Gravitation – Complete AMU 11 Notes

1) Newton's Universal Law of Gravitation

Every particle 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 centres.

Vector Form

The gravitational force on m₁ due to m₂ is: F₁₂ = -G(m₁m₂/r²) r̂₂₁
where r̂₂₁ is the unit vector from m₂ to m₁. The negative sign indicates attraction.

Characteristics of Gravitational Force:

• Always attractive (never repulsive) — unlike electrostatic force which can be repulsive.
• Central force — acts along the line joining the centres of the two bodies.
• Follows inverse square law — F ∝ 1/r².
• Independent of the medium — unlike electrostatic force, it doesn't depend on the intervening medium.
• Weakest force in nature — for small masses, gravitational force is negligible. For example, the gravitational force between two humans is about 10⁻⁷ N — too small to feel!
• Gravitational force is significant only for astronomical bodies (planets, stars, galaxies).
• Long-range force — its range is infinite (theoretically).

Importance of the Universal Law of Gravitation

Newton's Universal Law of Gravitation was a revolutionary breakthrough because it unified terrestrial physics (objects falling on Earth) with celestial physics (motion of planets). Before Newton, people thought different laws governed the heavens and the Earth. Newton showed that the same force that makes an apple fall also keeps the Moon in orbit!

Cavendish Experiment — Measuring G

In 1798, Henry Cavendish used a torsion balance to measure the gravitational force between two lead spheres. From this, he calculated G ≈ 6.67 × 10⁻¹¹ N m²/kg². This is why Cavendish is credited with "weighing the Earth" — once G was known, M = gR²/G gave the mass of Earth.

Why G is a Universal Constant

• G has the same value everywhere in the universe (hence "universal").
• G does NOT depend on medium, temperature, or any physical condition.
• G is a scalar quantity with unit N m²/kg².
• G is extremely small (6.67 × 10⁻¹¹), showing gravity is the weakest fundamental force.

2) Gravitational Field and Intensity

The gravitational field at a point is the gravitational force experienced by a unit mass placed at that point. It is a vector quantity.

So g is actually the gravitational field intensity at Earth's surface. That's why g = 9.8 N/kg = 9.8 m/s².

3) Kepler's Laws of Planetary Motion

Johannes Kepler (1571-1630) gave 3 laws based on Tycho Brahe's precise observations of planetary motion. Newton later used these laws to derive his Universal Law of Gravitation.

Law 1 — The Law of Orbits (Elliptical Orbit Law)

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

Law 2 — The Law of Areas (Equal Area Law)

The line joining the planet to the Sun sweeps out equal areas in equal intervals of time.

Consequence: When a planet is closer to the Sun (perihelion), it moves faster. When farther (aphelion), it moves slower. For Earth:

• Perihelion (early January): Earth is closest to Sun → moves fastest
• Aphelion (early July): Earth is farthest from Sun → moves slowest

Law 3 — The Law of Periods (Harmonic Law)

The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of its elliptical orbit.

4) Acceleration Due to Gravity (g)

The acceleration produced in a body due to the gravitational pull of Earth is called acceleration due to gravity (g).

Derivation of g = GM/R²

Mass of Earth from g:

5) Mass vs Weight

This is a very common AMU trick topic. Know the difference cold.

6) Variation of g with Height, Depth, and Latitude

The value of g is not constant everywhere on or above Earth. It varies with:

6.1 Variation with Height (above Earth's surface)

6.2 Variation with Depth (below Earth's surface)

6.3 Variation with Latitude (Rotation of Earth)

Comparison Table — Variation of g:

7) Gravitational Potential Energy (Basic Concept)

Gravitational potential energy (U) of a body at a point is the work done in bringing it from infinity to that point against the gravitational field. It is always negative — which means the force is attractive.

8) Free Fall (Detailed)

When a body falls only under the influence of gravity (no air resistance), it is called free fall.

Key Points about Free Fall:

• During free fall, all bodies accelerate downward at the same rate g, regardless of their mass.
• This was famously demonstrated by Galileo's Leaning Tower of Pisa experiment (and later on the Moon by Apollo 15 astronaut David Scott — a feather and a hammer hit the ground at the same time!).
• In free fall, the body experiences weightlessness — its weight appears to be zero.

Weightlessness

Weightlessness is the feeling of zero weight (g = 0). This happens when:

• A body is in free fall (falling freely under gravity).
• An astronaut in a satellite orbiting Earth (both satellite and astronaut are in free fall towards Earth).
• At the centre of the Earth (gravitational force cancels from all sides).

9) Escape Velocity

The minimum velocity required for a body to escape Earth's gravitational pull and never return is called escape velocity.

Derivation:

10) Orbital Velocity and Satellites

Orbital Velocity (v₀)

The velocity required for a satellite to revolve around Earth in a circular orbit is called orbital velocity.

Relation Between Escape Velocity and Orbital Velocity:

Time Period of a Satellite:

Types of Satellites:

Geostationary (Geosynchronous) Satellites:

• Orbit height ≈ 36,000 km above Earth's surface
• Time period = 24 hours (same as Earth's rotation period)
• Orbits in the equatorial plane, west to east
• Appears stationary from a fixed point on Earth
• Uses: Communication (TV, telephone), weather monitoring, broadcasting

Polar Satellites:

• Orbit at lower altitudes (500-800 km)
• Pass over the poles
• Time period ≈ 90 minutes
• Uses: Remote sensing, mapping, spying, environmental monitoring

11) Thrust and Pressure (Brief)

Thrust is the force acting perpendicularly to a surface. Pressure is thrust per unit area.

In the context of gravitation, the weight of a body (thrust) acting on a surface produces pressure. This is why:

• A sharp needle pricks easily (small area → large pressure).
• A camel has broad feet (large area → less pressure → doesn't sink in sand).
• Building foundations are broad (large area → less pressure on ground).

12) Thrust and Pressure (Brief)

Thrust is the force acting perpendicularly to a surface. Pressure is thrust per unit area.

In the context of gravitation, the weight of a body (thrust) acting on a surface produces pressure. This is why:

• A sharp needle pricks easily (small area → large pressure).
• A camel has broad feet (large area → less pressure → doesn't sink in sand).
• Building foundations are broad (large area → less pressure on ground).
• A tractor has wide tires to reduce pressure on soft ground.
• School bags have broad straps to reduce pressure on shoulders.

13) Applications of Gravitation in Daily Life

• Tides in the ocean — caused by the gravitational pull of the Moon (and Sun) on Earth's water.
• Satellites in orbit — communication TV, weather forecasting, GPS navigation — all depend on gravitational forces keeping satellites in orbit.
• Rockets must achieve escape velocity (11.2 km/s) to overcome Earth's gravity.
• Weightlessness experienced by astronauts — they are in continuous free fall around Earth.
• Formation of stars and planets — gravitational collapse of gas clouds leads to star formation.
• Earth's atmosphere is held by gravity — without it, all air would escape into space!

14) Why the Moon has No Atmosphere

The Moon's escape velocity is only 2.4 km/s, which is very small compared to Earth's 11.2 km/s. The average speed of gas molecules at normal temperatures is around 1-2 km/s. Many gas molecules on the Moon can easily exceed 2.4 km/s and escape into space.

Key point: For a planet or moon to retain an atmosphere, its escape velocity must be at least 6 times the root-mean-square speed of gas molecules. Since the Moon's escape velocity is too low, it couldn't hold onto an atmosphere.

15) Common Mistakes Students Make in Gravitation

13) Important Formulae — At a Glance

14) PYQ Battle — 16 AMU Past Questions on Gravitation

15) Practice Exercise — 15 Problems

Try these. Cover the answers — honest practice is the key.

16) Quick Revision — One-liners

• Newton's Law: F = G m₁ m₂ / r² (universal gravitational attraction)
• G = 6.67 × 10⁻¹¹ N m²/kg² (Cavendish determined it experimentally)
• Gravitational force is the weakest, longest-range, always attractive force
• Kepler's 1st: planets move in elliptical orbits with Sun at one focus
• Kepler's 2nd: equal areas in equal time → angular momentum conserved
• Kepler's 3rd: T² ∝ r³ (applies to all satellites, not just planets)
• g = GM/R² = 9.8 m/s² (acceleration due to gravity at Earth's surface)
• g decreases with height (g_h = g[R/(R+h)]²) and depth (g_d = g[1 - d/R])
• At centre of Earth: g = 0; at poles: g max; at equator: g min
• Mass = constant everywhere (kg); Weight = mg (N), varies with g
• Gravitational PE: U = -GMm/r (negative indicates attraction)
• Escape velocity: vₑ = √(2GM/R) = √(2gR) = 11.2 km/s (for Earth)
• Orbital velocity (near Earth): v₀ = √(gR) ≈ 7.9 km/s
• vₑ = √2 × v₀ (escape is 41% more than orbital speed)
• Geostationary: h ≈ 36,000 km, T = 24 h, equatorial orbit, appears stationary
• Free fall: all bodies accelerate at same g (in vacuum)
• Weightlessness: occurs in free fall/space (apparent weight = 0)
• Pressure = F/A, SI unit: Pa (N/m²)