Density and Pressure of a fluid at any point

Density

To find the density of fluid at any point , we isolate a small volume element V around that point and measure the mass m og of the fluid contained within that element. The density is then

In the theory , the density at any point in a fluid is the limit of this ratio as the volume element V at that point is made smaller and smaller. In practice, we assume that a fluid sample is large relative to atomic dimensions and thus is “ smooth “ ( with uniform density), rather than “lumpy” with atoms. This assum ption allows us to write as

Where m and V are the mass and volume of sample.

Density is a scalar property; its SI unit is the kilogram per cubic meter. In the table shows the densities of some substances and the average densities of some objects. Note that the density of a gas ( see Air ) varies considerably with pressure, but the density of a liquid ( see water ) does not; that is, gases are readily compressible but liquids are not.

Pressure

Let a small pressure – sensing device be suspended inside a fluid – filled vessel, as in Figure.

The sensor consists of a piston of surface area A riding in a close – fitting cylinder and resting against a spring. A readout arrangement allows us to record the amount by which the ( calibrated ) spring is compressed by the surrounding fluid , thus indicating the magnitude F of the force that acts normal to the piston . We define the pressure on the piston from the fluid as

In the theory , the pressure at any point in the fluid is the limit of this ratio as the surface area A of the piston , centered on that point, is made smaller and smaller . However , if the force is uniform over a flat area A , we can write as

Where F is the magnitude of the normal force on area A. ( when we say a force is uniform over an area, we mean that the force is evenly distributed over every point of the area )

We find by experiment that at a given point in a fluid at rest , the pressure p defined by equation has the same value no matter how the pressure sensor is oriented . Pressure ais a scalar, having no directional properties. It is true that the force acting on the piston of our pressure sensor is a vector quantity, but equation involves only the magnitude of the force , a scalar quantity.

The SI unit of pressure is the newton per square meter, which is given a special name , the pascal ( Pa ). In metric countries , tire pressure gauges ara calibrated in kilopascals. The pascal is related to some other common ( non – SI ) pressure unitsa as follows :

1 atm = 1.01 x 10⁵ Pascal = 760 torr = 14,7 lb/in²

The atmosphere ( atm ) is , as the name suggest, the approximate average pressure of the atmosphere at sea level. The torr ( named for Evangelista Torricelli , who invented the mercury barmoter in 1674 ) was formerly called the millimeter of mercury ( mm Hg ). The pound per square inch is often abbreviated psi. Table shows some pressure :

Sample problem

A living room has floor dimensions of 3.5 m and 4,2 m and a height of 2.4 m. ( a) Ehat does the air in the room weigh when the air pressure is 1.0 atm ? ( b ) What is magnitude of the atmosphere’s downward force on the top of your head , which we take to the have an area of 0.040 m² ?

Answer :

(a) The air’s weight is equal to mg , whre m is its mass. Mass m is related to the air density and the air volume V

Calculation : Putting the two ideas together and taking the density of air at 1.0 atm from table , we find :