Gravitation

Gravitation or gravity is the force of attraction between any two bodies. All the objects in the
universe attract each other with a certain amount of force, but in most cases, the force is too
weak to be observed due to the very large distance of separation. Besides, gravity’s range is
infinite, but the effect becomes weaker as objects move away.

This force of attraction was first observed by Sir Isaac Newton and was presented as Newton’s
law of gravitation in the year 1680. However, gravitation can generally exist in two main
instances.

1. Gravitation may be the attraction of objects by the earth

For example,

If a body (ball) is thrown upwards, it reaches a certain height and falls

downwards because of the gravity of the earth.

2. Gravitation may be the attraction of objects in outer space.

For example,

The force of attraction between the other planets and the sun.

What Is Gravitational Force?

Each body in this universe attracts other bodies towards itself with a force known
as Gravitational Force. Thus, gravitation is a study of the interaction between two masses. Out
of the two masses, the heavier one is called source mass, and the lighter one is called test mass.

Gravitational force is a central force which depends only on the position of the test mass from
the source mass and always acts along the line joining the centres of the two masses.

The core problem of gravitation has always been in understanding the interaction between the
two masses and their relativistic effects.
Newton’s Law of Gravitation

According to Newton’s law of gravitation, every particle in the universe attracts every other
particle with a force whose magnitude is,

 Directly proportional to the product of their masses, i.e., F ∝ (M1M2) . . . . (1)

 Inversely proportional to the square of the distance between their centre, i.e., (F ∝ 1/r2) . .
. . (2)
On combining equations (1) and (2), we get,

F ∝ M1M2/r2

F = G × [M1M2]/r2 . . . . (7)

Or, f(r) = GM1M2/r2 [f(r) is a variable, non-contact, and conservative force]

As f(r) varies inversely as a square of ‘r,’ it is also known as the inverse square law force. The
proportionality constant (G) in the above equation is known as the gravitational constant.

The dimension formula of G is [M-1L3T-2]. Also, the value of the gravitational constant,

 In SI units: 6.67 × 10-11 Nm2 kg-2,

 In CGS units: 6.67×10-8 dyne cm2 g-2

Gravitational Force Formula

Gravitational force is explained using Newton’s law of gravitation. Gravitational force decides
how much we weigh and how far a ball travels when thrown before it lands on the ground.

According to Newton’s law of gravitation, every particle in the universe attracts every other
particle with a force that is directly proportional to the product of their masses and inversely
proportional to the square of the distance between them.

Mathematically, it can be represented as,

F = Gm1m2/r2
Where,

 F is the gravitational force between two objects measured in Newton (N).

 G is the universal gravitational constant with a value of 6.674 × 10-11 Nm2kg-2.
 m1 is the mass of one massive body measured in kg.
  m2 is the mass of another massive body measured in kg.
 r is the separation between them measured in kilometre (Km).

Derivation of Newton’s law of Gravitation from Kepler’s Law

Suppose a test mass is revolving around a source mass in a nearly circular orbit of radius ‘r’ with
a constant angular speed (ω). Then, the centripetal force acting on the test mass for its circular
motion is,

F = mrω2 = mr × (2π/T)2

According to Kepler’s 3rd law, T2 ∝ r3

Using this in force equation, we get,

F = 4π2mr/Kr3 [Where, K = 4π2/GM]

⇒ F = GMm/r2, which is the equation of Newton’s law of gravitation.

Solved Examples

1. What is the force of gravity acting on an object of mass 2000 kg at the Earth’s surface?

Given:

Mass of Earth (m1) = 5.98 × 1024kg

Mass of object (m2) = 2000kg

The radius of the Earth (r)= 6.38 × 106m

Acceleration due to gravity (g) = 9.8 m/s2

Universal constant (G) = 6.67 x 10-11 N m2 / kg2

Solution:

F = Gm1m2/r2

F = ( 6.67 x 10-11) (5.98 × 1024)(2 x 103)/(6.38 × 106)2

F = (7.978 x 1017)/ (4.07044 × 1013)

F = 1.959 x 104 or F = 19.59 N

2. What is the force of gravity acting on an object of mass 1000 kg at 20,000 meters above
the Earth’s surface?

Given:

Mass of Earth (m1) = 5.98 × 1024kg

Mass of object (m2) = 1000kg

The radius of the Earth (r)= 6.38 × 106m

Acceleration due to gravity (g) = 9.8 m/s2

Universal constant (G) = 6.67 x 10-11 N m2 / kg2

h = 2 x 104 m

Solution:

F = Gm1m2/(r +h)2

F = ( 6.67 x 10-11) (5.98 × 1024)(1 x 103)/(6.38 × 106 + 2 x 104 )2

F = (3.988 x 1017)/(4.058 x 1013)

F = 9,827.50

F = 0.9827 x 104

What Is Acceleration Due to Gravity?

Acceleration due to gravity is the acceleration gained by an object due to gravitational force. Its
SI unit is m/s2. It has both magnitude and direction; hence, it’s a vector quantity. Acceleration
due to gravity is represented by g. The standard value of g on the surface of the earth at sea level
is 9.8 m/s2.

Formula of Acceleration Due to Gravity

Force acting on a body due to gravity is given by f = mg

Where f is the force acting on the body, g is the acceleration due to gravity, and m is the mass of
the body.

According to the universal law of gravitation, f = GmM/(r+h)2

Where,

 f = Force between two bodies

 G = Universal gravitational constant (6.67×10-11 Nm2/kg2)
 m = Mass of the object
 M = Mass of the earth
 r = Radius of the earth
 h = Height at which the body is from the surface of the earth
As the height (h) is negligibly small compared to the radius of the earth, we re-frame the
equation as follows:

f = GmM/r2

Now, equating both expressions,

mg = GmM/r2

⇒ g = GM/r2

Therefore, the formula of acceleration due to gravity is given by g = GM/r2

Note: It depends on the mass and radius of the earth.

This helps us understand the following:

 All bodies experience the same acceleration due to gravity, irrespective of their mass.
 Its value on earth depends upon the mass of the earth and not the mass of the object.
The value of acceleration due to gravity ‘g’ is affected by

 Altitude above the earth’s surface.

 Depth below the earth’s surface.
 The shape of the earth.
 Rotational motion of the earth.
Important Conclusions on Acceleration Due to Gravity

 For an object placed at a height of h, the acceleration due to gravity is less as compared
to that placed on the surface.
 As depth increases, the value of acceleration due to gravity (g) falls.
 The value of g is more at the poles and less at the equator

What does the value 9.8 m/s2 for acceleration due to gravity imply?
The value 9.8 m/s2 for acceleration due to gravity implies that for a freely falling body, the
velocity changes by 9.8 m/s every second.

Does mass have any effect on acceleration due to gravity?

The acceleration due to gravity is independent of the mass of the body.

What is the formula to calculate the force of attraction between two objects?
If m1 and m2 are the two masses separated by a distance r. According to the universal law of
gravitation, the force of attraction between them is
F = G (m1 m2/r2)

What is free fall?

If the object moves only under the influence of gravity, it is called free fall.

Density and Mass of Earth

Earth is the third smallest planet in the Solar System and is the third planet from the sun. It is the
densest planet and is the largest of the four terrestrial planets (Mercury, Venus, Earth and the
dwarf planet Ceres).

Earth’s mass is 5.515g/cm. Its volume is 1.08321 × 1012 km. It has a diameter of 8,000 miles or
13,000 km. The planet which closely resembles the Earth in its mass, density and volume is
Venus. Mercury has almost the same density as earth but is relatively smaller in mass.

Earth is not a perfect circle. It’s actually a spheroid. The reason for its circular shape is because
of its gravity. Gravity is the force which pulls objects towards its center. The earth is also not
perfectly round as it has a bulge in its center (the equator). This is due to the earth’s rotation.
The Earth’s mass was calculated through the efforts of Erasthothenes, Cavendish, Galileo and
Newton. As you can imagine there is no measuring tape long enough to go around the diameter
of the earth so mathematics was used in order to determine the size of the earth. Through the
combined efforts of these astronomers and mathematicians, overtime a formula was created in
order to find the mass of the earth.

This is the formula to find the gravitational pull of the earth: F=GmM/r2. F is the gravitational
force. G is a constant of proportionality. M is the two objects exerting the force and r is the
distance between the two objects. Take note that F= Ma. M is mass and A is its acceleration.
Once you figure out the force of gravity and divide it by the acceleration due to gravity you will
get the mass of the earth, which is 5.515 g/cm.

Why the weight of a body varies?

The force of gravity on a body varies slightly from place to place on the earth for two reasons.
First, the shape of the earth, and secondly, its rotation. The earth is not a perfect sphere, but
bulges at the equator, so that if a body is taken from a pole to the equator its distance from the
center of the earth will increase. Consequently, in accordance with Newton’s law of gravitation,
the gravitational pull on it will get less.

Weightlessness

Weightlessness is a term used to describe the sensation of a complete or near-complete absence

of weight. Astronauts orbiting the Earth often experience the sensation of weightlessness. These
sensations experienced by the orbiting astronauts are the same sensations experienced by anyone
who has been temporarily suspended above the seat on an amusement park ride. The causes of
the sensation of weightlessness in both these cases are the same.

Why Do We Feel Weightless?

Weightlessness is a sensation experienced by an individual where there are no external objects

touching one’s body. In other words, the sensation of weightlessness exists when all contact
forces are removed. These sensations are common to the state of free fall.

During free fall, the only force acting on the body is the force of gravity. As gravity is a non-
contact force, it cannot be felt without any opposing force. This is the reason why you feel
weightless when in a state of free fall.
It is important to remember that weightlessness is only a sensation and not a reality
corresponding to an individual who has lost weight. Weightlessness has a little to do with weight
and a lot to do with the presence and absence of contact forces.

Why Do Astronauts Feel Weightless in Space?

Astronauts orbiting in space feel a sense of weightlessness because there is no external contact
force in space pushing or pulling upon their bodies. Gravity is the only force acting upon their
body. Gravity being an action-at-a-distance force cannot be felt and therefore would not provide
any sensation of weight.

Do Astronauts Experience Weightlessness Because There Is No Force of Gravity in Space?

Many students are under the assumption that astronauts feel weightlessness because there is no
force of gravity in space. This is not true. If it were true then it would violate the circular motion
principles. If a person believes that the absence of gravity in space is the reason for
weightlessness then they would have to come up with a reason for how astronauts are orbiting in
space?

Is the Gravity in Space Lesser than the Gravity on the Earth?

The force of gravity acting on the astronaut in space is certainly lesser than the gravity on Earth’s
surface. But it is not small enough to account for a drastic reduction in weight. Let us consider
the space station to orbit at an altitude of approximately 400 km above the Earth’s surface, then
the value of g at that location will be reduced from 9.8 m/s2 to approximately 8.7 m/s2. While it
certainly reduces weight, it does not account for the absolutely weightless sensations that
astronauts experience. Their absolutely weightless sensations are because of the absence of a
surface to support them as they are free-falling towards the Earth.

What is a satellite?

A satellite is a body that orbits around another body in space. There are two different types of
satellites – natural and man-made. Examples of natural satellites are the Earth and Moon. The
Earth rotates around the Sun and the Moon rotates around the Earth. A man-made satellite is a
machine that is launched into space and orbits around a body in space. Examples of man-made
satellites include the Hubble Space Telescope and the International Space Station.

Man-made satellites come in many shapes and size and have different pieces of instruments on
them to perform different functions while in space. Satellites are built by engineers and take
months sometimes even years to build. The satellites have to endure many tests to make sure the
satellite can withstand the launch and the harsh environment of space.
NASA establishes missions for a specific purpose and the engineers develop a satellite to
perform the necessary functions for that mission. Once the satellite is launched into space, Space
Communications and Navigation (SCaN) provides the channel of communications for the data to
go to and from the Earth and the satellite. These communications include commands to the
spacecraft as well as the scientific data coming to Earth.

SCaN supports over 100 satellites including:

Satellites that observe Earth: Aqua and Aura

Satellites that observe the Sun and see the effect of solar winds on the Earth: Parker Solar Probe,
Solar Dynamics Observer (SDO) and Solar Terrestrial Relations Observatory (STEREO)

Satellites that observe the Moon and the planets: Lunar Reconnaissance Orbiter (LRO) and the
Mars Reconnaissance Orbiter (MRO)

Satellites that look at the origins of the universe: Hubble Space Telescope

› SCaN Missions – complete list of missions supported by SCaN and pictures of each satellite

How do Satellites Communicate?

Satellites communicate by using radio waves to send signals to the antennas on the Earth. The
antennas then capture those signals and process the information coming from those signals.
Information can include:

 scientific data (like the pictures the satellite took),

 the health of the satellite, and
 where the satellite is currently located in space.

Applications of artificial satellites

Following are the important applications of artificial satellites.

1. Communication: The geostationary satellites are used for communication purpose like long
distance telephone calls, telex, radio and T.V broadcasts etc. The satellites used for
communication purpose are called communication satellites.

2. Spying: Artificial satellites are used by military for spying of enemy troops to locate their
positions and movements.

3. Weather forecasting: Artificial satellites are used for weather forecasting. This is done by
fitting special instruments and powerful cameras in the satellites which monitor various climatic
factors such as air pressure, air temperature and humidity etc. The satellites used for weather
forecasting are called weather satellites.

4. Collection of information about other planets and outer space: The artificial satellites are used
for collecting information about other planets, outer space, stars, galaxies, asteroids etc.

5. Collection of information about natural resources of earth: Artificial satellites are also used for
collecting information about natural resources of earth like oil wells, underground water,
minerals, natural gas and coal deposits etc.