Revision: Gravitation JEE Main Gravitation

"Every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The direction of the force is along the line joining the particles."

or

"Every particle of matter attracts every other particle of matter with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."

The constant of proportionality in Newton's Law of Gravitation, whose value is 6.67430 × 10⁻¹¹ Nm²kg⁻², and which remains the same everywhere in the universe regardless of the medium or the nature of the bodies, is called the Universal Gravitational Constant.

The acceleration gained by a body due to the gravitational force of attraction of the Earth, whose value is constant at a given place but differs from place to place and is given by g = GM/R², is called Acceleration due to Gravity.

A satellite that orbits the Earth at a height of approximately 36,000 km above the equator with a period of revolution of 24 hours is called a geosynchronous satellite.

The objects that revolve around the Earth are called Earth satellites.

The minimum energy required by a satellite to escape from Earth’s gravitational influence is the binding energy of the satellite.

The exact horizontal velocity of projection that must be given to a satellite at a certain height so that it can revolve in a circular orbit round the Earth is called the critical velocity or orbital velocity (v_c).

A man-made object that orbits a planet or other celestial body is called an artificial satellite.

A satellite that travels over Earth's poles, passing close to the Earth's surface, with a period of revolution of nearly 85 minutes is called a polar satellite.

"The minimum energy required by a satellite to escape from Earth’s gravitational influence is the binding energy of the satellite."

OR

The energy that must be given to an orbiting satellite to make it escape to infinity (equal to the negative of total energy of the satellite), given by BE=GMm/2rBE=GMm/2r, is called Binding Energy.

"The minimum velocity with which a body should be thrown vertically upwards from the surface of the Earth so that it escapes the Earth’s gravitational field is called the escape velocity (vₑ) of the body."

The gravitational force of attraction (F) between two bodies of mass m₁ and m₂ separated by a distance r is:

• F: Gravitational Force of attraction (in Newtons, N).
• \[m_1, m_2\]: Masses of the two objects (in kilograms, kg).
• r (or d in the first part): Distance between the two objects (in meters, m).
• G: The constant of proportionality, called the Universal gravitational constant.
• Value in SI units: \[G=6.67\times10^{-11}\mathrm{N}\cdot\mathrm{m}^2/\mathrm{kg}^2\]
• Dimensions: \[[G]=[\mathrm{L}^3\mathrm{M}^{-1}\mathrm{T}^{-2}]\]

g_h = g\[\left(\frac{R^2}{(R+h)^2}\right)\] = g(1 - \[\frac {2h}{R}\])(for h < < R)

g_d = g(1 - \[\frac {d}{R}\])

g_λ​ = g − ω² R cos² λ

The formulas for acceleration due to gravity (g) are provided below:

On the Earth's Surface:

At height $h$ above the Earth's surface:

Simplified Formula for Small Altitudes ($h \ll R$):

Definition of Terms:

• g: Acceleration due to gravity on the Earth's surface.
• g_h: Acceleration due to gravity at height h above the Earth's surface.
• G: Universal Gravitational Constant.
• M: Mass of the Earth.
• R: Radius of the Earth.
• h: Altitude or height above the Earth's surface.

U(r) = -\[\frac {GMm}{r}\]

Where:

• U(r) = Gravitational potential energy at distance r from Earth's center
• G = Universal gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²)
• M = Mass of Earth (kg)
• m = Mass of the object (kg)
• r = Distance between the centers of mass of Earth and object (m)
• Negative sign = Shows that potential energy is negative (zero at infinity)

Binding Energy = +\[\frac {1}{2}\frac {GMm}{r}\]

Where:

• G = Universal Gravitational Constant
• M = Mass of the Earth
• m = Mass of the satellite
• r = Radius of the orbit (Distance from the center of the Earth)

\[v_e=\sqrt{\frac{2GM}{R}}\]

• v_e​ = Escape velocity (minimum speed needed to escape Earth’s gravity)
• G = Universal gravitational constant (6.674 × 10⁻¹¹ Nm²/kg²)
• M = Mass of the Earth (or celestial body)
• R = Radius of the Earth (or distance from the centre of the mass to the object)

1. Kepler’s first law: The orbit of a planet is an ellipse with the sun at one of the foci.
2. Kepler’s second law: The line joining the planet and the sun sweeps equal areas in equal intervals of time.
3. Kepler’s third law: The square of the orbital period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance of the planet from the Sun.

The universal law of gravitation states that every object in the universe attracts every other object with a force called the gravitational force. The force acting between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

F ∝ m M (masses of two bodies m and M)

F ∝ `1/"d"^2` (d is the distance between the two bodies.)

A and B are two bodies, the distance between which is given as d.

F = force with which one body is attracting the other.

G = proportionality constant.

Statement:

The law which states that every particle of matter attracts every other particle in the universe with a force whose magnitude is directly proportional to the product of masses and inversely proportional to the square of the distance between them is called Newton's Law of Gravitation.

Derivation:

Newton's Universal Law of Gravitation states that every particle of matter attracts every other particle of matter with a force which is the following:

• Directly proportional to the product of their masses: F ∝ m₁ ⋅ m₂
• Inversely proportional to the square of the distance between them: F ∝ \[\frac {1}{r^2}\]

Combining both, the gravitational force is expressed as follows:

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

where G is the universal gravitational constant, measured by Henry Cavendish using the Cavendish balance, with the value:

G = 6.67 × 10⁻¹¹ Nm²/kg²