Gravitation

🎯 Key Points

• F = GMm/r²; g = GM/R² at Earth's surface; g decreases both with height (g(1−2h/R)) and depth (g(1−d/R))
• Orbital velocity v₀=√(GM/r); escape velocity v_e=√(2GM/R)=√2 × v₀ (always, at the same r)
• Kepler's 3rd law: T² ∝ r³ - a direct consequence of equating gravitational force to centripetal force requirement
• Total energy of a satellite = −GMm/2r (always negative = bound orbit); Binding energy = +GMm/2r
• Astronauts feel "weightless" because they and their spacecraft share the same free-fall acceleration, not because gravity is zero up there

Newton's Law of Gravitation

• F = GMm/r² (attractive force between two masses)
• G = 6.67 × 10⁻¹¹ N·m²/kg²
• g = GM/R² (acceleration due to gravity at Earth surface)

Variation of g

• At height h: g_h = g(1 - 2h/R) for h << R
• At depth d: g_d = g(1 - d/R)
• Due to rotation: g is minimum at equator, maximum at poles

Orbital Mechanics

• Orbital velocity: v₀ = √(GM/r) = √(gR²/r)
• Time period: T = 2π√(r³/GM) (Kepler's 3rd law: T² ∝ r³)
• Escape velocity: v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s
• Geostationary orbit: T = 24 h, height ≈ 36,000 km

Gravitational Potential Energy

• U = -GMm/r (negative; zero at infinity)
• Total energy of satellite = -GMm/2r (negative, always)
• Binding energy = GMm/2r

Kepler's Laws of Planetary Motion

• Law of Orbits: every planet moves in an elliptical orbit with the Sun at one focus
• Law of Areas: the line joining the planet to the Sun sweeps out equal areas in equal times (areal velocity is constant); this is a direct consequence of conservation of angular momentum
• Law of Periods: the square of the time period is proportional to the cube of the semi-major axis, T² ∝ r³

Kepler’s laws: elliptical orbits with the Sun at one focus, equal areas A₁=A₂ swept in equal times, and T² ∝ a³. Image: Hankwang, CC BY 2.5, via Wikimedia Commons.

Gravitational Potential

• Gravitational potential: V = -GM/r (work done per unit mass to bring it from infinity to that point)
• Relation to potential energy: U = mV
• Gravitational potential inside a uniform solid sphere is constant in form but field varies linearly; on the surface V = -GM/R

Weightlessness in Satellites

• An astronaut in orbit experiences apparent weightlessness because both the astronaut and satellite have the same centripetal acceleration (= g at that height) and fall freely around Earth together
• The normal force between astronaut and satellite floor becomes zero, not because gravity vanishes

Relation Between Escape and Orbital Velocity

• v_e = √2 × v₀ (escape velocity is √2 times the orbital velocity at the same radius)
• If orbital speed is increased to √2 times its value, a satellite in circular orbit escapes the gravitational field entirely

Gravitational Field Intensity and Superposition

• Gravitational field intensity: E = F/m = GM/r² (force per unit mass placed at a point; a vector directed towards the source mass)
• Superposition principle: the net gravitational force (or field) due to several masses is the vector sum of the forces (or fields) due to each mass taken individually
• Field inside a uniform solid sphere at distance r from centre: E = GMr/R³ (increases linearly with r); outside: E = GM/r²
• Field due to a uniform spherical shell is zero everywhere inside it

Acceleration due to Gravity: Rotation and Shape of Earth

• Due to Earth's rotation, effective g at latitude λ is g' = g − ω²R·cos²(λ); g is minimum at the equator and maximum at the poles
• If Earth's rotation stopped, g at the equator would increase by ω²R (≈ 0.034 m/s²)
• Because Earth bulges at the equator (R_equator > R_pole), g is also slightly larger at the poles due to the smaller radius there

Types of Satellites: Geostationary and Polar

• Geostationary satellite: orbits in the equatorial plane with period 24 h, same direction as Earth's rotation, at height ≈ 36,000 km; appears fixed in the sky (used for communication)
• Polar satellite: low-altitude (few hundred km) satellite in a north–south orbit passing over the poles; used for weather imaging and remote sensing, scanning the whole globe strip by strip

🚀 JEE Advanced Edge

Variation of g with latitude: Due to Earth's rotation, the effective g at latitude λ is g' = g − ω²R cos²λ, where ω is Earth's angular velocity. This is why g is minimum at the equator (λ=0°, full subtraction) and maximum at the poles (λ=90°, no subtraction since rotation provides no centripetal requirement there).

Energy required to move a satellite between orbits: ΔE = E_final − E_initial = (−GMm/2r_f) − (−GMm/2r_i) - since total energy becomes LESS negative (increases) as r increases, moving a satellite to a HIGHER orbit always requires a net energy INPUT, even though its speed there is actually LOWER (counterintuitive - higher orbit means lower orbital speed but higher total energy, because potential energy increases more than kinetic energy decreases).

Worked problem: Calculate the height of a geostationary satellite above Earth's surface (R=6400 km, g=9.8 m/s², T=24h). Approach: Equate gravitational force to centripetal requirement: GM/r² = ω²r → r³ = GM/ω². Using GM=gR², and ω=2π/T, solve for r (orbital radius from Earth's centre) ≈ 42,300 km, then height = r − R ≈ 35,900 km, matching the well-known ~36,000 km figure.

Inertial and Gravitational Mass

• Inertial mass is defined through Newton's second law, m = F/a, and measures a body's resistance to acceleration.
• Gravitational mass is defined through the law of gravitation and measures the strength of a body's gravitational interaction.
• Experiments show the two are equal to very high precision; this equivalence is the basis of Einstein's principle of equivalence.
• Because they are equal, all bodies fall with the same acceleration g in a given gravitational field, independent of their mass.
• This is why a feather and a coin fall together in an evacuated tube.

Binding Energy of a Satellite

• The total energy of a satellite in a circular orbit is E = minus GMm/(2r), which is negative, showing the satellite is bound.
• Binding energy is the minimum energy needed to remove the satellite from orbit to infinity: BE = plus GMm/(2r).
• Kinetic energy KE = GMm/(2r) and potential energy PE = minus GMm/r, so total E = KE plus PE = minus KE.
• The magnitude of potential energy is twice the kinetic energy, a result linked to the virial theorem.
• For a body resting on Earth's surface, binding energy equals GMm/R, related to escape velocity by BE = (1/2)m(v_e)^2.

Effect of Orbit Radius on Satellite Motion

• Orbital speed v = square root of (GM/r) decreases as the orbit radius r increases, so higher satellites move slower.
• The time period T = 2 pi times square root of (r³/GM) increases with r, consistent with Kepler's third law T² proportional to r^3.
• A satellite in a lower orbit has greater kinetic energy but smaller (more negative) total energy than one in a higher orbit.
• Due to atmospheric drag in low orbits, a satellite loses energy, spirals inward, and paradoxically speeds up as it descends.
• The near-Earth orbital speed is about 7.9 km/s and the period is roughly 84 minutes.

Gravity Inside the Earth and a Tunnel

• Assuming uniform density, the value of g at depth d below the surface is g_d = g(1 minus d/R).
• At the centre of the Earth (d = R), the value of g becomes zero, since equal mass surrounds the point in all directions.
• Only the mass of the sphere below the point contributes; the outer shell exerts zero net force on an interior point.
• A body dropped in a frictionless tunnel through the Earth executes simple harmonic motion about the centre.
• The time period of oscillation in such a tunnel is T = 2 pi times square root of (R/g), about 84 minutes, the same as a near-Earth orbit.