GRAVITATION

Newton’s First law of Motion does not allow any change in the state of the body unless there is net external force acting on it. This means that things falling towards the earth must be due to the some force. Newton generalized the idea and said that every object in this universe attracts every other object with a certain force. Heavier objects exert larger force on each other. The force with which the two objects attract each other due to their masses is called force of gravitation or gravitational force. The force of gravitation exerted by the earth on the bodies is called GRAVITY.

UNIVERSAL LAW OF GRAVITATION OR NEWTON’S LAW OF GRAVITATION

“Every body in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.”

Let us consider two bodies A and B of masses m1 and m2 respectively which are separated by a distance of r, then the force of gravitation F acting on the two bodies is given by:

F α m1 x m2

And F α 1 / r2

OR

where G is a constant and is called Universal gravitational constant .

Since the force of gravitation between the two bodies is inversely proportional to the square of the distance between them

F α 1 / r2

THEREFORE THIS LAW IS ALSO CALLED INVERSE SQUARE LAW

FORCE OF GRAVITATION IS A VECTOR QUANTITY

1) If distance = doubled ; Force of gravitation becomes one- fourth.2) If distance = halved , Force of gravitation becomes four times.3) If mass= doubled ; force of gravitation is doubled.4) Both masses = doubled; force of gravitation becomes four times.

UNIVERSAL GRAVITATIONAL CONSTANT ‘G’

If m1 = 1 unit and m2 = 1 unit and r = 1 unit

Therefore F = G Numerically , Universal gravitational constant (G), is defined as the force of attraction between two bodies of unit masses separated by a unit distance apart.

UNIT OF G

OR

On substituting the SI unit of force: Newton; distance : metre and mass : kilogram,

The unit of G becomes

POINTS TO REMEMBER

Universal Gravitation Constant G is a SCALAR QUANTITY. The value of G is 6.67 x 10-11 Nm2 / Kg2 and is same through out the universe, so it is called

universal gravitational constant. The value of G does not depend upon the nature, size or masses of the bodies. The value of G does not depend upon the nature of the medium between the two bodies. Value of G was determined in laboratory by SIR HENRY CAVENDISH. Since the value of G is very small, so the gravitational force is a very weak force.

( What is the value of G on Jupiter?)

NEWTONS THIRD LAW OF MOTION AND GRAVITATION

According to Newton’s Third Law Of Motion: “ To every action there is an equal and opposite reaction “. Hence two bodies when separated by a distance r experiences equal and opposite forces.

Q A Stone attracts the earth and earth also attracts the stone towards its center. Then, why only stone falls towards the earth but the earth is not moved towards the stone?

The gravitational forces acting on both the stone and the earth are equal in magnitude. Since, the mass of the stone is very small as compared to the mass of the earth, so the acceleration ( a = f / m) produced in the stone is very large and the acceleration produced in the earth is negligible. That is why earth is not seen being pulled towards the stone.Q Calculate the gravitational force between the stone of mass 1 Kg and the earthSolution Distance between the stone and the earth i.e radius of the earth ( r ) = 6.4 x 106 m Mass of the stone (m) = 1 Kg Mass of the earth (M) = 6 x 1024 Kg

G= 6.67 x 10-11 Nm2/kg2

= 9. 8 NAcceleration produced in stone =

Acceleration produced in earth =

So we see stone moving towards the earth and not earth towards the stone

FREE FALLAny object when dropped from a height fall towards the earth. This happens due to the gravitational force exerted by the earth. If a feather and a stone are dropped from the same height, then it is observed that the feather reaches later than the stone. It may be concluded from this heavier objects fall more rapidly than the lighter ones. But Galileo later showed that all bodies whether light or heavy, fall at the same speed towards the earth.He explained that the feather experiences greater air resistance due to its larger surface area.Due to this opposing force feather takes longer time to reach the ground than the stone. But if this air resistance is eliminated (vacuum), both feather and the stone will reach the ground simultaneously.

Q Why does crumpled sheet of paper falls faster than the plain sheet of paper, when dropped from the same height?Q A coin and a piece of paper are dropped simultaneously from the same height. Which of the two will reach the ground first? They are dropped in i) air ii) in vacuum? Give reasons for your answer

CONCLUSION

If air resistance is neglected, then the only force acting on the body is the force of gravitation of the earth. This force of gravitation of the earth is constant and hence produces a constant acceleration. Since this acceleration is produced by the gravitational force of the earth. Hence is known as acceleration due to gravitational force of earth or acceleration due to gravity

DEFINITION OF ACCELERATION DUE TO GRAVITY

The acceleration with which a body falls towards the earth only due to the gravitational pull of the earth is known as acceleration due to gravity.

It is denoted by ‘g’ Its value on the surface of the earth is 9.8 m/s2

All bodies irrespective of their masses fall down with constant acceleration of 9.8m/s2

DEFINITION OF FREE FALL

The falling body on which only gravitational pull of the earth acts is known as freely falling body and such a fall is called FREE FALL.

A freely falling body has acceleration equal to the acceleration due to gravity , that is 9.8m/s2.

Q What is the value of acceleration for a freely falling body?

DERIVATION OF THE EXPRESSION FOR THE ACCELERATION DUE TO GRAVITY (g)

Consider an object of mass m lying on or near the surface of the Earth. Let Me be the mass of the Earth and Re be its radius i.e., Re is the distance between the object and the centre of the Earth.

According to Newton's law of gravitation, the force of attraction (F) between the Earth and the object is

According to Newton's second law of motion this force produces acceleration (g) in the object.

F=ma (a=g)

F=mg

Substituting the value of F in eq (2) we get,

From eq (3) it is very clear that acceleration due to gravity does not depend on the mass m of the object. It only depends on the mass of the Earth (Me) and the distance from the centre of the Earth to the object.

Q Derive a mathematical expression to show that acceleration due to gravity is independent of the mass of the falling body, it only depends on mass of the earth.

FACTORS ON WHICH THE VALUE OF ‘g’ DEPENDS

Directly proportional to the mass of the earth ( Me)

Inversely proportional to the radius of the earth

‘ g ‘ IS A VECTOR QUANTITY

Numerical value of acceleration due to gravity on Earth

Gravitational Constant (G) = 6.6734x10-11 Nm2/kg2

Mass of the Earth (Me) = 6x1024 kg

Radius of the Earth (Re) = 6.4x106 m

VARIATION IN THE VALUE OF ‘g’

o Due to the shape of the eartho Due to altitudeo Due to depth

o Variation of 'g' with altitude

Variation of 'g' with altitude

Let a body of mass m be placed on the surface of the Earth, whose mass is M and radius is R.

Let the body be now placed at a height h above the Earth's surface. Let the acceleration due to gravity at that position be g|.

For comparison, the ratio between g’ and g is taken

(R + h) > R therefore g’ / g < 1 OR g’ < g

The value of ‘ g ‘ decreases with the height above the surface of the earth

o Variation of 'g' with depth

Consider a body of mass m, lying on the surface of the Earth of radius R and mass M. Let g be the acceleration due to gravity at that place.

Let the body be taken to a depth d from the surface of the Earth. Then, the force due to gravity acting on this body is only due to the sphere of radius (R- d). If g’ is the acceleration due to gravity at depth 'd'

The value of ‘ g ‘ decreases with the depth below the surface of the earth

AT THE CENTER OF THE EARTH d = R therefore g = 0

o Variation due to the shape of the earth

Earth is not spherical in shape but egg shaped. Therefore the radius of the earth (R ) is not same everywhere

Value of g at poles is given by

Value of g at equator is given by

Therefore,

Since , Re > Rp , therefore

VALUE OF ‘g’ IS MORE AT EQUATOR AND LESS AT POLES

MASS and WEIGHT MASS

Mass of the body is defined as the quantity of matter contained in the body .Since mass is the measure of inertia of the body so mass is also called inertial mass

SI UNIT OF MASS ---- Kg

CHARACTERISTICS OF MASS Is proportional to the amount of matter contained in the body Does not depend on the shape , size and the state of the body REMAINS SAME AT ALL THE PLACES , that is Constant through out the universe IS A SCALAR QUANTITY Is measured with the help of the beam balance It can never be zero. WEIGHT

The force with which a body is attracted towards the center of 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 acceleration due to gravity( g )

Rp

Re

Pole

Equator

Thus the force which the earth exerts is given by

F = m x gThis force is known as the weight of the body , denoted by WTherefore,

W = m x g Hence weight depends on

Mass of the body

Value of acceleration due to gravity ( g)

SI UNIT OF WEIGHT-------- NEWTON (N)

CHARACTERISTICS OF WEIGHT At a given place the value of g is constant. Therefore W α m at a given place. Hence the weight

of the object is equal to the mass of the body , at a given place Weight is a VECTOR QUANTITY Is measured by spring balance. Since the value of g is different at different places . hence the weight of the body is different at

different places

CHANGE IN Weight , With CHANGE In Acc due to Gravity (g)

1) Since Therefore

Body weighs more at the poles and less at the equator

2) The value of g decreases with increase in height from the surface of the earth, therefore weight of the body decreases with increase in the height WEIGHT OF A PERSON IS LESS ON MOUNT EVEREST THAN AT SEA LEVEL

3) Value of g decreases with increases in depth Therefore AT CENTER OF THE EARTH d= 0 , therefore W = 0

4) The value of g =0, in interplanetary space W=0

Q A body of mass 10 kg is taken to the center of the earth. What will be its i) Mass ii) weight?

Relation between the weight of the body on earth and the weight of the body on moon

Where Me and Re are the mass and radius of the Earth respectively.

Where Mm and Rm are the mass and radius of the moon respectively.

Divide eq (1) by eq (2)

We know that mass of the Earth is 100 times that of the moon and its radius is four times that of the moon.

i.e.

Me = 100 Mm

Re = 4 Rm.

Which means that acceleration due to gravity on moon is 1/6th that on the Earth.

W α g, therefore