UNIT 4
Density and Pressure
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7

 

Density and Pressure in Fluids

Fluids

All fluids

— can flow. Some like pitch flow slowly.
— assume the dimensions of their containers.
— cannot withstand a shearing stress.
—exert a force in the direction perpendicular to its surface.
— do not have orderly structure of constinuent atoms.
— are liquids and gases.

Density

— density ρ = [Δm/Δv] .
— we assume that a fluid sample is large relative to atomic dimensions and thus with uniform density) so density becomes ρ = [m/v] where m and V are the mass and volume of the sample.
— is scalar.
— unit is kgm-3
— density of gases varies considerably with pressure, but the density of liquids and solids do not because gases are readily compressible but liquids and solids are not.

Pressure

— pressure of uniform force on flat area.
— p = F/A where F is the magnitude of the normal force on area A.
— Pressure is a scalar.
— Unit of pressure is the Newton per square meter Nm-2, called the pascal (Pa).
— Normal atmospherica pressure = 1.01 X 105 Pa.

Fluids at Rest

— pressure increase with depth. Divers know that pressure increases with depth in water.
— Pressure decreases with altitude in the atmosphere.
— In fluids at rest pressure in fluids is called hydrostatic pressure.


pressure
Fig 1

Absolute and Gauge Pressure

Consider a cylinder of water bounded on the top by y1 and the bottom by y2 in Fig 1.

— The forces on the cylinder of water are:

— F1 on the top surface.
— F2 on the bottom surface.
— mg is the weight of the cylinder (caused by gravity).

— The pressures on the cylinder of water are:

— P1 on the top surface.
— P2 on the bottom surface.

— The dimensions of the cylinder:

— y1 - y2 is the height
— A is the area of the top or bottom surface.

Note: The water is at rest (no translation or rotation) so it is in static equilibrum.
ΣF = 0
F2 + (-F1) + (-mg) = 0
F2 = F1 + mg --- Eq1

By definition F2 = P1 A and F2 = P2 A
So Eq 1 becomes P2A = P1A + mg --- Eq2
Mass of water in cylinder = ρV where V is the volume of the cylinder
Mass of water in cylinder = ρ [A (y1 - y2)]
So Eq2 becomes P2A = P1A + ρg [A (y1 - y2)]
P2 = P1 + ρg (y1 - y2)
This becomes the famous equation P= ρgh where h is the depth of the liquid if y1 becomes 0 . This is called gauge pressure as it is the pressure difference between the suface and the point as measured by a gauge.
If the pressure at the surface is p0 then the absolute pressure is p0 + ρgh
Gauge Pressure = ρgh
Absolute Pressure = p0 + ρgh

Divers experience p0 + ρgh
Implication: Pressure at a point in a fluid in static equilibrium depends on depth at that point and is the same on a particular horizontal level.

pressure

The Mercury Barometer

— long glass tube is filled with mercury and inverted with its open end in a dish of mercury.
— space above the mercury column contains only mercury vapor.
— pressure of mercury vapor is so small at ordinary temperatures that it can be neglected.

The weight of the mercury column is supported by atmospheric pressure so:

Po = ρgh

Note: The vertical height h of the mercury column

— does not depend on the cross-sectional area of the vertical tube.
— depends on the value of g at the location of the barometer
— on the density of mercury, which varies with temperature
— corrections are made for g and temperature.

pressure

The Open-Tube Manometer

— a U-tube containing a liquid, with one end of the tube connected to the gas supply and the other end open to the atmosphere.
— measures gauge pressure of a gas.
— Pg = ρgh
— gauge pressure can be positive or negative e.g. positive in a tyre andnegative in a vacuum cleaner input. A good example is pressure in your lungs which is positive when exhaling and negative when inhaling.

 

Pascal’s principle

If an incompressible fluid experiences a change in the pressure, then that pressure is transmitted undiminished to every portion of the fluid and to the walls of its container.

 

   

 

 

 

 

 

   
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Concept by Kishore Lal. Programmed by Kishore Lal... Copyright © 2015 Kishore Lal. All rights reserved.