Gravitation - Notes, Topics, Formulas, Books, FAQs

Gravitation Class 11th Topics(NCERT Syllabus)

1. Introduction

We observe that all objects fall towards the Earth due to gravity, a fact studied by Galileo through experiments. Ancient astronomers proposed geocentric and heliocentric models to explain planetary motion. Later, Copernicus supported the Sun-centered model, while Tycho Brahe’s observations helped Kepler frame his three famous laws of planetary motion. These ideas finally led Newton to propose the Universal Law of Gravitation.

2. Kepler's Laws

Johannes Kepler studied planetary motion using Tycho Brahe's data and proposed three famous laws, known as Kepler's Laws of Planetary Motion:
1. Law of Orbits (First Law):

Every planet moves around the Sun in an elliptical orbit, with the Sun at one focus.
2. Law of Areas (Second Law):

A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
This means planets move faster when closer to the Sun and slower when farther away.
3. Law of Periods (Third Law):

The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of its orbit:

$
T^2 \propto a^3
$

• Kepler's Law of Planetary Motion

3. Universal 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.

$F=G \frac{m_1 m_2}{r^2}$

Where,
$F=$ Gravitational force between the two bodies
$m_1, m_2=$ Masses of the bodies
$r=$ Distance between their centers
$G=$ Universal Gravitational Constant $\left(6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^2 \mathrm{~kg}^{-2}\right)$

• Universal Law of Gravitation

4. The Gravitational Constant (G)
In Newton's law of gravitation, the proportionality constant is called the Universal Gravitational Constant (G).

$
F=G \frac{m_1 m_2}{r^2}
$

Definition:
The gravitational constant $G$ is defined as the force of attraction between two unit masses ( 1 kg each) placed 1 meter apart in vacuum.

Value of G: $G=6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^2 / \mathrm{kg}^2$

• The Gravitational Constant (G)

5. Acceleration due to Gravity of the Earth (g)

Every object near the surface of the Earth experiences a constant acceleration towards its center, called acceleration due to gravity (g).
From Newton's law of gravitation:

$
F=G \frac{M m}{R^2}
$

and from Newton's second law:

$
F=m g
$

Equating,

$
g=\frac{G M}{R^2}
$

where
$M=$ mass of Earth
$R=$ radius of Earth
$G=$ gravitational constant
Value at Earth's surface:

$
g \approx 9.8 \mathrm{~m} / \mathrm{s}^2
$

• Acceleration due to Gravity of the Earth (g)

6. Acceleration due to Gravity Below and Above the Surface of Earth

The value of acceleration due to gravity is not the same everywhere. It changes when we go above the surface (at some height) or below the surface (at some depth).
(a) Above the Surface of Earth

At a height $h$ above the surface, the distance from the Earth's center becomes $(R+h)$. The acceleration due to gravity is:

$
g_h=\frac{G M}{(R+h)^2}
$

For small heights ( $h \ll R$ ):

$
g_h \approx g\left(1-\frac{2 h}{R}\right)
$

Thus, $g$ decreases with increase in height.
(b) Below the Surface of Earth

At a depth $d$ below the surface, only the mass of the Earth up to radius $(R-d)$ is effective. In this case:

$
g_d=g\left(1-\frac{d}{R}\right)
$

This shows that $g$ decreases linearly with depth and becomes zero at the center of the Earth.

7. Gravitational Potential Energy

The gravitational potential energy $(\mathbf{U})$ of a body is the energy possessed by it due to its position in the gravitational field of the Earth.

For a mass $m$ at a distance $r$ from the Earth's center, the potential energy is:

$
U=-\frac{G M m}{r}
$

The negative sign shows that the gravitational force is attractive.
At infinity ( $r \rightarrow \infty$ ), the potential energy is taken as zero.
On the Earth's surface $(r=R)$ :

$
U=-\frac{G M m}{R}
$

• Gravitational Potential Energy

8. Escape Speed

The escape speed is the minimum speed with which a body must be projected from the Earth's surface so that it can overcome the Earth's gravitational pull and never return.

It is obtained by equating kinetic energy with gravitational potential energy:

$
\begin{gathered}
\frac{1}{2} m v^2=\frac{G M m}{R} \\
v_{e s c}=\sqrt{\frac{2 G M}{R}}=\sqrt{2 g R}
\end{gathered}
$

For Earth:

$
v_{e s c} \approx 11.2 \mathrm{~km} / \mathrm{s}
$

Escape speed is independent of the mass of the body and depends only on Earth's mass and radius.

• Escape Speed

9. Earth Satellites

A satellite is a body that revolves around a planet under the influence of its gravitational force. The Moon is a natural satellite of Earth, while artificial satellites are launched for communication, navigation, and research.

For a satellite of mass $m$ revolving close to Earth's surface at radius $r$ :
The gravitational force provides the centripetal force:

$
\begin{gathered}
\frac{G M m}{r^2}=\frac{m v^2}{r} \\
v=\sqrt{\frac{G M}{r}}
\end{gathered}
$

where $v=$ orbital speed.
Time period (T):

$
T=\frac{2 \pi r}{v}=2 \pi \sqrt{\frac{r^3}{G M}}
$

Satellites can be geostationary (fixed position relative to Earth, $T=24 \mathrm{~h}$ ) or polar (covering entire Earth surface).

10. Energy of an Orbiting Satellite

A satellite of mass $m$ revolving around the Earth in a circular orbit of radius $r$ has both kinetic energy (K.E.) and gravitational potential energy (P.E.).
1. Kinetic Energy:

$
K . E .=\frac{1}{2} m v^2=\frac{G M m}{2 r}
$

2. Gravitational Potential Energy:

$
\text { P.E. }=-\frac{G M m}{r}
$

3. Total Mechanical Energy (E):

$
E=K . E .+P . E .=-\frac{G M m}{2 r}
$

The negative total energy indicates that the satellite is bound to Earth.

• Time Period And Energy Of A Satellite

Gravitation - Important Formulas

1. Universal Law of Gravitation

$
F=G \frac{m_1 m_2}{r^2}
$

$F=$ Gravitational force, $m_1, m_2=$ masses, $r=$ distance,
2. Acceleration due to Gravity on Earth

$
g=\frac{G M}{R^2}
$

$M=$ Mass of Earth,$R=$ Radius of Earth, $G=$ Gravitational constant

Above surface:

$
g_h=g\left(1-\frac{2 h}{R}\right)
$

Below surface:

$
g_d=g\left(1-\frac{d}{R}\right)
$

3. Gravitational Potential Energy

$
U=-\frac{G M m}{r}
$

At Earth's surface: $U=-\frac{G M m}{R}$
Work done from infinity $\rightarrow$ potential energy at distance $r$.
4. Escape Speed

$
v_{e s c}=\sqrt{\frac{2 G M}{R}}=\sqrt{2 g R}
$

Minimum speed to leave Earth's gravity without further propulsion
5. Orbital Speed of Satellite

$
v=\sqrt{\frac{G M}{r}}
$

$r=$ radius of orbit from Earth's center
6. Time Period of Satellite

$
T=2 \pi \sqrt{\frac{r^3}{G M}}
$

7. Energy of an Orbiting Satellite

$
K . E .=\frac{G M m}{2 r}, \quad P . E .=-\frac{G M m}{r}, \quad E=K . E .+P . E .=-\frac{G M m}{2 r}
$

8. Relation for Planetary Motion (Kepler's 3rd Law)

$
T^2 \propto r^3
$

$T=$ Time period of revolution, $r=$ Semi-major axis of orbit

Real-Life Examples of Gravitation

1. Objects Falling to the Ground: If we drop a stone, fruit, or pen, it falls directly towards the Earth. This is due to the gravitational pull of Earth.
2. Rainfall: When water droplets form in clouds, gravity pulls them down as rain.
3. Tides in Oceans: The gravitational attraction of the Moon and the Sun causes tides in seas and oceans.
4. Standing and Walking: We can stand and walk on the surface of the Earth because gravity holds us down.
5. Planetary Motion: The Earth and other planets revolve around the Sun due to the Sun's gravitational force.
6. Moon's Revolution: The Moon revolves around the Earth because of Earth's gravity.
7. Weight of a Body: The weight of a body is nothing but the force of gravity acting on it.
8. Atmosphere around Earth: The layer of air surrounding the Earth is held in place by Earth's gravity.
9. Thrown Objects: A ball thrown upwards comes back down because Earth pulls it towards the ground.
10. Formation of Celestial Bodies: Stars, planets, and galaxies are formed when gases and matter come together due to gravity.

Exam-wise weightage of Gravitation

Summary

According to Sir Isaac Newton, gravity is that force which pulls two masses. The law of gravitation as given by Newton states that: this force is directly proportional to the product of mass and inversely proportional to distance squared. Gravitation is weak but universal; it affects all objects in the universe equally regardless of their size or composition. This force obeys Newton’s third law of motion, i.e., for every action, there is an equal but opposite reaction; it also varies with different shapes of the earth such that it is stronger at the poles than at the equator.

NCERT Notes Subject Wise Link:

• NCERT notes Class 11 maths
• NCERT notes Class 11 Physics
• NCERT notes Class 11 Chemistry
• NCERT notes Class 11 Biology

NCERT Exemplar Solutions Subject-wise link:

• NCERT exemplar solutions for class 11 Physics.
• NCERT exemplar solutions for class 11 Chemistry.
• NCERT exemplar solutions for class 11 mathematics.
• NCERT exemplar solutions for class 11 Biology