Pressure

Pressure
Pressure is defined as the force per unit area. It is a scalar quantity. Its SI unit is N/m² or Pascal (Pa). If the force of different magnitude is applied on the same area, the pressure varies. Pressure is  more when more force acts on the body. Likewise, if same force acts on two different areas, the pressure varies. Pressure goes on increasing when force is increased and vice-versa. When force is applied to the small area the pressure and increases when force is applied to a large area.
Let us consider a force 'F' is applied to a certain surface area 'A' , then pressure is directly proportional to the force and inversely proportional to the area of the surface in which the force acts normally. Mathematically,

$P\alpha F$..................(a)

$P\alpha \frac{1}{A}$....................(b)

Combining equation (a) and (b), we get, 

$P\alpha \frac{F}{A}$...............(c)

$P=k\frac{F}{A}$.................(d)

[Where, k is proportionality constant and its value is 1 because when 1 N force is applied in 1 m², the pressure is 1N/m²]

Therefore, 

$P=\frac{F}{A}$....................(e)

Pressure in liquid: 

Let us consider a container with base area 'A' filled with liquid of density 'd'. Let the depth of liquid inside the container be 'h'. 'F' be the force exerted by the liquid on the base of the container which is equal to the weight of the liquid inside the column.

By the definition of pressure,

$P=\frac{F}{A}$..............(a)

$P=\frac{mg}{A}$..............(b) [As, F=mg]

$P=\frac{dvg}{A}$...............(c) [As, d=m/v and m = dv]

$P=\frac{d(A\times h)g}{A}$.........(d) [As, V=A×l]

$P=dgh$.........(e)      

Equation (e) shows that pressure exerted by liquid in a column depends on: 

- density of liquid 'd'

- height of liquid column 'h'

- acceleration due to gravity 'g'

Equation (e) shows that pressure exerted by the liquid is directly proportional to the height of the liquid column, density of the liquid and acceleration due to gravity.

Laws of liquid pressure: 

• Liquid pressure is directly proportional to its density
• liquid pressure at a point is same in all directions within the liquid. 
• Liquid pressure is independent of the shape of the vessel. 
• Liquid pressure is directly proportional to the depth.

Pascal's law:

Pascal's law states that 'Pressure is equally exerted perpendicularly on all directions as pressure is applied at a point on liquid contained in a enclosed container.'
Pascal's law is based on the principle that liquids are in-compressible and liquid transmits pressure equally in all directions. With the help of Pascal's law we can produce large force by applying small forces. In other words Pascal's law amplifies the applied force and helps in doing work. Hydraulic machines were developed on the basis of Pascal's law. Examples of hydraulic machines are hydraulic machine, hydraulic jack and hydraulic lift etc.

Let us consider a F1, F2, F3, and F4 be the forces applied at pistons p1, p2, p3 and p4, at the same time P1,P2, P3 and P4 are the pressures in the respective pistons. According to Pascal's law, pressure at piston p1 is equally distributed in all other pistons. Mathematically,

p1 = p2 = p3 = p4

Similarly, 

$\frac{{{F}_{1}}}{{{A}_{1}}}=\frac{{{F}_{2}}}{{{A}_{2}}}=\frac{{{F}_{3}}}{{{A}_{3}}}=\frac{{{F}_{4}}}{{{A}_{4}}}$

Hydraulic press:

Hydraulic press is a machine that works under the principle of Pascal’s law. It is usually ‘U’ shaped tube filled with liquid and fitted with air tight pistons. It magnifies the applied force. In other words it converts small applied force into large force. When a small force is applied in a piston of small cross-section area, pressure is equally transmitted in all directions; a large force appears over the piston with larger cross-section area.

Hydraulic press

Hydraulic press as a force multiplier:

Let us consider a hydraulic press filled with water and fitted with air pistons P1 and P2 with cross-section area A1 and A2 respectively. Here, area A1 is smaller than area A2.

When force F1 is applied on a small piston in downward direction , it produces force F2 on piston P2 which moves upward direction due to equal transmission of pressure in all direction according to Pascal’s law.

According to Pascal’s law,

Pressure on small cylinder with piston P1 = Pressure on big cylinder with piston P2.

So, P₁ = P₂

Where P₁  and P₂ are pressure on small and big piston.

Or, $\frac{{{F}_{1}}}{{{A}_{1}}}=\frac{{{F}_{2}}}{{{A}_{2}}}$

Or, ${{F}_{2}}=\frac{{{F}_{1}}\times {{A}_{2}}}{{{A}_{1}}}$

Since, A₂ >A₁ then F­₂ >F₁.

Therefore, hydraulic press or hydraulic lift acts as force multiplier. 

Hydraulic brake

A hydraulic brake is a mechanical component used mostly in vehicles which works on Pascals’s law. In consists of a master cylinder filled with a fluid which is attached to the wheel cylinder with the help of a pipe. The wheel cylinder also consists of the same fluid inside it that is connected to the brake shoe. When foot pedal is pressed the piston in master cylinder is pushed inward along with the special fluid. Then the pressure in the master cylinder is transmitted to the wheel cylinder. The pistons in the wheel cylinder apply force to the brake shoe that widens the brake shoe. Widening of break shoe come in contact with the wheel and produces friction. As a result vehicle stops.

Hydraulic brake

Upthrust: 
Thrust is the force acting perpendicular to the surface. Its unit is Newton. Only liquid and gas exerts up-thrust.
We feel easier to lift a bucketful of water until it is under the surface of the water but becomes heavy when it is out of the surface of water. Similarly, we have experienced during swimming it is easier to lift  a friend inside the water but becomes heavy out of the water.  These examples show that when a body is wholly or partially immersed in water, water pushes the body up with certain force. This force is called upthrust.
Upthrust is defined as the upward force exerted by liquid on an object immersed in in the liquid. It is also called buoyant force or bouncy.

How to measure the upthrust:

Let a stone be tied on a string and suspend it with a spring balance, the weight of the stone in the air be w1 N. The weight of the stone will be w2 N when completely immersed inside the water contained in beaker. There is difference between w1 and w2 due the up-thrust exerted by water in upward direction. This difference in weight gives the up-thrust. Mathematically,
Weight of stone air =w1
Weight of stone inside the water = w2
Then,
Up-thrust = Difference in weight in air and water
                = (w1 - w2)N

Archimedes' Principle: 
Archimedes' principle states that when a body is wholly or partially immersed in a liquid, it experiences a loss in wight due to up-thrust which is equal to the weight of the liquid displaced by it.
This principle can be applied for both liquid and gas.

Properties of Archimedes' Principle: 

• This law can be applied for both liquid and gas.
• Archimedes' principle holds true when an object is wholly or partially immersed in liquid. 
• Up-thrust is independent of weight of an object.

Verification of Archimedes' principle: 
Let us tie a stone by a thread with a spring balance. Measure the weight of the stone in the air, let the weight be w1. Insert the stone completely inside the overflow can (Eureka can) filled with water upto the spout. Due to the up-thrust there is loss in weight of the stone, the weight of the stone be w2. At same time place a empty beaker in pan balance just below the spout of overflow can. Consider the weight of the of the beaker be w3. When the stone is inserted into the water it displaces some water which flows out from the spout into the beaker. Let the weight of the displaced water and beaker be w4. Then,

Loss in weight of the stone in water = w1 - w2
Weight of water displaced by water = Weight of beaker and water - weight of the beaker
                                                          = w4 - w3
The loss in weight of the stone w1 - w2 = Weight of water displaced by the stone w4 - w3
                                               Up-thrust  = Weight of liquid displaced

Hence, it found that up-thrust is equal to the weight of water displaced by it.

Density: 
Density is defined as the mass per unit volume. Its is scalar quantity. Its SI unit is kg / m3. Mathematically,
                                             Density $D=\frac{M}{V}$
Relative density: 
Relative density is defined as the ratio of density of a substance to the density of water at 4°C.
Mathematically,
                                             Relative density \[\text{R}\text{.D}=\frac{\text{Density of substance}}{\text{Density of water at 4 }\!\!{}^\circ\!\!\text{ C}}\]
Relative density has no unit because it the ratio between two densities.

Meaning of Relative Density:

• If a substance has relative density less then 1, it will float in the liquid taken.
• If a substance has relative density equal to 1, it will float with completely immersed in the liquid. 
• If a substance has relative density greater then 1, it will sink in the liquid.  

Law of flotation: 
When a body is immersed  in a liquid experiences two types of forces. They are:

        i.The force of gravity directed vertically downward.
        ii.The up thrust directed vertically upward.

Due to the action of these two forces, a body moves in the direction of greater force. There will be three possible cases if an object is immersed in liquid:

1. If the weight of an object is greater than up-thrust, the body will sink to the bottom.
2. If the weight of an object is equal to the up-thrust, the body will remain anywhere inside the liquid.
3. If the weight of an object is less than up-thrust, the body will rise to the surface of the liquid and floats. 

Principle of flotation:
Principle of flotation states that a body floats in liquid if it can displace the liquid equal to its won weight i.e
Weight of floating body = Weight of the liquid displaced = Up-thrust.

When a body is immersed in a liquid, it experiences two type of forces:
       a. Up-thrust: Acting in upward direction.
       b. Gravity: Acting in downward direction.
Due to the action to these two forces, a body will move in the direction of greater force. According to the resultant of the forces, there are three cases:
      i. If the weight of an object is less than up-thrust the body will floats on the surface
         of the liquid in which it is immersed.
      ii. If the weight of an object is equal to the up-thrust, the body can be
          in equilibrium condition at any point in liquid.
      iii. If the weight of an object is greater than up-thrust, the body
           sinks to the bottom
Hydrometer: 

Hydrometer is an instrument used to measure the density of various liquid, based on the principle of flotation. There are two types of hydrometers.They are constant immersion hydrometer and constant weight hydrometer. 
a. Constant weight or variable immersion hydrometer: The hydrometer which measures the density by observing the scale of he floating neck is called constant weight hydrometer. It consists of a bulb containing mercury of lead shots to keep it upright while floating. The stem is unevenly calibrated in kg/m3 or g/cm3 units of density. The numbers denoting density in the stem are bigger at the bottom, decreasing upward because a hydrometer sinks less in liquids having more density and sinks more in liquids of less density. 
b. Constant immersion or variable weight hydrometer: The hydrometer which measure the density by adding the weights to immerse to the constant level is called constant immersion hydrometer. It measures the relative density. It is also called the Nicholson hydrometer. It is made to sink to the same level in the water as well as in the liquid by placing weights on its pan at the top so that it displaces the same volume of liquids.
Atmosphereic pressure: 
The pressure exerted by column of air on unit surface area is called atmospheric pressure. It is measured by barometer. Atmospheric pressure can not be changed by us. The atmospheric pressure at sea level  is considered as normal or slandered pressure. Its value at sea level is about 101300 N per meter or 760 mm of Hg.
Air pressure: 
It is the pressure exerted by the air enclosed in a container. It is measured by manometer. Air pressure inside the enclosed container can be changed.
Barometer:
A barometer is an instrument that is used to measure the atmospheric pressure of certain place. There are two types of barometers, a. Mercury barometer (or  Fortin's barometer )  and   b. Aneroid barometer.
A simple barometer consists of a long glass tube filled with mercury and turned upside down in a container of mercury. It works by balancing the mercury level inside the glass tube against the air pressure outside. The level of mercury in the tube gives the measurement of the air pressure.
Syringe: 
A syringe is a medical instrument which is used to draw blood out from a body and to inject medicine through blood. It consists of three parts; piston, barrel and needle. It is made up of glass or plastic. It has a scale on the barrel that indicates the volume of liquid inside it.
When piston is pulled outward keeping needle inside the medicine or liquid, a partial vacuum is created inside the barrel. As a result, there is low pressure inside it so liquid or medicine flows into the the syringe. Likewise, When the piston is pushed inside, there is high pressure in the barrel and pushes the liquid or medicine out of the syringe.
Air Pump: 
Air pump is an instrument used to pump air inside a tube.
Water Pump:
A water pump is an instrument that is used to pull under-ground water. 

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Force

Force

Force is the external agent which changes or tries to change the state of rest or motion of a body. It is also called pull or push. Its SI unit is Newton (N). It is derived physical quantity. Since it has both magnitude and direction it is a vector quantity.

Force can produce the following effects on a body:

• It can change the direction of motion.
• It can change the speed of a body.
• It can change the state of rest or motion of a body.
• It can change the shape and the size of a body.

Examples of forces are frictional force, gravitational force, magnetic force, centripetal and centrifugal force, nuclear force etc.
The mutual force of attraction between any two heavenly bodies in the universe is called force of gravitation or simply gravitation. Due to this force earth revolve around the sun. Before 1542 AD it was believed that the earth is the center of the solar system and all other planets and sun revolve around the earth which is called geocentric model, proposed by ancient Greek. Later Nicholas Copernicus introduced new model called a heliocentric model in which sun is the center of the solar system and earth revolve around the sun including remaining planets.

Consequences of gravitational force:

• Existence of atmosphere
• Existence of solar system.
• Revolution of satellites around the planets.
• Revolution of artificial satellites around the earth
• Revolution of planets around the planets.

Newton’s Universal law of gravitation:

Newton’s law of gravitation states that “The force of gravitation between two masses or bodies anywhere in the universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers”.

It is called universal law because this law holds true for all the bodies, whether it is massive or microscopic placed anywhere in the universe. The effect of gravitation is more in liquid than in solid because inter-molecular force of attraction is less in liquid and it is weaker than in solid.

Consider, two bodes with mass m₁ and m₂ which are 'd' distance apart from their centers.

According to Newton’s law of gravitation,

$F\alpha {{m}_{1}}{{m}_{2}}$..............................1

$F\alpha \frac{1}{{{d}^{2}}}$...............................2

Combining equation 1 and 2 we get,

Or, $F\alpha \frac{{{m}_{1}}{{m}_{2}}}{{{d}^{2}}}$.....................3

Therefore, $F= \frac{G{{m}_{1}}{{m}_{2}}}{{{d}^{2}}}$......................................4

Where, G is a proportionality constant which is commonly known as universal gravitational constant and its value is 6.67×10 -11 Nm²kg ^{– 2} .

From equation (4) it is found that there exists a force of attraction between any two bodies in the universe separated by some distance. The force acting between the bodies has the following features:

• This force is attractive in nature
• The direction of force is along the line joining their centers.

Definition of G:

According to Newton’s law of gravitation,

$F=\ \frac{G{{m}_{1}}{{m}_{2}}}{{{d}^{2}}}$

If we put the value of m₁ =1kg and m₂ = 2 kg and d = 1m in above expression, we get

F = G ……………………5

From equation (iv) we can define gravitational constant as ‘The force of attraction between two bodies, each with a mass of 1kg kept at a distance of 1 meter apart.

Value of G = 6.67×10 -11 Nm²kg ^{– 2} was found experimentally by Henry Cavendish. This value of G shows that gravitational force is very weak force. The value of G is independent of the nature and composition of masses and medium in which they remain. It cannot be noticed if the two masses are small but becomes noticeable when the two bodies are extremely large.

G is called a universal gravitational constant because it exist everywhere in the universe and it is applicable to all the objects in the universe whether the objects are terrestrial or celestial or of space.

Gravity:
Gravity of a planet or satellite is defined as 'The gravitational force between a planet or satellite and a body on or near to ifs surface.' It is always directed towards the centere of the object. Due to gravity an object acquires weight. 

Let us consider the mass of earth be 'M' and an object is place on the surface of  the earth. The distance between their centers be 'R' which is nearly equal to the radius of the earth.

According to Newton's law of gravitation;

 $F=\ \frac{G{{m}_{1}}{{m}_{2}}}{{{R}^{2}}}$  .........i

or, $mg=\frac{GMm}{{{R}^{2}}}$                      [$\because $F=W=mg]

Where, 
            M=mass of earth
            R=Radius of earth
            W=Weight of the object
            G=Gravitational constant= 6.67×10 -11 Nm²kg ^{– 2}

or, $g=\frac{GM}{{{R}^{2}}}$

or, $g\alpha \frac{GM}{{{R}^{2}}}$...................ii

Here, equation (ii) shows that gravity of a planet is directly proportional to the Mass of the planet or satellite (M) that means if the radius of the planet is increased keeping mass M constant, the gravity will be less and if the radius is decreased, the gravity will increase. 
Equation (ii) also shows that acceleration due to gravity is independent of mass of and object.

Differences between gravity and gravitation:

Sn	Gravity	Gravitation
1	Gravity is the force between a planet or a satellite and a body on or near to the surface	Gravitation is the mutual force of attraction between two heavenly bodies in the universe
2	It is not universal force.	It is universal force.
3	It is expressed as $F=\frac{GMm}{{{R}^{2}}}$	It is expressed as $F=\frac{G{{M}_{1}}{{M}_{2}}}{{{d}^{2}}}$
4	Due to gravity all the objects fall towards the earth or planet.	Due to gravitation, the earth and other planets revolve around the sun.

Acceleration due to gravity: 

Before the Galileo's idea of falling bodies, Greek philosopher, Aristotle believed that when heavy body and light body fall form the same height, the heavier one falls faster on the earth than the lighter one which which was prove to be wrong after Galileo's experiment. According to Galileo the heavier and the lighter objects fall simultaneously on the earth surface. In other words, the acceleration produced on freely falling objects does not depend on the mass.
When Galileo performed the experiment with feather and coin, he found that feather fall more slowly than the coin. It was found that feather fell slowly due to the air resistance when Newton performed the experiment in both in vacuum and air. In vacuum where there is no air resistance both the coin and feather fell simultaneously.

Feather and coin experiment in air

Every object thrown upward falls towards the center of the earth with increase in velocity and hits the ground. The increase in the velocity of an object with produces acceleration on it. The acceleration produced on a freely falling body due to the force of gravity is called acceleration due to gravity. It is denoted by 'g' and its unit is m/s².

Feather and coin experiment in vacuum

Variation of acceleration due to gravity: 
i. Variation of acceleration due to gravity due to the shape of earth:
Earth is not perfectly spherical in shape. It is bulged out in the equatorial region and flattened in the polar regions. So, the radius towards the pole is smaller than the radius towards the equatorial region i.e R_e>R_p. Hence the value of acceleration due to gravity varies from place to place on earth. The value of acceleration due to gravity is 9.78 m/s² at equator and 9.83 m/s² at poles.

ii. Variation of acceleration due to gravity with altitude: 
If we consider the radius of the earth be 'R' at the surface 'h' be the distance at certain height above the surface of the earth. Then the total distance between the center of the earth and the point at certain height 'h' be (R+h). Then,

Acceleration at surface of the earth is $g=\frac{GM}{{{R}^{2}}}$  ........................a

Acceleration at certain height 'h' is ${{g}^{'}}=\frac{GM}{{{(R+h)}^{2}}}$......................b
Putting the value of GM=gR2 from equation (a) in equation (b) we get,

$g={{g}^{'}}\frac{{{R}^{2}}}{{{(R+h)}^{2}}}$

iii) Variation of acceleration due to gravity with depth: 

If we consider the earth to be a perfect sphere of mass M and radius R. Let 'P' be any point 'x' under the surface of the earth.Then acceleration due to gravity at point 'p' will be,

$g=\frac{GM}{{{(R-x)}^{2}}}$

Here, R-x is less than R, so the value of 'g' under the surface is less than at the surface.

Differences between Acceleration due to gravity (g) and Gravitational constant (G):

S.no	Acceleration due to gravity (g)	Gravitational constant (G)
1	It is acceleration produced on a freely falling body due to gravity.	It is the mutual force of attraction between two bodies with unit mass kept at 1m distance.
2	It is vector quantity.	It is scalar quantity.
3	It differs from place to place.	It remains constant everywhere.
4	It unit is m/s2	Its unit is Nm2/Kg2.

Differences between Acceleration due to gravity and gravity: 

S.no	Acceleration due to gravity (g)	Gravity
1	It is acceleration produced on a freely falling body due to gravity.	It is the force of attraction between planet or satellite and an object on or near to the surface.
2	It is the effect of gravity.	It is the cause of the acceleration due to gravity.
3	It unit is m/s2	Its unit is N.

Free fall:
 The falling of an object towards the center of another object without any external resistance is called free fall. In other words, the falling of an object with an acceleration equal to the acceleration produced by the pull of the earth is called free fall. E.g. The falling of an object on the surface of the moon.
Weightlessness: 
The condition in which the weight of an object seems to be zero is called weightlessness. It is also called as the apparent loss of weight of an object. The state of weightlessness can be observed in the following conditions:

• A body at the centere of the earth.
• A body in a freely falling elevator or lift.
• A body in the neutral point in space.
• A body inside the satellite orbiting a planet. 

Differences between free fall and weightlessness:

S.no	Free fall	Weightlessness
1	The motion possessed by a body without any external resistance is called free fall.	The condition in which the weight of an object seems to be zero.
2	It is cause of weightlessness.	It is the effect of free fall.

Differences between Weightlessness during free fall and Weightlessness in space: 

S.no	Weightlessness in free fall	Weightlessness in space
1	It occurs during free fall of an object.	It occurs when acceleration due to gravity is zero.
2	In this condition, a = g.	In this condition, g = 0.
3	It occurs due to gravity	It does not occur under the influence of gravity.
4	It occurs within the gravitational field.	It occurs in the outer space.
5	It is apparent weightlessness	It is true weightlessness.

Mass: The quantity of matter contained in a body is called mass of that object. Its SI unit is 'Kg'. It  is scalar quantity and measured by beam balance. It is a constant quantity. It can be zero.

Weight: It is the measurement of force of gravity acts on a body. It is a vector quantity with SI unit 'N'. It is measured by spring balance. Weight of an object will be zero, at the centere of the earth or at the place where R = 0.  
Differences between Mass and Weight: 

S.no	Mass	Weight
1	It is the quantity of matter contained in a body.	It is the amount of force of gravity acted on a body.
2	Mass of an object does not vary from place to place	Weight of an object varies from place to place.
3	Mass is scalar quantity	Wight is the vector quantity.
4	Mass is measured by beam balance.	It is measured by spring balance.
5	There is no place or condition under which mass of an object be zero.	Weight of an object will be zero, at the centere of the earth or at the place where R = 0.
6	The SI unit of mass is kilogram(Kg).	The SI unit of weight is Newton(N).

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