Board exam focus — Gravitation (CBSE & HBSE)

Gravitation explains how every mass in the universe attracts every other mass through a single inverse-square law. This chapter builds from Kepler's empirical planetary laws to Newton's universal law of gravitation, then derives acceleration due to gravity (g) and its variation with altitude, depth, latitude and Earth's shape. It develops gravitational potential and potential energy, and applies energy conservation to escape velocity, orbital velocity, time period of satellites, and the special case of geostationary satellites. The chapter is high-weightage in CBSE and HBSE, with frequent numericals on g-variation, escape/orbital speeds and Kepler's third law.

Universal Law of Gravitation and Acceleration Due to Gravity

Kepler's Laws of Planetary Motion

Before Newton, Kepler summarised planetary motion in three laws (deduced from Tycho Brahe's data):

1. Law of Orbits: Every planet moves in an elliptical orbit with the Sun at one focus.
2. Law of Areas: The line joining a planet to the Sun sweeps out equal areas in equal intervals of time — areal velocity is constant. This is a direct consequence of conservation of angular momentum.
3. Law of Periods: The square of the time period is proportional to the cube of the semi-major axis: T² ∝ a³.

The law of areas means a planet moves faster near perihelion (closest to Sun) and slower near aphelion (farthest).

Newton's Universal Law of Gravitation

Every particle 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:

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

• G = universal gravitational constant = 6.67 × 10⁻¹¹ N m² kg⁻²
• G is a scalar, has the same value everywhere in the universe, and was measured by Cavendish.
• The force is attractive, acts along the line joining the masses, and obeys Newton's third law.

Acceleration Due to Gravity (g)

For a body of mass m on Earth's surface, the gravitational force equals mg. Equating with the universal law:

$$g = \frac{GM}{R^2}$$

where M is Earth's mass and R its radius. Its value at the surface ≈ 9.8 m/s².

g is independent of the mass of the falling body — this is why all bodies fall with the same acceleration in vacuum.

Relation between g and density: Since M = (4/3)πR³ρ,

$$g = \frac{4}{3}\pi G R \rho$$

Variation of g with Altitude, Depth, Latitude and Shape

Variation with Altitude (Height h)

At a height h above the surface, the value of g decreases:

$$g_h = g\left(\frac{R}{R+h}\right)^2$$

For small heights (h << R), using binomial approximation:

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

Fractional decrease in g with height = 2h/R.

Variation with Depth (Depth d)

At a depth d below the surface, only the inner sphere of radius (R − d) attracts the body:

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

• At the centre of the Earth (d = R), g = 0.
• g decreases linearly with depth but non-linearly with height.

Variation with Latitude (Rotation of Earth)

Due to Earth's rotation, the effective value of g at latitude λ is:

$$g_\lambda = g - R\omega^2\cos^2\lambda$$

• At the equator (λ = 0°): g is minimum, reduced by Rω².
• At the poles (λ = 90°): g is maximum, unaffected by rotation.

If Earth stopped rotating, g at the equator would increase; if Earth's rotation speeded up, bodies at the equator could become weightless.

Variation with Shape

Earth is an oblate spheroid: equatorial radius > polar radius. Since g = GM/R², g is greater at the poles and smaller at the equator — both shape and rotation make g maximum at the poles.

Gravitational Potential Energy, Escape and Orbital Velocity, Satellites

Gravitational Potential and Potential Energy

Gravitational potential (V) at a point is the work done in bringing unit mass from infinity to that point:

$$V = -\frac{GM}{r}$$

Gravitational potential energy (U) of a mass m at distance r from M:

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

The negative sign shows the force is attractive and U increases (toward zero) as r increases. U = 0 at infinity (reference).

For a body near the surface raised by height h, the change in PE ≈ mgh (only for h << R).

Escape Velocity

The minimum velocity needed to project a body so it escapes Earth's gravity (reaches infinity with zero KE):

$$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$$

• For Earth, v_e ≈ 11.2 km/s.
• Escape velocity is independent of the mass and direction of projection of the body.

Orbital Velocity

The velocity required for a satellite to revolve in a circular orbit at height h:

$$v_o = \sqrt{\frac{GM}{R+h}}$$

For a satellite close to the surface (h ≈ 0):

$$v_o = \sqrt{gR} \approx 7.9\ \text{km/s}$$

Important relation: v_e = √2 × v_o (escape velocity is √2 times the orbital velocity of a near-surface satellite).

Time Period of a Satellite

$$T = 2\pi\sqrt{\frac{(R+h)^3}{GM}}$$

Geostationary and Polar Satellites

Energy of a Satellite

• Kinetic energy: KE = GMm / 2(R+h)
• Potential energy: PE = − GMm / (R+h)
• Total energy: E = − GMm / 2(R+h) (negative → bound system).

Binding energy of a satellite = + GMm / 2(R+h); for a body at rest on the surface, binding energy = GMm/R.

Frequently asked questions

Are these Gravitation notes free?

Yes — the Gravitation notes for Physics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Gravitation notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Gravitation chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Gravitation.