Gravitation Chapter - Physics JEE Main

Newton in 1686 stated that each particle of matter gets attracted to every other particle in the universe. This universal attractive force is called gravitation. Newton gave the following law about gravitation.

The force of attraction between any two material particles is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them. It acts along the line joining the two particles. 

Consider two particles of masses m1 and m2 are positioned at a distance ‘r’ apart. If the force of attraction acting between them is F, then according to Newton’s law of gravitation, we have 

F = G\[\frac{m_1m_2}{r^2}\]

The proportionality constant G is called ‘gravitational constant’. Its value is the same for all pairs of particles in the universe. Hence G is a ‘universal constant’.

Although Newton's law of gravitation applies strictly to particles (point-masses), it can be applied to extended objects as well, provided the sizes of the object are small compared to the distance between them. For example, the moon and the earth are located far enough from each other so that, to a good approximation, we can treat both the bodies as point masses and apply the law of gravitation. Spherically symmetric bodies tend to attract external particles as if their entire mass were concentrated at their centres of mass. Hence the law of gravitation can be applied to them, irrespective of whatever their size be.    

JEE Main Physics Chapters 2024 

What is Gravitation?

Gravitation, as defined by Sir Isaac Newton, is the force of attraction between any two objects with mass. This force, though invisible, is incredibly powerful, capable of holding together galaxies and shaping the very fabric of spacetime. The strength of gravitational attraction depends on the masses of the objects involved and the distance between them.

\text{Newton's Law of Universal Gravitation, mathematically expressed as } F = \frac{G(m_1m_2)}{r^2}, \text{ quantifies the gravitational force between two objects. Here, } F \text{ represents the force of gravitation, } G \text{ is the universal gravitational constant, } m_1 \text{ and } m_2 \text{ are the masses of the two objects, and } r \text{ represents the distance between their centers.}

\text{The concept of gravitational potential, denoted by } V, \text{ describes the potential energy per unit mass of an object in a gravitational field. It essentially measures the work done against gravity to move an object from infinity to a specific point in the field. The formula for gravitational potential is } V = -\frac{GM}{r}, \text{ where } M \text{ is the mass of the object creating the gravitational field.}

\text{Gravitational acceleration, represented by } g, \text{ is the acceleration experienced by an object in a gravitational field. It is the force per unit mass acting on the object. The formula for gravitational acceleration is } g = \frac{GM}{R^2}, \text{ where } R \text{ is the distance from the object to the center of the gravitational field.}

\text{Escape velocity, denoted by } v, \text{ is the minimum speed an object needs to attain to escape the gravitational pull of a celestial body. The formula for escape velocity is } v = \sqrt{\frac{2GM}{R}}, \text{ where } M \text{ is the mass of the celestial body and } R \text{ is its radius.}

Gravitation Class 11 Formulas

In the realm of Class 11 Physics, the study of gravitation holds paramount importance. Understanding the gravitational forces that govern the motion of celestial bodies is fundamental to comprehending the universe's workings. This introduction delves into the gravitational formulas encapsulated in the Class 11 syllabus, serving as a compass for students navigating the intricacies of gravitational physics. From the universal law of gravitation to formulas elucidating planetary motion, this guide encompasses all the essential gravitation formulas essential for Class 11 students. Let's embark on a gravitational journey, unraveling the mathematical intricacies that unveil the secrets of celestial mechanics.

Here are some important gravitation formula list for class 11:

Newton's Law of Universal Gravitation:

F = \frac{G \cdot (m_1 \cdot m_2)}{r^2}

where:

\( F \) is the gravitational force between two objects (N),

\( G \) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2\)),

\( m_1 \) and \( m_2 \) are the masses of the two objects (kg),

\( r \) is the distance between the centers of the two objects (m).

Gravitational Potential:

V = -\frac{G \cdot M}{r}

where:

\( V \) is the gravitational potential at a point (J/kg),

\( G \) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2\)),

\( M \) is the mass of the object creating the gravitational field (kg),

\( r \) is the distance from the point to the center of the object (m).

Gravitational Acceleration:

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

where:

\( g \) is the gravitational acceleration at a point (\( \text{m/s}^2 \)),

\( G \) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2\)),

\( M \) is the mass of the object creating the gravitational field (kg),

\( R \) is the distance from the point to the center of the object (m).

Escape Velocity:

v = \sqrt{\frac{2GM}{R}}

where:

v is the escape velocity from an object (m/s)

G = 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2

M is the mass of the object (kg)

R is the radius of the object (m)

Gravitation class 11 all formulas can be accessed through Vedantu’s page for Important Gravitation Formulas.

Law of Gravitation in Vector Form

We are considering two-point masses m1 and m2 at a distance ‘r’ apart. Let r12 be a unit vector pointing from mass m1 to mass m2, and r21 a unit vector pointing from m2 to m1. Then the gravitational force F12 exerted on m1 by m2, is given in magnitude and direction by the vector relation:

F12 = -G\[\frac{m_2m_1}{r^2}r^{21}\]

The minus sign indicates that F12 points in a direction opposite to r21, that is gravitational force is attractive, m1 experiencing a force directed towards m2.

The force exerted on m2 by m1 is similarly

F21 = -G\[\frac{m_2m_1}{r^2}r^{12}\]

Because r21 = - r12, equation 1 and 2 show that 

F12 = -F21

Gravitational Constant G

Newton’s law of gravitation is

                                                       F = G\[\frac{m_2m_1}{r^2}\]

If we put m1 = m2 = 1 and r  = 1, then G = F

Thus, the gravitational constant G is numerically equal to the force with which two particles, each of the unit mass and placed a unit distance apart, attract each other. 

The SI units of force F, distance r and mass m1 (or m2) are Newton (N), meter (m) and kilogram (kg) respectively. Therefore, the SI unit of G is Nm2 kg-2 or m3 kg-1 s-1 (because N = kg m s-1).    

Newton's Law of Gravitation Formula

This law states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

The law stated by Newton can be expressed in a mathematical formula which was given by Johannes Kepler in the 17th century. The formula of gravitation can be stated as:

$F = \dfrac{G \cdot (m_1 \cdot m_2)}{R^2}$

Where:

F is the gravitational force between two objects,

G is the gravitational constant, a universal constant that is approximately equal to $6.67430 \times 10^{-11} N(m/kg)^2$

$m_1$ and $m_2$ are the masses of the two objects,

r is the distance between the centers of the two objects.

Understanding the Law:

• Inverse Square Law: The most important aspect of Newton's law of gravitation is the inverse square relationship. It means that as the distance between two objects increases, the gravitational force between them decreases rapidly. If you double the distance between two objects, the gravitational force becomes one-fourth of its original strength.

• Proportional to Mass: The law also states that the gravitational force is directly proportional to the product of the masses of the two objects. This means that objects with greater mass will exert a stronger gravitational force on each other.

• Universal Applicability: One of the remarkable features of this law is its universality. It applies to all objects with mass, from small particles to celestial bodies like planets, stars, and galaxies. This single law explains the motion of planets around the sun, the moon's orbit around the Earth, and many other gravitational interactions in the universe.

• Mutual Attraction: Importantly, Newton's law of gravitation implies that every particle in the universe attracts every other particle. This means that there is a mutual attraction between all objects with mass, and this force of attraction acts along the line joining the centers of the two objects.

Gravitational Field:

The gravitational field at a point is a vector quantity that describes the gravitational force per unit mass at that point. It is given by:

$E = \dfrac{F}{m}$

The gravitational field intensity depends on the mass of the object causing the gravitational field and the distance from that object. It is inversely proportional to the square of the distance from the object, as described by Newton's law of universal gravitation.

Relation Between Gravitational Force and Gravitational Field:

The gravitational field intensity at a point is related to the gravitational force experienced by an object at that point through the following equation:

$F = m \cdot E$

This equation shows that the gravitational force experienced by an object is directly proportional to its mass and the gravitational field intensity at that location. Essentially, the gravitational field intensity tells us how much force a unit mass would experience at a particular point, and multiplying it by the mass of the object gives us the actual gravitational force acting on that object due to gravity.

Gravitational Potential Energy:

Gravitational potential energy (GPE) is the energy possessed by an object due to its position in a gravitational field. It represents the potential for the object to do work as a result of its position relative to another massive object, typically the Earth. GPE is a form of potential energy and is proportional to an object's mass, the acceleration due to gravity, and its height above a reference point.

$U=mgh$

Where,

U is the gravitational potential energy.

m is the mass of the object.

g is the acceleration due to gravity (approximately 9.81 m/s² on Earth's surface).

h is the height or distance above a reference point (usually the Earth's surface)

The Concept of Escape Velocity:

Escape velocity is the minimum velocity required for an object to break free from the gravitational pull of a massive body (like the Earth) and move into space without any additional propulsion. It's a critical concept in the context of space travel and satellite launches.

$v_e = \sqrt{\dfrac{2GM}{R}}$

Where:

$v_e$  is the escape velocity.

G is the gravitational constant (approximately $6.67430 \times 10^{−11}$  m$^3$  kg$^{−1}$ s$^{ −2}$).

M is the mass of the massive body (e.g., the Earth).

R is the distance from the center of the massive body to the object's initial position.

Escape velocity depends on the mass of the body and the distance from its center. For Earth, the escape velocity is approximately 11.2 kilometers per second (km/s). This means that an object, like a spacecraft, needs to reach this velocity to overcome Earth's gravitational pull and venture into space.

Kepler’s Laws of Planetary Motion

Our solar system consists of a sun that is stationary at the centre of the universe and nine planets that revolve around the sun in separate orbits. Also, there are celestial bodies that move around the planets. These are called satellites. For example, the moon revolves around the earth, hence the moon is a satellite of the earth. Similarly, Mars has two satellites, Jupiter has sixteen satellites, Saturn has nineteen and so on. 

Kepler found important regularities in the motion of the planets. These regularities are known as ‘Kepler’s three laws of planetary motion’.

1. Law of Orbits: All the planets move around the sun in elliptical orbits having the sun at one focus of the orbit.

2. Law of Areas: The line joining any planet to the sun sweeps out equal areas in equal times, that is, the areal velocity of the planet remains constant. 

3. Law of Periods: The square of the period of revolution of any planet around the sun is directly proportional to the cube of its mean distance from the sun. 

Acceleration Due to Gravity:

The acceleration due to gravity is the acceleration experienced by an object in a gravitational field. It is a vector quantity that points in the direction of the net gravitational force on the object. The magnitude of the acceleration due to gravity is given by Newton's law of universal gravitation,

$g = \dfrac{G  M}{r^2}$

where:

g is the acceleration due to gravity

G is the gravitational constant

M is the mass of the object creating the gravitational field

r is the distance between the object and the object creating the gravitational field

Variation of 'g' with Altitude and Depth:

The acceleration due to gravity decreases with altitude and increases with depth. This is because the mass of the Earth is not uniformly distributed, and the distance between the object and the center of the Earth increases with altitude.

The variation of g with altitude can be calculated using the following formula:

$g = g_0 \dfrac{1 - 2h}{R_e}$

where:

g is the acceleration due to gravity at altitude h

g_0 is the acceleration due to gravity at sea level

h is the altitude

R_e is the radius of the Earth

The variation of g with depth can be calculated using the following formula:

$g = g_0 \dfrac{1 + h}{ R_e}$

where:

Calculation of 'g' on Different Celestial Bodies:

The acceleration due to gravity on different celestial bodies can be calculated using Newton's law of universal gravitation. For example, the acceleration due to gravity on the moon is approximately 1.67 m/s^2, which is about one-sixth of the acceleration due to gravity on Earth.

Geostationary and Polar Satellites:

Geostationary orbit is a circular orbit 35,786 kilometers (22,236 miles) above Earth's equator, where the satellite's orbital period is equal to the Earth's rotation period, 24 hours. A satellite in geostationary orbit appears motionless over a single point on Earth's surface. This makes geostationary satellites ideal for applications such as telecommunications, weather forecasting, and remote sensing.

Polar orbit is an elliptical orbit that passes over both poles of Earth. Satellites in polar orbits typically have orbital periods of 90-120 minutes and altitudes of 160-1,000 kilometers (100-620 miles). Polar satellites provide global coverage of Earth's surface, making them useful for applications such as mapping, environmental monitoring, and disaster response.

Escape Velocity:

Escape velocity is the minimum velocity required for an object to escape the gravitational pull of a massive body. It is given by:

$v_e = \sqrt{\dfrac{2GM}{r}}$

where:

$v_e$ is the escape velocity

G is the gravitational constant

M is the mass of the massive body

r is the distance from the center of the massive body

Escape velocity is significant because it determines the minimum energy required to launch an object into space or to escape the gravitational pull of a planet or star.

Orbital Velocity:

The orbital velocity of an object in a circular orbit is given by:

$v_o = \sqrt{\dfrac{GM}{r}}$

where:

$v_o$ is the orbital velocity

The orbital velocity is related to the escape velocity by the following equation:

$v_o = \dfrac{1}{\sqrt{2}} v_e$

This means that the orbital velocity is equal to the escape velocity divided by the square root of 2.

Weightlessness and Free Fall: 

Weightlessness is the state of experiencing no apparent gravitational force. It is also referred to as microgravity or zero gravity. Astronauts experience weightlessness when they are in orbit around the Earth. This is because the gravitational force is accelerating the astronauts and the spacecraft at the same rate. As a result, the astronauts feel like they are floating.

Free fall is the condition in which an object is falling under the influence of gravity alone. This means that there is no other force acting on the object, such as air resistance or friction. When an object is in free fall, it accelerates at the same rate as all other objects, regardless of their mass. This is known as the principle of equivalence.

Gravitational Potential Energy in Orbits:

The gravitational potential energy of an object in orbit is the energy that the object has due to its position in the gravitational field of another object. It is given by:

$U = \dfrac{-GMm}{r}$

The gravitational potential energy of an object in orbit is negative. This is because the gravitational force is attractive, and the object has to do work against the force to move away from the central object.

JEE Main Gravitation Solved Example 

Example 1) The acceleration due to gravity at the moon’s surface is 1.67m s-2. If the radius of the moon is 1.74 x 106m, calculate its mass. Take G = 6.67 x 10-11N m2 kg-2

Solution 1) Let Mm be the mass, Rm the radius of the moon, and g be the acceleration due to gravity on its surface. Then, 

Mm = \[\frac{gR_m^2}{G}\] = \[\frac{1.67Nkg^{-1} × (1.74×10^6m)^2}{6.67 × 10^{-11}N m^2kg^-2}\] = 7.58 × 1022kg

Example 2) A body of mass 100 kg falls on Earth from infinity. What will be its velocity on reaching the earth? Calculate its Energy. The radius of the Earth is 6400 km and g = 9.8 m s-2.Air friction is negligible. 

Solution 2) A body projected up with the escape velocity ve will go to infinity. Therefore, the velocity of the body falling on the earth from infinity will be ve. Now, the escape velocity of the earth is 

ve = \[\sqrt{2gR_e}\]= \[\sqrt{2 × (9.8 m s^{-2}) × (6400 × 10^3m)}\]

=1.12 × 104ms-1 = 11.2 kms-1.

The kinetic energy required by the body is 

K = ½ m ve2 = ½ × 100 kg × (11.2 × 103m s-1)2 = 6.27 × 109J

JEE Main Physics Gravitation Study Materials

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JEE Main Physics Study and Practice Materials

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Benefits of Using Vedantu for JEE Main 2024 - Physics Gravitation

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Conclusion

In the realm of JEE Main, the chapter on "Gravitation" holds immense significance as it unlocks the mysteries of the universe's fundamental force. Understanding the principles of gravitational interactions, celestial motion, and planetary dynamics is crucial not only for excelling in the exam but also for comprehending the cosmos. This chapter, while delving into Newton's law of universal gravitation and Kepler's laws, offers a glimpse into the harmony of the celestial bodies. It not only empowers students with the knowledge to tackle complex problems but also fosters a sense of wonder and curiosity about the cosmos, making it an essential part of the JEE Main curriculum.