Gravitation Class 9 Notes Science Chapter 9

Gravitation

Orbit : The paths followed by the planets around the sun and by the satellites around the planets are known as orbits.

According to Newton's first law of motion, these planets and satellites can move in circular orbits only if some force is acting on them. The mystery of the motion of all these heavenly bodies (i.e. planets and satellites) in nearly circular orbits was solved by Issac Newton when he observed that an apple fall from a tree towards the earth. Therefore, he proposed that all particles or objects in the universe attract each other in the same manner as the earth attracted the apple.

"The force of attraction between any two particles in the universe is called gravitation or gravitation force".

Centripetal force :- The force act towards the centre is called centripetal force.

Example: Consider a girl whirling a stone along a circular path. If the girl releases the stone at some point, the stone flies off along the tangent, at that point on the circular path. Let us discuss this observation carefully.

Before the release of thread, the stone was moving with a certain uniform speed and changed its direction at every point. Because of the change in direction, it moved with a variable velocity and has some definite acceleration. The force that causes this acceleration and makes the stone move along the circular path, acts towards the centre, i.e., towards the hand of the girl. This force is called centripetal force.When the thread is released, the stone does not experience the centripetal force and flies off along a straight line. This straight line is always tangent to the circular path.

Newton's Universal law of gravitation

" Every particle in the universe attracts every other particle with of force which is directly proportional to the product of two masses and inversely proportional to the square of the distance between them ". 

The direction of the force is along the line joining the two masses.

If m₁ and m₂ are the masses of two bodies separated by a distance d and F is the force of attraction between them, then

F µ m₁m₂ F µ 

F µ

or F = where G is a constant of proportionality and known as the constant of universal gravitation
and is equal to

Fd² = Gm₁m₂ G = 

If m₁ = m₂ = 1kg and d = 1m, then G =  or G = F

i.e. Universal gravitational constant is the force of attraction (in newton) between two bodies of mass 1(kg) each lying 1(m) distance apart.

Characteristics of Gravitational force :-

1. Gravitational force between two bodies or object does not need any contact between them. It means, gravitational force is action at a distance.

2. Gravitational force between two bodies varies inversely proportional to the square of the distance between them. Hence, gravitational force is an inverse square force.

3. The gravitational forces between two bodies or objects form an action-reaction pair. If object A attract object B with a force F₁ and the object

B attracts object A with a force F₂, then

F₁ = F₂

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Unit of gravitational constant

G = 

In SI unit G =  = Nm² kg^-2

In CGS unit of dyn cm² g^-2.

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Do you know ?

1. The value of G = 6.67 × 10^-11 N m² kg^-2 or 6.67 × 10^-8 dyn cm² g^-2

2. The value of G is same throught the universe and hence G is known as universal grvitaional constant.

3. Value of G was determined in laboratory by sir henry cavendish.

4. Sinc the value of G is very small, so the gravitational force is a very weak force.

5. Gravity : The gravtional force between a body on the earth is called gravity.

IMPORTANCE OF THE UNIVERSAL LAW OF GRAVITATION

The Universal law of gravitation given by Newton has explained successfully several phenomena. For example:

1. The gravitational force of attraction of the Earth is responsible for binding all terrestrial objects on the Earth.

2. The gravitational force of the Earth is responsible for holding the atmosphere around the Earth.

3. The gravitational force of the Earth is also responsible for the rainfall and snowfall on the Earth.

4. The flow of water in rivers is also due to gravitational force of the Earth on water.

5. The moon revolves around the Earth on account of gravitational 'pull of the Earth on the Moon.

Even all artificial satellites revolve around the Earth on account of gravitational pull of the Earth on the satellites.

6. The predictions about solar and lunar eclipses made on the basis of this law always come out to be true.

The gravitational force plays an important role in nature.

7. All the planets revolve around the sun due to the gravitational force between the sun and the planets. The force required by a planet to move around the sun in circular path (known as centripetal force) is provided by the gravitational force of attraction between the planet and the sun. Thus, gravitational force is responsible for the existence of the solar system.

8. Tides in oceans are formed due to the gravitational force between the moon and the water in oceans.

9. Gravitational force between a planet and its satellite (i.e., moon) decides whether a planet has a moon or not. Since the gravitational force of the planets like mercury and venus is very small, therefore, these planets do not have any satellite or moon.

10. We stay on the earth due to the gravitational force between the earth and us.

GRAVITATIONAL FORCE BETWEEN LIGHT OBJECTS AND HEAVY OBJECTS

The formula applied for calculating gravitational force between light objects and heavy objects is the .

same, i.e., F =. Let us take three cases:

1. When two bodies of mass 1 kg each are 1 metre apart.

Sol. i.e., m₁= m₂ = 1 kg, r = 1 m

Taking G = 6·67 × 10^-11 Nm²/kg², we obtain gravitational force of attraction,

F =  = = 6·67 × 10^-11 N

which is extremely small. Hence, we conclude that though every pair of two objects exert gravitational pull on each other, yet they cannot move towards eachother because this gravitational pull is too weak.

2. When a body of mass 1 kg is held on the surface of Earth.

Sol. Here, m₁ = 1 kg

m₂ = mass of Earth = 6 × 10²⁴ kg

r = distance of body from centre of Earth = radius of Earth = 6400 km = 6.4 × 10³km = 6·4 × 10⁶ m

Gravitational force of attraction between the body and Earth,

F = =  = 9.8 N

It means that the Earth exerts a gravitational force of 9·8 N on a body of mass one kg. This force is much larger compared to the force when both the bodies are lighter. That is why when a body is dropped from a height, it falls to the Earth.

3. When both the body are heavy.

Sol. Let us calculate gravitational force of attraction between Earth and the Moon.

Mass of Earth, m₁ = 6 × 10²⁴ kg

Mass of Moon, m₂ = 7·4 × 10²² kg

Distance between Earth and Moon, r = 3·84 × 10⁵ km = 3:84 × 10⁸ m

Gravitational constant, G = 6·67 × 10^_11 Nm²/kg²

The gravitational force between Earth and Moon,

F = == 2·01 × l0²⁰ N

which is really large. It is this large gravitational force exerted by Earth on Moon, which makes the Moon revolve around the Earth.

KEPLER'S LAWS OF PLANETARY MOTION

Johannes Kepler was a 16^th century astronomer who established three laws which govern the motion of planets (around the sun). These are known as Kepler's laws of planetary motion. The same laws also describe the motion of satellites (like the moon) around the planets (like the earth). The Keplar's laws of planetary motion are given below.

1. Kepler's first law : The planets move in elliptical orbits around the sun, with the sun at one of the two foci of the elliptical orbit.

2. Kepler's second law : Each planet revolves around the sun in such a way that the line joining the planet to the sun sweeps over equal areas in equal intervals of time.

3. Kepler's third law : The cube of the mean distance of a planet from the sun is directly proportional to the square of time it takes to move around the sun.The law can be expressed as :

r³ µ T²

or r³ = constant × T²

or  = constant

where r = Mean distance of planet from the sun

and T = Time period of the planet (around the sun)

Through Kepler gave the laws of planetary motion but he could not give a theory to explain the motion of planets. It was Newton who showed that the cause of the motion of planets is the gravitational force which the sun exerts on them. In fact, Newton used the Kepler's third law of planetary motion to develop the law of universal gravitation.

Newton's inverse-square rule

The force between two bodies is inversely proportional to the square of distance between them' is called the inverse-square rule.

Consider a planet of mass m moving with a velocity (or speed) v around the sun in a circular orbit of radius r, A centripetal force F acts on the orbiting planet (due to the sun) which is given by :

F = 

The mass m of a given planet is constant 

If the planet takes time T to complete one revolution (of 2pr) around the sun, then its velocity v is given by:

v = 

The factor 2p is a constant 

Now, taking square on both sides 

If we multiply as well as divide the right side of this relation by r 

The factor  is constant by Kepler's third law. 

by putting in place of v² in relation   or 

Newton's third law of motion and gravitation

The Newtons third law of motion also holds good for the force of gravitation. This means that when earth exerts a force of attraction on an object, then the object also exerts an equal force on the earth, in the opposite direction.

According to Newton's second law,

Force = Mass × Acceleration

F = ma

Acceleration = 

or a = 

The mass of earth is very very large and acceleration produced in the earth very very small and cannot be detected with even the most accurate instrument available to us.

Free Fall :-

Any object dropped from some height always falls towards the earth. If a feather and a stone are dropped from the top of a tower, it is observed that feather falls onto the ground much later than the stone. So, it was thought that object of different masses dropped from same height take different times to reach the ground.

However, Galileo dropped three iron balls of different masses simultaneously from the top of the tower of Pisa and found that all the three balls reached the earth's surface at the same time.

Galileo explained that the feather suffered much air resistance during fall because of its large surface area. Due to this opposing force, feather takes longer time to reach the ground than the stone. He further explained that if air resistance is eliminated, both feather and the stone will reach the ground simultaneously.

Conclusion :- Galileo concluded that the bodies of different masses dropped simultaneously from the same height hit the ground at the same time, if air resistance is neglected.

Definition of Free Fall :-

The falling body on which only force of gravitaion of the earth acts is known as freely falling body and such fall of a body is known as free fall. A freely falling body has acceleration equal to acceleration due to graveily(g).

Experiment verification :-

This fact was verified experimentally by Robert Boyle just after the death of Galileo. Robert Boyle used his newly invented vacuum pump to evacuate the air from a long jar containing a lead bullet and a feather. Then he inverred the jar and found that both the bullet and the reached the bottom of the jar at the same time.

If the air resistance is neglected or not taken into account, then the only force acting on the falling body is the force of gravitation of the earth. This force of gravitation of the earth is constant and hence produces a constant acceleration in the body. Since this acceleration is produced by the gravitational force of the earth and hence known as acceleration due to gravitational force of earth or acceleration due to gravity.

Acceleration due to gravity :-

The acceleration with which a body falls towards the earth due to earth's gravitational pull is known as acceleration due to gravity. It is denoted by 'g'.

Thus, all bodies irrespective of their masses fall down with constant acceleration.

Determination of value of g

When a body of mass m is dropped from a certain distance R from the centre of earth of mass M, then the force exerted by the earth on the body is

F =  ............(i)

Let this force produces an acceleration a in mass m.

F = ma or F = mg ... (ii)

From (i) and (ii),

For bodies falling near the surface of earth, this acceleration is called acceleration due to gravity and is represented by g.

  ... (A)

where M is the mass of the earth i.e., 6 × 10²⁴ kg and R, the radius of the earth i.e., 6.4 × 10⁶ m

Value of g on moon

Mass of moon = 7.4 × 10²² kg and its radius = 1,740 km

or R = 1,740,000 m = 1.74 × 10⁶ m

g = = 1.63 ms-²

Mass of the earth

We can determine mass of the earth from equation (A)

Average density of the earth

It can also be determined from equation (A) above.

g =       Volume = 

Mass = Volume × density

d = 

Taking the earth to be a sphere of radius R

 d = 

or d = 5.5 × 10³ kg m^-3

Calculation of acceleration due to gravity on the moon and to prove that it is 1/6th of the acceleration due to gravity on the earth.

Mass of the moon (M) = 7.4 × 10²² kg Radius of the moon (R) = 1.74 × 10⁶ m

Gravitational constant (G) = 6.7 × 10^_11 Nm²/kg²

Acceleration due to gravity on the moon, g = 

HIGH ORDER THINKING SKILL

Variation in the value of 'g' :-

1. Variation in the value of 'g' with the shape of the earth.

The acceleration due to gravity 'g' on the surface of the earth is given by

g =  ... (1)

This expression for 'g' is calculated by considering the earth as a spherical body.

In fact, the earth is not sphereical in shape but it is egg shaped as shown in figure.

Therefore, the radius of the earth (R) is not constant throughout. Hence, the value of 'g' is different at different points on the earth.

The equatorial radius (R_E) of the earth is about 21 km longer than its polar radius R_P).

Now from equation (1), value of 'g' at equator is given by g_e =  ... (2)

Value of 'g' at pole is given by g_P =  Dividing equation (3) by equation (2), we get

 ... (A)

Since R_E > R_P  g_P > g_E

Thus, value of 'g' is more at equator than at poles.

2. Variation in the value of 'g' with the altitude (or height) above the surface of the earth.

We know, acceleration due to gravity on the surface of the earth. The distance of the body from the centre of the earth = (R h).

Therefore, acceleration due to gravity at height 'h' is given by

gh =  ... (2)

Dividing (2) by (1), we get

or  ... (3)

Since (R h) > R

 < 1 or gh < g

This shows that the value of 'g' decreases as we go higher and higher.

Thus, value of 'g' decreases with the height from the surface of the earth.

3. Variation in the value of 'g' with depth below the surface of the earth.

The value of 'g' decreases with depth below the surface of the earth.

The value of 'g' at depth d below the surface of the earth is given by

g_d = g g_d = g

g = 

m =  p R³ r

g =  ... (1)

g' = 

M' = p (R d)³ r

g' = 

g' =  pG (R d) r ... (2)

division (2) by (1)

At the centre of the earth d = R g' = g₀

g₀ = g  = 0

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Do you know ?

1. The acceleration due to gravity of a planet depends on its mass and its radius. Its value is high if mass is large and radius is small.

2. The value of g at the surface of earth is 9.8 ms^_2 on an average.

3. The value of g decreases with height. 

4. The value of g decreases with depth.

5. The value of g is more at poles and less at equator.

6. The value of g is zero at the centre of the earth.

7. The value of acceleration due to gravity is minimum at planet mercury and maximum at planet jupiter.

8. Acceleration due to gravity is independent of mass shape size etc of falling body i.e. there will be equal acceleration in a light and heavy falling body.

9. The rate of decrease of the acceleration due to gravity with height is twice as compared to that with depth.

10. If the rate of rotation of earth increases the value of acceleration due to gravity decreases at all places on the surface of the earth except at of poles.

11. If earth stops rotating there will be increase in the value of acceleration due to gravity at equator by a value =RW² = 0.034 m/s² but there will be no change in the value of g at poles.

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Gravitation and gravity

DIFFERENCE BETWEEN 'g' AND 'G'

 
Equation of motion of freely falling bodies

When the bodies are falling under influence of gravity, they experience acceleration g i.e., 9.8 ms^-2. However, when these are going up against gravity, they move with retardation of 9.8 ms^-2. All the equations of motion already read by us are valid for freely falling body with the difference that a is replaced by g. For motions vertically upwards (a) is replased by (-g).

The equation of motion

v = u at

Replace a = g

 when body falls in downward

 when body through upward

s = ut at²

Replace a = g & s = h

v² _ u² = 2as

Replace s = h

MASS

The amount of matter contained in a body is called its mass

or

The measure of the quantity of matter In a body is called its mass.

The mass of a body is a scalar quantity. It is independent of surroundings and theposition of the body. It is a constant quantity for a given body.

Mass is measured in kilograms (kg) in SI system.

Characteristics of mass of a body :-

1. Mass of a body is proportional to the quantity of matter contained in it.

2. Mass of a body does not depend on the shape, size and the state of the body.

3. Mass of a body remains the same at all place. This means, the mass of a body will be same throughour the universe. This is because the quantity of matter contained in the body does not change throughout the universe.

4. Mass of a body does not change in the presence of other bodies near it.

5. Mass of a body is a scalar quanity.

6. Mass of a body can be measured with the help of a beam balance.

7. Masses of object or bodies are added algebrically.

Weight

The force with which a body is attracted by the earth is known as the weight of the body.

When the earth attracts a body with a gravitational force, the body accelerates towards the earth with an acceleration due to gravity (g).

Thus, the force with which body of mass m is attracted by the earth is given by

F = ma = m × g = mg

This force is known as the weight of the body. Weight of a body is denoted by W.

Weight, W = mg

Weight has both magnitude and direction. Hence weight is a vector quantity.

Unit of Weight :-

SI unit of weight is same as that of the force i.e., newton (N).

Variation in the weight of a body

Weight of the body is given by

W = mg

So the weight of a body depends upon (i) the mass of the body and (ii) value of acceleration due to gravity.

The mass of a body remains the same throughout the universe, but the value of 'g' is different places. Hence the weight of a body is different at different place.

1. The value of 'g' is more at poles and less at the equator. Therefore, weight of a body is more at the poles and less at the equator. In other words, a body weighs more at the poles and less at the equator.

2. The value of 'g' on the surfaces of different planets of the solar system is different, therefore, the weight of a body is different on the different planets.

3. The value of 'g' decreases with height from the surface of the earth. Therefore, the weight of a body also decreases with height from the surface of the earth. That is why, the weight of a man is less on the peak of Mount Everest than the weight of the man at Delhi.

4. The value of 'g' decreases with depth from the surface of the earth. Therefore, the weight of a body decreases with depth from the surface of the earth.

5. The value of 'g' at the centre of the earth is zero, hence weight (= mg) of the body is zero at the centre of the earth.

PRACTICAL UNITS of weight

In SI, the weight is also measured in kg f or kg wt.

Therefore, kilogram force or kilogram weight is force with which a mass of 1 kg is attracted by centre of earth.

1 kg f = 1 kg wt = 1 kg × 9.8 m/s² = 9.8 kg ms² = 9.8 N 

In CGS, the practical unit of weight is grams force or g wt or 1g f or 1g wt is force with which a mass of 1g is attracted by the centre of the earth.

g = 9.8 ms²

g = 9.8 × 100 cm/s²

1 g f = 1 g wt = 1 g × 980 cm s² = 980 dyne 

Thrust and pressure

Thrust : Force acting normally on a surface is called the thrust.

Thrust is a vector quantity and is measured in the unit of force, i.e., newton (N).

Pressure : The thrust acting on unit area of the surface is called the pressure.

If a thrust F acts on a area A, then pressure (P) = 

Pressure is directly proportional to the force.

Pressure in inversely proportional to the area.

For the examples :

Ex. A sharp knife cuts easily than a blunt knife by applying the same force.

Ex. A sharp needle pressed against our skin pierces it. But a blunt object with a wider contact area does not affect the skin when pressed against it with the same force.

Some interesting aspects of pressure

1. The foundation of a building or a dam has a large surface area so that the pressure exerted by it on the ground is less. This is done to prevent the sinking of the building or the dam into the ground.

2. The tyres of a bus or a truck have larger width than those of a car. Further, the number of tyres of heavy vehicles is more than four. This is done to enable the tyres to carry more weight and to prevent sinking

into ground.

3. A sleeping mattress is so designed that when you lie on it, a large area of your body comes in its contact. This reduces the pressure on the body and sleeping becomes comfortable.

4. Railway track are laid on large sized wooden or iron sleepers.

We know, Pressure = 

The weight (i.e., thrust) of the train is spread over a large area of the sleepers. Therefore, the pressure acting on the ground under the sleepers is reduced. This prevents the sinking of the ground under the weight of the train.

5. A sharp knife is more effective in cutting the objects than a blunt knife.

The pressure exerted = 

The area under the shrap knife is less than the area under the blunt knife. Hence, the pressure exerted by the sharp knife is more than the pressure exerted by the blunt knife on an object. Therefore, the sharp knife penetrates easily into the object than the blunt knife when same force is applied in both the cases. Hence, a sharp knife cuts the objects easily than a blunt knife.

6. A camel walks easily on the sandy surface than a man inspite of the fact that a camel is much heavier than a man.

This is because the area of camel's feet is large as compared to the area of man's feet. So the pressure exerted by camel on the sandy surface is very small as compared to the pressure exerted by man. Due to large pressure, sand under the feet of a man yields (i.e. sink) and hence he cannot walk easily on the sandy surface.

7. A sharp needle, pierce the skin easily but not a blunt needle although the force applied on both the needles is same.

Pressure exerted = 

The area under the pointed end of the sharp needle is very small as compared to the area under the pointed end of the blunt needle. So pressure exerted by the sharp needle is much more than the pressure exerted by the blunt needle. Hence a sharp needle pierces the skin easily than the blunt needle.

8. It is painful to hold a heavy bag having strap made of a strong and thin string.

We know, Pressure = 

When we hold a heavy bag having strap made of a strong and thin string, then the are under the strap is small. Hence, large pressure is exerted by the strap on our fingers or shoulder. Due to this large pressure, the strap tends to cut the skin and hence pain is caused.

9. The army tank has a large weight. Therefore to avoid large pressure on the ground its weight is distributed throughout the tank. This is done by making the tank run on a steel track rather than on wheels. The steel

tracks reduce the pressure of the ground.

Units of Pressure. The SI unit of pressure is called pascal (Pa) in honour of Blaise Pascal.

I Pa = 1 N/m²

One pascal is defined as the pressure exerted on a surface area of 1 m² by a thrust of 1 N (acting normally on it).

Other units of pressure are bar and millibar where

1 bar = 10⁵ N/m² and 1 millibar = 10² N/m²

It is a common practice in meteorology to measure atmospheric pressure in bars and millibars. Further, 
1 atmospheric pressure (1 atm) = 101·3 k Pa = 1·013 bar = 1013 m bar

Density

Density of a substance is defined as its mass per unit volume.

density(d) = 

Unit of density :- Since mass (M) is measured in kilogram (kg) and the volume (V) is measured in metre³ (m³), the unit of density is kg/m³. In cgs system, the unit of density is g cm³.

These units are related as : 1 g cm³ = 1000 kgm³.

RELATIVE DENSITY

Relative density (R.D.) of a substance is the ratio of the density of the substance to the density of water at 4°C.

Thus, Relative density =

Unit of Relative Densit

Since relative density is a ratio of two similar quantities, it has no unit.

Further, relative density =

= 

If the volume of a given substance is equal to the volume of water at 4°C,

relative density = 

Relative density can also be defined as the ratio between the mass of the substance and the mass of an equal volume of water at 4°C. 

PRESSURE IN FLUIDS

A substance which can flow is called a fluid. All liquids and gases are thus fluids. We know that a solid exerts pressure on a surface due to its weight. Similarly, a fluid exerts pressure on the container in which it is contained due to its weight. However, unlike a solid, a fluid exerts pressure in all directions.

A fluid contained in a vessel exerts pressure at all points of the vessel and in all directions.

All the streams of water reach almost the same distance in the air.

PASCAL'S LAW

In an enclosed fluid, if pressure is changed in any part of the fluid, then this change of pressure is transmitted undiminished to all the other parts of the fluid.

BUOYANCY

When a body is partially or wholly immersed in a liquid, an upward force acts on it which is called upthrust or buoyant force. The property of the liquids responsible for this force is called buoyancy.

Buoyancy is a familiar phenomenon : a body immersed in water seems to weigh less than when it is in air. When the body is less dense than the fluid, then it floats. The human body usually floats in water, and a helium filled balloon floats in air.

When a body is immersed in a fluid (liquid or gas) it exerts an upward force on the body. This upward force is called upthrust or buoyant force (U or F_B) and the phenomenon is termed as buoyancy. Thus buoyancy or upthrust is the upward force exerted by a fluid (liquid or gas) when a body is immersed in it.

It is a common experience that when a piece of cork is placed in water it floats with two-fifth of its volume inside water. If the cork piece is pushed into water and released it comes to the surface as if it has been pushed by someone from inside due to the buoyant force exerted by fluid.

Factor on which buoyancy depends :-

Upthrust depends upon the following two factors :

(a) The volume of the body submerged in the fluid. :- It is found that greater the volume of a body greater is the upthrust it experience when inside a fluid.

(b) The density of fluid in which the body is immersed :- It is also found that greater the density of the fluid greater is the upthrust it applies on the body.

Effect on buoyancy :-

When a body is partially or fully immersed in a fluie, then following vertical forces are experienced by it.

(a) Its weight (W) acting vertically downwards through the centre of the body.

(b) Force of buoyancy (U or B or F_B) or upthrust, acting vertically upwards through the centre of gravity of the body.

The following three cases arise :-

(1) When W < U, the body floats : In this case the body will rise above the surface on the liquid to the extent that the weight of the liquid displaced by its immersed part equals the weight of the body. Then the body will float with only a part of it immersed in the liquid. In this case Vrg < Vsg or r < s. Thus if a cork, which has a density less than that of water will rise in water till a portion of it is above water. Similarly a ship floats in water since its density is less than the density of water.

(2) When W > U, the body sinks :- If r and s present the densities of the body and the fluid respectively and V the volume of the body (which is also the volume of the fluid displaced) then Vrg > Vsg or r > s i.e., the body sinks in the fluid in case its density is greater than the density of the fluid. An iron nail has greater density than water, therefore it sinks in water.

(3) When W = U :- The resultant force acting on the body when fully immersed in the fluid is zero. The body is at rest anywhere within the fluid. The apparent weight of the body is zero for all such positions.

Thus, we find that a body will float when its weight is equal to the weight U of the fluid displaced i.e. the upthrust.

ARCHIMEDES PRINCIPLE

Archimedes principle states that :- "Anybody completely or partially submerged in a fluid is buoyent up by a force equal to the weight of the fluid displaced by the liquid".

In other words :- "When a body is partially or completely immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body.

Experimental verification of archimedes principle :-

Consider a container C₁ filled with water upto the level from where pipe P extends out. The other end of pipe P opens to a small container C₂ placed on a weighing balance which measure 00.00 [after the placement of the container C₂]. A block B hangs on a spring balance S which shows a reading of 7 kg.

(a) If we partially immerse the block in water we observe some water flows out from C₁ to C₂ through P. The weighing machine shows a reading 1 kg and the loss of reading in spring balance is 7 - 6 = 1kg. This means that weight of water displaced by the block is equal to loss in weight of block.

(b) Now we completely immerse the block in water, we observe that the weight of water displaced by the block is 5 kg and the reading in spring balance is 2 kg. The loss of weight of block is 7 - 2 = 5 kg. Again we reach the same conclusion that weight of water displaced by the block is equal to the loss in weight of block.

(c) What happens when the block is further immersed? No more water will be displaced by the block and therefore reading shown by weighing machine and spring balance remains unchanged.

Why the spring balance shows a loss in weight of the block when Fig. A block of 7 kg hanging on a the block is immersed in water? This is because of buoyant force on spring balance acting vertically upwards. The loss in weight is equal to the buoyant force.

Conclusion 1:

Buoyant force µ volume of liquid displaced (V). If two bodies of different material have same volume, the buoyant force acting on them, when completely immersed in water, is same.

Instead of water if we take a liquid lighter than water then the volume displaced by the block on complete immersion will be the same but the buoyant force will be less. This is because the density of lighter liquid is less than that of water.

We know that Density = 

Mass = Density × Volume

For lighter liquids, the mass of the liquid displaced is less even when the volume displaced is the same.

Conclusion 2:

Buoyant force µ Density of liquid

i.e., Buoyant force µ d

It has also been found that buoyant force also depends on the acceleration due to gravity.

Conclusion 3 :

Buoyant force µ g

If we combine all the three, we get

Buoyant force µ V dg

Þ Buoyant force µ mg [mass = V × d]

Þ Buoyant force µ Weight of the liquid displaced

Note: A body placed in a gaseous medium is also acted by the upthrust equal to the weight of the gas displaced.

The above facts has been summarised in Archimedes principle which states that the upward force acting on a solid body which is partially or completely immersed in a fluid, is equal to the weight of the fluid displaced.

This upward force is called buoyant force or upthrust.

Applications of Archimedes Principle

1. In designing ships and submarines.

2. Lactometer is based on the Archimedes principle. It is used to determine the purity of a sample of milk.

3. Hydrometer is also based on the Archimedes principle. It is used to determine the density of liquids.

(a) Ships :

Although it is made of iron and steel which are materials denser than water a ship floats in water. This is due to the fact that a floating ship displaces a weight of water equal to its own weight including that of the cargo. The volume of the ship is much larger than the volume of the material with which it is made. Since the empty space in the ship contains air therefore its average density is less than the density of water. Thus a ship floats with a small section under water.

A ship has to move in waters of different seas which have different densities. As a result it sinks more in water with less density than in water with more density. Therefore ships are marked with white lines on its sides called load lines of pimsoll marks. The load lines (called tha Plimsoll mark) on the side of a ship of a ship show how low in the water it can lie and still be safely and legally loaded under different conditions.

(b) Submarines :

A submarine, figure sinks by taking water into its buoyancy tanks. Once submeged, the upthrust is unchanged but the weight of the submarine increases with the inflow of water and it sinks faster. To surface, compressed air is used to blow the water out of the tanks.

Each submarine is provided with ballast tanks. If the submarine has to submerge these tanks are filled with water. This makes the average density of the submarine greater than that of water as a result it sinks. When the submarine has to be surfaced, compressed air is blown into these tanks to expel the water. Again the average density of the submarine becomes less than that of water, hence it floats.