Gravitation

Gravitation is a fundamental natural phenomenon by which all physical bodies possessing mass or energy are attracted toward one another. It is the weakest of the four fundamental forces of nature (the others being the electromagnetic force, strong nuclear force, and weak nuclear force), yet it dominates the macroscopic structure of the universe, governing the orbits of planets, the lifecycle of stars, and the formation of galaxies.

Newton’s Law of Universal Gravitation

Formulated by Sir Isaac Newton in 1687, this law states that every particle of matter 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 their centers.

Mathematical Representation

Where:

• F: The gravitational force of attraction between the two bodies (N).
• m₁, m₂: The masses of the interacting bodies (kg).
• r: The distance between the centers of the masses (m).
• G: The Universal Gravitational Constant.

The Universal Gravitational Constant (G)

Unlike local gravitational acceleration, G is a scalar constant that remains identical throughout the entire universe, independent of time, location, medium, or the nature of the masses.

• Value of G: 6.674 × 10^-11 N·m²/kg² (experimentally determined by Henry Cavendish using a torsion balance).
• Dimensional Formula: [M^-1 L³ T^-2].

Key Structural Properties of Gravitational Force

• Universal Action: It is a purely attractive force; a repulsive gravitational force does not exist in classical physics.
• Central Force: It acts directly along the straight line joining the centers of mass of the two interacting bodies.
• Inverse-Square Law: The force diminishes rapidly as distance increases (F ∝ 1/r²). If the distance between two objects is doubled, the gravitational pull drops to one-fourth of its original strength.
• Action-Reaction Consistency: It forms an action-reaction pair in strict compliance with Newton’s Third Law. The Earth pulls an apple downward with a force exactly equal in magnitude to the upward force the apple exerts on the Earth.

Acceleration Due to Gravity (g)

Acceleration due to gravity is the uniform acceleration experienced by a freely falling body near the surface of a massive celestial object due to its gravitational pull.

Relationship Between g and G

By equating Newton’s Second Law (F = mg) with the Law of Gravitation (F = GMm/R²), the mass of the falling object (m) cancels out:

Where M is the mass of the Earth (or celestial body) and R is its radius. This indicates that g is completely independent of the mass of the falling object.

• Standard Value on Earth: Approximately 9.8 m/s² or 32.2 ft/s² at sea level.
• Dimensional Formula: [M⁰ L¹ T^-2].

Factors Causing Variations in g

The value of g is not absolute and fluctuates based on geographic location, altitude, and depth.

Shape of the Earth

The Earth is not a perfect sphere but an oblate spheroid, flattened at the poles and bulging at the equator. Consequently, the equatorial radius (R_e) is roughly 21 km larger than the polar radius (R_p). Because g ∝ 1/R²:

• The value of g is maximum at the poles.
• The value of g is minimum at the equator.

Altitude (Height above Surface)

As an object ascends above the Earth’s surface to a height h, the distance from the center of mass increases. The modified acceleration g’ is given by:

Therefore, g decreases with increasing altitude.

Depth below Surface

As an object goes deeper inside the Earth to a depth d, the shell of earth mass above it exerts no net gravitational force inside, reducing the effective mass pulling it downward. The modified acceleration is given by:

Therefore, g decreases with increasing depth, becoming precisely zero at the center of the Earth.

Axial Rotation of the Earth

The rotation of the Earth on its axis generates an outward centrifugal effect that opposes gravity. The effective acceleration at a latitude φ is:

• At the equator (φ = 0°), the centrifugal counteraction is at its maximum, reducing g further.
• At the poles (φ = 90°), the rotation has zero impact on g.
• If the Earth were to stop rotating, the value of g would increase everywhere except at the poles.

Mass vs. Weight

Parameter	Mass	Weight
Fundamental Nature	The fundamental measure of the quantity of matter contained within an object.	The gravitational force exerted on an object by a celestial body.
Quantity Type	Scalar Quantity.	Vector Quantity (directed toward the center of mass).
Formula	Expressed fundamentally via inertia.	W = mg
SI Unit	Kilogram (kg).	Newton (N) or Kilogram-force (kgf).
Variability	Constant everywhere in the universe.	Changes depending on the local value of g (e.g., an object weighs 1/6th its Earth weight on the Moon).
Zero Condition	Can never be zero for physical matter.	Becomes zero in states of weightlessness or at the center of the Earth.

Kepler’s Laws of Planetary Motion

Johannes Kepler formulated three empirical laws describing the orbits of planets around the Sun, which Newton later validated mathematically using his gravitational theory.

First Law (The Law of Orbits)

Every planet moves in an elliptical orbit around the Sun, with the Sun situated at one of the two foci of the ellipse.

Second Law (The Law of Areas)

The radius vector drawn from the Sun to any planet sweeps out equal areas in equal intervals of time.

• Physical Insight: This means that the areal velocity of a planet remains completely constant (dA/dt = constant). This is a direct consequence of the conservation of angular momentum.
• Orbit Velocity: A planet moves faster when it is closer to the Sun (Perihelion) and slower when it is farthest away (Aphelion).

Third Law (The Law of Periods)

The square of the time period (T) of revolution of a planet around the Sun is directly proportional to the cube of the semi-major axis (r) of its elliptical orbit.

Satellite Mechanics: Orbital and Escape Velocity

Satellites are natural or artificial bodies that orbit around a primary celestial mass due to centripetal containment provided by gravity.

Orbital Velocity (v_o)

The specific horizontal velocity required to put a satellite into a stable, continuous circular orbit around a celestial body. For a satellite orbiting very close to the Earth’s surface (h ≈ 0):

• Value near Earth’s surface: Approximately 7.92 km/s.

Escape Velocity (v_e)

The minimum speed with which an unpowered projectile must be launched from the surface of a body to break completely free from its gravitational field without ever falling back down.

• Value for Earth: Approximately 11.2 km/s.
• Value for the Moon: Approximately 2.38 km/s (its low escape velocity makes it impossible for the Moon to retain an atmosphere, as gas molecules easily exceed this speed).

Mathematical Relationship

If a satellite’s orbital speed is increased by roughly 41.4% (√(2) – 1), it will completely break out of its orbit and escape into deep space.

Classification of Artificial Satellites

Geostationary (Geosynchronous) Satellites

Satellites that appear completely stationary relative to a fixed observer on Earth because their orbital parameters match the Earth’s rotation.

• Time Period: Exactly 24 hours.
• Direction of Motion: West to East, orbiting directly above the Earth’s equator.
• Orbital Altitude: Exactly 35,786 km above the surface.
• Primary Uses: Telecommunications, television broadcasting, and global weather monitoring.

Polar Satellites (Low Earth Orbit – LEO)

Satellites that travel in north-south paths, crossing directly over or near the geographical poles.

• Time Period: Roughly 90 to 100 minutes.
• Orbital Altitude: Lower altitudes, ranging between 500 km and 800 km.
• Primary Uses: Remote sensing, environmental mapping, spy/reconnaissance operations, and high-resolution meteorology (since the Earth rotates beneath it, the satellite can scan the entire planet strip by strip).

Core Scientific Facts and Trivia for Prelims

Tidal Phenomena

Ocean tides are primarily caused by the differential gravitational attraction exerted on the Earth’s oceans by the Moon and, to a lesser extent, the Sun. The pull creates two tidal bulges on opposite sides of the planet simultaneously.

Weightlessness in Orbit

Astronauts floating inside the International Space Station (ISS) experience weightlessness not because there is “no gravity” in space (gravity at that altitude is still roughly 90% of surface gravity), but because the station and the astronauts are in a continuous state of free fall together around the Earth, meaning the normal reaction force drops to zero.

Gravitational Waves

First predicted by Albert Einstein in 1916 within his General Theory of Relativity, gravitational waves are ripples or distortions in the fabric of spacetime caused by violent, massive cosmic events, such as the collision of black holes or neutron stars. They travel at the speed of light and were first directly detected experimentally by LIGO in 2015.

Lagrange Points

Specific locations in space where the combined gravitational forces of two large bodies (like the Sun and the Earth) equal the centripetal force required for a small third object to move with them. These points allow spacecraft to maintain a stable position with minimal fuel consumption. For instance, India’s Aditya-L1 solar observatory is positioned at Lagrange Point 1.

Last Modified: May 27, 2026