Pressure

Pressure:

A fluid at rest exerts a force on the surface of contact. The surface may be a wall or the bottom of an open container of the fluid. The normal force (F) exerted by a fluid at rest per unit surface area (A) of contact is called the pressure (p) of the fluid. 

P = F/A

Thus, an object having small weight can exert high pressure if its weight acts on a small surface area. The SI unit of pressure is N/m2. The dimension of pressure is L-1M1T-2. Pressure is a scalar quantity. 

Pressure Due to a Liquid Column:

A vessel is filled with a liquid. Let us calculate the pressure exerted by an imaginary cylinder of cross sectional area A inside the container. Let the density of the fluid be ρ, and the height of the imaginary cylinder be h as show in the figure. The liquid column exerts a force F = mg, which is its weight, on the bottom of the cylinder. This force acts in the downward direction. Therefore, the pressure p exerted by the liquid column on the bottom of cylinder is, 

Atmospheric Pressure:

Earth's atmosphere is made up of a fluid, namely, air. which exerts a downward force due to its weight. The pressure due to this force is called atmospheric pressure. Thus, at any point, the atmospheric pressure is the weight of a column of air per unit cross section starting from that point and extending to the top of the atmosphere. Clearly, the atmospheric pressure is highest at the surface of the Earth, i.e., at the sea level, and decreases as we go above the surface as the height of the column of air above decreases. The atmospheric pressure at sea level is called normal atmospheric pressure. The region where gas pressure is less than the atmospheric pressure is called vacuum. Perfect or absolute vacuum is when no matter, i.e., no atoms or molecules are present. 

Absolute Pressure and Gauge Pressure:

Consider a tank filled with water as shown in the figure. Assume an imaginary cylinder of horizontal base area A and height x1 - x2 = h. x1 and x2 being the heights measured from a reference point, height increasing upwards: x1 > x2 . The vertical forces acting on the cylinder are:

1. Force F1 acts downwards at the top surface of the cylinder, and is due to the weight of the water column above the cylinder

2. Force F2 acts upwards at the bottom surface of the cylinder, and is due to the water below the cylinder

3. The gravitational force on the water enclosed in the cylinder is mg, where m is the mass of the water in the cylinder.

As the water is in static equilibrium, the forces on the cylinder are balanced. The balance of these forces in magnitude is written as, F2 = F1 + mg 

Substituting Eq. (2.4) and Eq. (2.5) in F2 we get, 

p2 A = p1 A + ρAg (x1 - x2 ) p2 = p1 + ρg (x1 - x2 ) --- (2.6) 

This equation can be used to find the pressure inside a liquid (as a function of depth below the liquid surface) and also the atmospheric pressure (as a function of altitude or height above the sea level). 

To find the pressure p at a depth h below the liquid surface, let the top of an imaginary cylinder be at the surface of the liquid. Let this level be x1 . Let x2 be some point at depth h below the surface as shown in Fig. Let p0 be the atmospheric pressure at the surface, i.e., at x1 . Then, substituting x1 = 0, p1 = p0 , x2 = -h, and p2 = p in Eq. (2.6) we get, p = p0 + ρhg 

The above equation gives the total pressure, or the absolute pressure p, at a depth h below the surface of the liquid. The total pressure p, at the depth h is the sum of:

1. p0 , the pressure due to the atmosphere, which acts on the surface of the liquid, and

2. hρg, the pressure due to the liquid at depth h

In general, the difference between the absolute pressure and the atmospheric pressure is called the gauge pressure. Gauge pressure at depth h below the liquid surface can be written as, p - p0 = hρg