Pressure - Pressure in a fluid, Absolute and gauge pressure, practice problems, FAQs

Shivam and Rani are playing in the ground. Rani gives a challenge to shivam to stand on the ground for the three different cases. First normally on both the feet, then on the single foot and then on his toes. In all the cases she notes down the timing. Shivam was able to stand for a longer time in the first case, then for the second case, and the least for the third time. As in all the cases the weight of shivam was equal, then why was he not able to stand for equal time? This can be understood by the concept of pressure. Pressure is inversely proportional to the area of surface on which the force is applied perpendicularly. In the first case the area of contact with the ground was more, so the pressure on legs was lesser and he was able to stand for longer time. In the last case the area of contact was lesser so pressure on legs is larger, consequently he falls sooner. In this topic we will learn about the pressure.

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Table of content

What is pressure?
Atmospheric Pressure
Pressure in a fluid
Absolute and gauge pressure
Practice problems
FAQs

What is pressure?

The pressure is defined as the force applied perpendicularly to an object's surface divided by the surface area over which it is applied. It is denoted by P.

Consider an small elemental area dA of an object, if a small force dF is applied perpendicular to the surface, then pressure is,

$P = \frac{d F_{p e r p e n d i c u l a r}}{d A}$

If a finite planar surface with size A has constant pressure at all locations, then

$P = \frac{F}{A}$

The Nm-2 or pascal (Pa) is the SI unit for pressure. And it is a scalar quantity.

Atmospheric Pressure

The atmospheric air around the planet is surrounded by a layer of gases, and as a result, this air exerts a pressure on the earth surface. This pressure is called the "atmospheric pressure." It is also known as the “Barometric pressure” .Atmospheric Pressure is denoted by (Patm). At sea level, it has a value of 101325 Pa.

Pressure in a fluid

Take into account a fluid at rest in a container with a small cubical volume element at a depth h from the free surface inside of it, as illustrated in the figure.

Let width of the element be dh and area of the element be dA. If is the density of the fluid, then mass of the fluid dm=dAdh . The pressure force on the top surface PdA and bottom of the surface is (P+dP)dA. As the fluid is at rest, balancing the force in the vertical direction.

PdA+W=(P+dP)dA

PdA+dm g=PdA+dPdA

PdA+dAdh g=PdA+dPdA dm=dAdh

dA dh g=dP dA

$\frac{d P}{d h} = ρ g$

The pressure gradient in the vertical direction is expressed in this way. The variation of pressure with height is positive which means that on going downward in the fluid its pressure increases.

dP=gdh

The pressure at any depth h can be found by integrating the above equation. Let at free surface (h=0) pressure is Po and at a depth h pressure is P then we get,

PoPdP=0hgdh

(P-Po)=gh

P=Po+gh

This is the absolute pressure of the fluid at a depth h. In general the water surface is exposed to the atmosphere, the value of Po=Patm , so the above equation can be written as,

P=Patm+gh

Absolute and gauge pressure

The total pressure at a point in the fluid is called absolute pressure. The pressure relative to a complete vacuum is measured by absolute pressure. Therefore, absolute pressure is positive for all pressures above a complete vacuum, zero for a perfect vacuum, and never negative.

For above expression of pressure in fluid the absolute pressure is

Pabs=Patm+gh

The excess pressure above or below the atmospheric pressure is usually known as the gauge pressure. When the gauge pressure is above atmospheric pressure, it is positive; it is zero, at atmospheric pressure; and it is negative, below atmospheric pressure.

For above expression of pressure in fluid the gauge pressure can be found by subtracting atmospheric pressure from the absolute pressure

Pabs-Patm=gh

Pgauge=gh

The absolute and gauge pressure can be understood by the following figure.

Practice problems

Q. An area of 50 m2 is evenly subjected to a 250 N force. What is the pressure measured in pascals?

A. Given A=50 m2

F=250 N

The pressure is given as,

$P = \frac{F}{A}$

$P = \frac{250}{50}$

P=5 Pa

As a result, the area is under 5 Pa of pressure.

Q. Calculate the gauge pressure at the bottom of a container that is 50 cm high and filled with water.

A. Given h=50 cm=0.5 m

The gauge pressure in a fluid is give by,

Pgauge=gh

Density of the water $ρ = 1000 k g / m^{3} a n d g = 9.81 m / s^{2}$

$P_{g a u g e} = 1000 \times 9.81 \times 0.5 P_{g a u g e} = 4905 P a$

The gauge pressure at the bottom of the fluid is 4905 Pa.

Q. Calculate the pressure a scuba diver will feel at a depth of 10 metres below the ocean's surface.

A. Given h=10 m

The atmosphere pressure $P_{a t m} = 101325 P a$

The pressure felt by the scuba diver is,

$P = P_{a t m} + ρ g h P = 101325 + 1000 \times 9.81 \times 10 P = 199425 P a$

The pressure on the scuba diver is 199425 Pa .

Q. A nail is having tip area 0.002 m2 exerting a pressure of 24525 Pa on the wall. What is the force applied by the hammer?

A. Given A=0.002 m2 and P=24525 Pa

The force can be calculated as,

F=PA

F=245250.002

F=49.05 N

The force by the hammer is 49.05 N.

FAQs

Q. Why are the edges of the knife made sharp?
A. The sharp edge will have lesser thickness and hence lesser area. As we know the pressure is inversely proportional to the area. So by making the edge sharp, the greater pressure can be applied by the knife.

Q. Why is the gauge pressure important?
A. As all the equipment in the fluid system and our world functions in the atmospheric pressure. The atmospheric pressure is almost constantly present and doesn't vary greatly. Therefore, incorporating it in your measurements sometimes seems like a waste of time. Because of this reason we use gauge pressure.

Q. What is Pascal's law?
A. According to Pascal's law, every change in pressure applied to an enclosed incompressible fluid is equally transmitted to all of the fluid's components and the vessel's walls.

Q. What variables affect the pressure in a fluid?
A. The pressure depends upon fluid density, the point's depth into the fluid and gravitational acceleration.