Centripetal and Centrifugal Force

When a body moves around a circular path, it experiences an inward force which tends to pull a body towards its
centre. This force applied on a body is called centripetal force. At the same time, a force acts on the same body
which tends to pull it away from the centre. This force is called centrifugal force.
Due to these forces (centripetal and centrifugal), moon and other satellites move around the planets. Similarly, earth
and other planets revolve around the sun due to balanced centripetal and centrifugal force.
Gravitation
The force of attraction between any two or more than two bodies in the universe is called gravitation. Force of
gravitation is an invisible force. If masses of the objects are small, then gravitational force between them is small
and cannot be detected easily. If one of the object is massive, then force of attraction becomes large. Gravitational
force of attraction between any two objects is always mutual, which means that if earth attracts an apple, the apple
also attracts the earth towards with an equal force.
Universal Law of Gravitation
In the year 1687, Sir Isaac Newton proposed a theory in relation with gravitational force, mass and the distance
between their centres.
Newton’s law of gravitation states that everybody in this universe attracts every other body with a force which is:
i) Directly proportional to the product of the masses two bodies and
ii) Inversely proportional to the square of distance between their centres.
Newton’s law of gravitation is applicable for both terrestrial or celestial bodies in the universe. So, this law is also
called Universal law. This force is always attractive; it is never repulsive.
Importance of Newton’s Law of Gravitation
This law explains successfully several unconnected phenomena. Some of these are:
1) It explains about the force which holds us on earth.
2) It explains the motion of moon or other satellites around their planets.
3) It also explains the motion of earth and other planets around the sun.
4) It explains about the occurrence of tides in the ocean. The tide is due to the gravitational force between moon
and the large mass of water in the ocean.
Verification of Newton’s Law of Gravitation

Consider the two bodies of the mass m1 and m2 with force F acting between them towards their centre. If the
distance between their centers is r then,

According to Newton’s universal law of gravitation,

We have F ∝ m1m2……(i) and
1
F ∝ 2 …………(ii)
r
m1 m2
Combining (i) and (ii) we get F ∝
r2
m1 m2
F=G
r2
Where G is proportionality constant which is known as universal gravitational constant.
 The numerical value of gravitational constant (G) is 6.67x10-11.
 From the equation given above, we come to know that, masses of the bodies and the distance between their centres
are the factors affecting the gravitational force.

Universal Gravitational constant (G)

The value of Gravitational constant was found out by Henry Cavendish by using a sensitive balance.
Universal gravitational constant is defined as the force of attraction between any two bodies each of unit masses (1
kg each) placed at a unit distance (1 metre).
Units of G
m1 m2
We know that, F = G
r2
Fr2
We have, 𝐺 =
m1 m2
Nm2
In SI unit, 𝐺= = Nm2kg-2
kgkg
dyne cm2
In C.G.S unit, 𝐺 = = dyne cm2g-2
g.g
Hence, the value of gravitational constant is 6.67 x 10-11 Nm2kg-2 in SI unit.
Value of G is 6.67 x 10-11 Nm2kg-2 everywhere. It is independent of the temperature, pressure, nature of the
intervening medium and chemical composition of the masses of the two bodies.
What happens to the force of attraction between two objects, if:
i. The mass of one object is doubled?
We know that,
m1 m2
F=G
r2
When mass of one object is doubled,
m1’ → 2m1
2m1 m2
F’ = G
r2
m1 m2
Or, F’ = 2 (G )
r2
m1 m2
Or, F’= 2F [ where, F = G ]
r2
∴ Force becomes double its original value.
ii. The distance between them is doubled?
m1 m2
F=G
r2
When distance between them is doubled,
r’ → 2r
m 1 m2
F’ = G
(2r)2
m1 m2
Or, F’ = G
4r2
’ 1 m1 m2
Or, F = [G ]
4 r2
1 m1 m 2
Or, F’= 4 F [where, F = G ]
r2
∴ Force becomes one-fourth its original value.
iii. The distance between them is tripled?
m1 m2
F=G
r2
When distance between them is tripled,
r’ → 3r
m1 m2
F’ = G
(3r)2
m1 m2
Or, F’ = G
9r2
 1 m1 m2
Or, F’ = [G ]
9 r2
1 m1 m2
Or, F’= 9 F [where, F = G ]
r2
∴Hence, force becomes one-ninth its original value.
iv. The masses of both the objects are doubled?
We know that,
m1 m2
F=G
r2
When mass of both the objects are doubled,
m1’ → 2m1
m2’ → 2m2
2m1 2m2
F’ = G
r2
m1 m2
Or, F’ = 4 (G )
r2
m1 m2
Or, F’= 4F [ where, F = G ]
r2
∴ Force becomes four times its original value.

v. Mass of both objects are doubled and distance between them is tripled?
m1 m2
F=G
r2
When distance between them is tripled,
m1’ → 2m1
m2’ → 2m2
r’ → 3r
2m1 2m2
F’ = G
(3r)2
m1 m 2
Or, F’ = 4[G ]
9r2
4 m1 m2
Or, F’ = [G ]
9 r2
4 m1 m 2
Or, F’= 9 F [where, F = G ]
r2
∴Hence, force becomes four-ninth its original value.

Force of Gravitation of the earth (gravity)

Force of attraction with which earth (or other planets) attract every other body, in the gravitational field, towards its
centre is called gravity.
Attraction between a planet (earth) or its satellite and a body, having masses of widely different orders is called
gravity. Forces involved are large and body moves towards the planet. Gravity is a special case of gravitation in
which small bodies move towards huge planet.
Force of gravity is responsible for holding the atmosphere above the surface of the earth, flow of water in rivers,
rainfall, etc. we can keep us firmly on ground is due to gravity.
Acceleration due to Gravity
Acceleration produced in a body due to the force of gravity of the earth or other planets and satellites, is called
acceleration due to gravity. It is denoted by ‘g’and its SI unit is ms-2. When a body falls from a certain height, its
acceleration due to gravity is positive and when a body is thrown upwards, its acceleration due to gravity is negative.
Relation between acceleration due to gravity and radius of the earth
Let ‘M’ be the mass and ‘R’ be the radius of the earth and ‘m’ be the mass of the
body kept at the surface of the earth.
According to the Newton’s law of gravitation the force of attraction between them is
given by:
GMm
F= …………(i)
R2

Also the body is attracted towards the centre of the earth with a force given by
F=mg………………(ii)
From (i) and (ii),
GMm
mg =
R2
GM
g= ………. (iii)
R2

G and M are constant whereas R varies because radius of the earth is more at the equator than at poles.
From equation (iii), We can say that the acceleration due to the gravity is independent to the mass of the body but
depends on the mass and radius of the earth.
To calculate the value of g, we should substitute the values of universal gravitational constant (G), mass of the earth
(M) and radius of the earth (R).
We know that,
Mearth = 6x1024 kg, R= 6.4x106 m (average) and G=6.67x10-11 Nm2kg-2
GM
g=
R2
6.67x10−11 x6x1024
or, g = (6.4x106 )2

or, g = 9.8 ms-2

Variation of the value of g:
We know that acceleration due to gravity depends on radius of the earth or the distance between the centre of earth
and the centre of a body.
𝟏
a. Variation due to the shape of the earth: 𝐠 ∝ 𝑹𝟐
Since, our earth is not perfectly round, its equatorial radius is more than that of polar radius i.e Re > Rp. Acceleration
due to gravity is inversely proportional to the radius of the earth. Hence, the value of the ‘g’ maximum at poles and
minimum at equator i.e gp > ge
𝑹 𝟐
b. Variation due to the height from the surface of the earth: 𝐠 𝒉 = (𝑹+𝒉) where, h is the height from the
surface.
If we increase the height from the surface of the earth the quantity in the bracket becomes less than 1.
And the acceleration due to the gravity decreases as height from the surface is increased.
𝐠(𝑹−𝑿)
c. Variation from the depth of earth surface: 𝐠 𝒅 = where, gd is acceleration due to gravity at the depth
𝑹
and X is depth of the earth from the surface.
(𝑅−𝑋)
Since, the quantity 𝑅 is less than 1,the value of g inside the earth decreases with increase in depth.
At centre of the earth x=R. So, gravity at the centre of the earth is 0.
Comparison between G and g
S.No. G g
1. It is gravitational constant. It is acceleration due to gravity.
2. It is same everywhere. Its value changes from place to place.
 3. Its numerical value in SI unit is very small and Its average numerical value on the surface in SI unit is
is 6.67x10-11 Nm2kg-2. 9.8 ms-2.

Mass and Weight

Mass: Mass of a body is defined as the quantity of matter contained in the body. It is represented by ‘m’. The value
of mass is always constant i.e, it does not change from place to place. It is a scalar quantity. Mass of a body is
measured by beam balance or pan balance by comparing the unknown mass with known standard mass.
Weight: Weight of a body is defined as the force with which the object is attracted towards the centre of the earth
or other planets and satellites. It is represented by ‘W’. Its value differs according to the place. The value of weight
depends on acceleration due to gravity of that place. Weight of the body increases with the increase in acceleration
due to gravity and vice-versa keeping the mass constant. It is a vector quantity having direction towards the centre
of the earth.
Mathematically, Weight= mass x acceleration due to gravity i.e, W=mxg. Weight depends upon acceleration due
to gravity and acceleration due to gravity depends on the distance.
E.g. As the stone is away from the earth surface the weight of the stone is less than the stone present in the bottom.
Differences Between Mass and Weight
S.No. Mass Weight
Mass is the quantity of matter contained in Weight of a body is defined as the force with which
1. the body Mass of a body is defined as the the object is attracted towards the centre of the earth
quantity of matter contained in the body. or other planets and satellites.
2. It is a scalar quantity. It is a vector quantity.
3. Its SI unit is kilogram (kg). Its SI unit is Newton (N).
Mass of a body remains constant Weight of a body changes from place to place.
4.
everywhere.
Mass of a body is never zero. Weight of a body becomes zero at the centre of the
5.
earth and in outer space.
Mass is measured by beam balance or pan Weight is measured by a spring balance.
6.
balance.
Weight of a body on the moon
We know that,
W= m x g
Weight on the moon 𝑀𝑔 𝑔 1
= 𝑀𝑔𝑚 = 𝑔𝑚 . Since, the acceleration due to gravity of the moon is 6 th the gravity of the earth,
Weight on the earth 𝑒 𝑒
1
weight of a body on the moon is also 6 th the weight of the earth.
Free fall.
When an object is falling towards the surface of the earth only under the influence of gravity without external
resistance, the fall of the object is free fall. Acceleration produced in the freely falling bodies is same for all bodies
and is independent of the mass of the falling body. It means acceleration produced on a body depends on air
resistance and not on the mass of the falling body.