Chapter 7: Gravitation | Summary

Newton's Universal Law of Gravitation states that every body in the universe attracts every other body with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Mathematically, F = Gm₁m₂/r², where G is the universal gravitational constant with the accepted value of 6.67 × 10⁻¹¹ N m²/kg².

For extended objects, the gravitational force on any mass is the vector sum of forces due to all constituent point masses of the other body. Two important results follow: a uniform spherical shell attracts an external point mass as if its entire mass is concentrated at its centre, and exerts zero force on a point mass inside it.

The gravitational constant G was first measured experimentally by Henry Cavendish in 1798 using a torsion balance apparatus. The setup consists of a bar with two small lead spheres at its ends, suspended by a fine wire. Large lead spheres brought close to the small ones produce a torque that twists the wire. By measuring the angle of twist and the restoring couple per unit twist, G can be calculated.

The currently accepted value of G is 6.67 × 10⁻¹¹ N m²/kg². Knowing G, the acceleration due to gravity g, and the radius of the Earth RE allows one to estimate the mass of the Earth using g = GME/RE², which is why Cavendish's experiment is often described as 'weighing the earth'.

The acceleration due to gravity at the surface of the Earth is given by g = GME/RE². Since the Earth can be treated as a collection of concentric spherical shells, a point outside the Earth experiences a gravitational force as if the entire mass of the Earth is concentrated at its centre.

For a point inside the Earth at distance r from the centre, only the mass enclosed within the sphere of radius r contributes to the gravitational force. The outer shells exert no net force on the interior point, a consequence of the shell theorem. This leads to the result that gravitational force inside the Earth varies linearly with distance from the centre.

At a height h above the Earth's surface, the acceleration due to gravity is g(h) = GME/(RE + h)². For h much smaller than RE, this approximates to g(h) ≈ g(1 − 2h/RE), showing that g decreases with altitude. At a depth d below the surface, g(d) = g(1 − d/RE), showing that g also decreases as one goes deeper.

A remarkable feature of Earth's gravitational acceleration is that it is maximum at the surface and decreases both upward and downward from the surface. The rate of decrease is different in the two directions: above the surface, g decreases more rapidly (inversely as square of distance) compared to below the surface where it decreases linearly.

Gravitational potential energy is the energy stored in a body due to its position in a gravitational field. Since gravity is a conservative force, the work done is path-independent. For points near the Earth's surface, the potential energy is W(h) = mgh. For arbitrary distances from the Earth's centre, it is W(r) = −GMEm/r, with zero potential energy chosen at infinity.

The gravitational potential due to the Earth is the potential energy per unit mass: U(r) = −GM/r. For a system of particles, the total gravitational potential energy is the sum over all pairs, by the principle of superposition. Only differences in potential energy are physically meaningful, but the convention V = 0 at r → ∞ is universally adopted.

Escape speed is the minimum initial speed required for an object projected from the Earth's surface to escape Earth's gravitational field and reach infinity. Using energy conservation, the escape speed is ve = √(2GME/RE) = √(2gRE), which evaluates to approximately 11.2 km/s for the Earth.

Escape speed does not depend on the mass of the projected object or the direction of projection. For the Moon, the escape speed is about 2.3 km/s — about five times smaller than Earth's — because both the Moon's mass and radius are smaller. This explains why the Moon has no atmosphere, as gas molecules with sufficient thermal speeds can escape the Moon's gravity.

Earth satellites are objects revolving around the Earth, obeying Kepler's laws just as planets do around the Sun. For a satellite in a circular orbit at height h, the orbital speed is V = √[GME/(RE + h)] and the time period is T = 2π(RE + h)^(3/2) / √(GME). The speed decreases and the period increases with increasing orbital height.

The Moon is the only natural satellite of Earth with a period of approximately 27.3 days. Since 1957, many artificial satellites have been launched for applications in telecommunication, geophysics, and meteorology. For a satellite at height close to Earth's surface, the time period is approximately 85 minutes.

For a satellite in a circular orbit at distance (RE + h) from the Earth's centre, the kinetic energy is KE = GMEm / [2(RE + h)], the potential energy is PE = −GMEm / (RE + h), and the total mechanical energy is E = −GMEm / [2(RE + h)]. The total energy is negative, indicating the satellite is in a bound state.

The magnitude of kinetic energy is exactly half the magnitude of potential energy. When the orbit becomes elliptical, both KE and PE vary from point to point, but the total energy remains constant and negative. A positive or zero total energy would mean the object is unbound and escapes to infinity, which is not the case for any satellite.

Chapter 7 establishes gravitation as a universal and fundamental interaction governing the motion of everything from falling apples to orbiting planets. Starting from Kepler's empirical laws and Newton's universal law of gravitation, the chapter builds a coherent framework covering the gravitational constant, variations of g with altitude and depth, gravitational potential energy, escape speed, and the mechanics of Earth's satellites. The negative total energy of bound orbiting objects and the inverse-square nature of gravitational force are recurring themes that unify the entire chapter.

For board examinations, this chapter is critically important. Derivations such as g = GME/RE², g at height and depth, orbital speed, time period and energy of satellites, and escape speed are frequently tested. Kepler's laws and their physical basis — especially the connection between the law of areas and conservation of angular momentum — are conceptual staples. Numericals involving satellite energy, escape speed, and applications of Kepler's third law appear regularly in CBSE board papers and competitive examinations.