fluids

I. Density

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Density, denoted by the lower case Greek letter r, is defined as the ratio of mass, M, to volume, V.

Thus, we write: r= M/V

In most cases the object being studied will be either a sphere or cube. If it is a sphere its Volume is given by:
V= 4*p*R³/3
where R is the radius of the sphere.

If it is a cube its Volume is given by:
V=a³

where "a" is the side length.

Interactive Example
Iron has a density of 7800 kg, what is the weight to the nearest newton of an iron ball with diameter 10 cm. Use 9.8 for g.
Enter you answer and click outside

II. Pressure

Pressure is Force distributed over an Area and has the units of N/m². It is closely related to stress which we encountered in the last chapter. Stress is used with solids and Pressure with fluids (liquids and gases).

Atmospheric Pressure is around 1.0x 10⁵ N/m². In fact, Atmospheric Pressure,P_o, is sometimes measured in terms of a unit called atmosphere (atm).

1 atm= 1.013x10⁵ N/m².

Pressure increases with depth, h, in a fluid (usually a liquid) in steps of r*g. So that in a body of a liquid at a depth "h" the pressure,P, is given by

P=P_o+rgh

Interactive Example
Water has a density of 1000 kg/m³ using 10 for g what is the pressure at a depth of 100 m and surface?
Enter you answer and click outside

III. Fluid Statics

Problems in Fluid Statics usually hinge around three facts:

Objects that float displace their weight in water.
Objects that sink displace their volume in water.
The Buoyant Force on an object immersed or partly immersed in a liquid is equal to the weight of the displaced liquid.

Interactive Activity

Study the movie at the left. A Gray Cylinder (outlined in green) is lowered into a container of water. Notice that as the cylinder is submerged it displaces a volume of water (caught by an overflow container of the same total volume) equal to its submerged volume.

The reading on the scale is equal to the weight of the cylinder minus the buoyant force, if any, of the liquid.
Note, also, that the scale reading decreases by exactly the amount of the weight of the displaced water.

Note if you don't have the quicktime plugin you can view a GIF animation version of the movie here .

Interactive Example
Iron has a density of 7800 kg, what is the apparent weight, W', ( scale reading ) to the nearest newton of an iron ball with diameter 10 cm if it is immersed in water. Use 9.8 for g.
Enter you answer and click outside

IV. Fluid Dynamics

Fluid flow is either laminar (smooth) or turbulent. We will only be concerned with the first. An example of turbulent flow is cream stirred into coffee and an example of the laminar flow would be cream stirred into molasses. Laminar flow is characterized by a lack of whirlpools and eddies. The approach taken here, strictly only applies to incompressible fluids so it is only approximately true for gasses, such as, air.

The governing equations are, The Equation of Continuity,

v₁A₁=v₂A₂

which usually applies to flow through pipes (A=p*R²) or ducts (A=a*b) in that case, "v" refers to the speed at any point and "A" the cross sectional area at that point. However these "pipes" can be very general. For example, a hole in a water tank can be thought of as a very short small pipe attached to a very large pipe. The "Equation of Continuity" is essentially a statement of Conservation of Matter.

The other equation is called Bernoulli's Equation.

P+.5*rv²+rgh=constant.
Which in words says that the

The Pressure at any point

plus one-half the density times the velocity squared at that point

plus the density times "g" times the height there

equals a constant

"Bernoulli's Equation" is essentially a restatement of Conservation of Energy as it applies to fluids.

In any flow problem you will need to use one or both of these equations.

Tips

If a cross sectional area or diameter or radius is mentioned that is an indication that you need to use the first equation to find something like a missing velocity necessary for use in the second equation. Do this first!
In the second equation make sure to convert pressures to N/m² and velocities to m/s.
You use the second equation just as you did Conservation of Energy problems. Usually that means evaluating it at the beginning and setting it equal to its value at the end of the problem statement, that is,
P₁+.5*rv₁²+rgh₁ = P₂+.5*rv₂²+rgh₂

Interactive Examples

Water with a gauge pressure of 3 atm and a velocity of 4 m/s is piped through a 4 cm diameter pipe at ground level to a height of a height of 10 m in a 8 cm pipe? What is its final velocity?
Enter you answer and click outside

What is the final absolute pressure to the nearest N/m²? Use 10 for g. and 101000 for an atm.
Enter you answer and click outside

Fluids

I. Density

Note you must be using a 4 level browser to use all the features on this page. You should see a lowercase alpha here: a. If not you need to get a browser that supports greek fonts such as NetScape 4.5. Also a square root sign should be here: √, if not you need to upgrade your browser

Interactive Example

II. Pressure

Pressure is Force distributed over an Area and has the units of N/m2. It is closely related to stress which we encountered in the last chapter. Stress is used with solids and Pressure with fluids (liquids and gases).

Interactive Example

III. Fluid Statics

Interactive Activity

Note if you don't have the quicktime plugin you can view a GIF animation version of the movie here .

Interactive Example

IV. Fluid Dynamics

In any flow problem you will need to use one or both of these equations.

Tips

Interactive Examples

Pressure is Force distributed over an Area and has the units of N/m². It is closely related to stress which we encountered in the last chapter. Stress is used with solids and Pressure with fluids (liquids and gases).