Gravitation

Sir Isaac Newton proposed that every object in the universe attracts every other object, and he expressed this attraction in a single, exact formula.

• The force of attraction acts along the line joining the centres of the two bodies.
• This force is called gravitational force and it is always attractive, never repulsive.

• F = G·m₁·m₂/r² — Force = (Gravitational constant × Mass 1 × Mass 2) ÷ (distance)².
• Where F = gravitational force, m₁ and m₂ = the two masses, r = distance between their centres, and G = the universal gravitational constant.
• Force is directly proportional to the product of the masses: F ∝ m₁·m₂.
• Force is inversely proportional to the square of the distance: F ∝ 1/r² — this is called the inverse-square law.
• If the distance is doubled, the force becomes one-fourth; if the distance is halved, the force becomes four times.

• Gravitation is a universal force — it acts between all bodies, big or small, everywhere in the universe.
• It is the weakest of the four fundamental forces of nature, yet it governs the motion of planets, stars and galaxies.
• Gravitational force does not need a medium and acts even through empty space (vacuum).

The constant G in Newton's law is the same everywhere in the universe, which is why it is called "universal," and its value was first measured experimentally by Henry Cavendish.

• G is numerically equal to the force of attraction between two unit masses (1 kg each) placed at unit distance (1 m) apart.
• The value of G is the same everywhere in the universe — it does not change with place, time, mass or medium.
• G was first measured by Henry Cavendish using a torsion balance experiment.
• The SI unit of G is N·m²/kg² (newton metre-squared per kilogram-squared).
• The very small value of G shows that gravitational force is extremely weak between ordinary objects.

When a body falls freely towards the Earth, it speeds up due to the Earth's pull, and the rate at which its velocity increases is called the acceleration due to gravity.

• The average value of g on Earth's surface is 9.8 m/s² (approximately 10 m/s² in rough calculations).
• g is directed towards the centre of the Earth (vertically downward).
• g is a vector quantity and its SI unit is m/s².
• g does not depend on the mass of the falling body — a heavy and a light body fall with the same acceleration (in the absence of air resistance).

• g = GM/R² — where M = mass of the Earth, R = radius of the Earth, and G = gravitational constant.
• This shows g depends on the mass and radius of the Earth, not on the mass of the falling body.
• The value of g is greater on a planet with larger mass or smaller radius.

Although their symbols look similar, G and g are completely different quantities — one is a universal constant while the other changes from place to place.

As we rise above the Earth's surface, we move farther from its centre, so the pull of gravity — and therefore g — becomes weaker.

• As height above the surface increases, g decreases because the distance from the Earth's centre increases.
• g is inversely proportional to the square of the distance from the centre: g ∝ 1/r².
• At very large distances from the Earth, g becomes almost zero.
• This is why the value of g is less on the top of a high mountain than at sea level.

As we go below the Earth's surface, only the mass of the inner sphere pulls us, so g keeps decreasing until it becomes zero at the very centre of the Earth.

• g is maximum at the surface of the Earth and decreases on going below it.
• At the centre of the Earth, the value of g is zero.
• g decreases both with altitude (going up) and with depth (going down) — it is greatest at the surface.
• The weight of a body becomes zero at the centre of the Earth because g = 0 there.

• g is maximum at the Earth's surface (at the poles).
• g is zero at the centre of the Earth.
• g decreases as you move away from the surface in either direction (up or down).

The Earth is not a perfect sphere — it is slightly flattened at the poles and bulging at the equator — so the value of g is not the same everywhere on its surface.

• The radius of the Earth is smallest at the poles and largest at the equator.
• Since g = GM/R², smaller radius means larger g, so g is greatest at the poles.

• The value of g is maximum at the poles because the polar radius is smallest.
• The value of g is minimum at the equator because the equatorial radius is largest.
• The rotation of the Earth also reduces g at the equator (the effect of centrifugal force is greatest there).
• A body weighs slightly more at the poles than at the equator because g is larger at the poles.

Mass and weight are two different physical quantities that are often confused — mass is the amount of matter in a body, while weight is the gravitational pull on that body.

• Mass is a scalar quantity and its SI unit is the kilogram (kg).
• The mass of a body remains constant everywhere — on Earth, on the Moon, or in space.
• Mass is measured by a beam balance (a physical or common balance).

• Weight is the force with which the Earth attracts a body towards its centre.
• W = mg — Weight = Mass × Acceleration due to gravity.
• Weight is a vector quantity and its SI unit is the newton (N).
• The weight of a body changes from place to place because g changes.
• Weight is measured by a spring balance.

When the only force acting on a body is gravity, it is said to be in free fall, and in this state the body feels no weight — a condition called weightlessness.

• During free fall, the only force acting is gravity, and the body falls with acceleration g.
• In free fall, all bodies fall at the same rate, regardless of their mass (when there is no air resistance).

• Weightlessness is the state in which the apparent weight of a body becomes zero.
• A body in free fall is in a state of weightlessness — it feels no weight even though gravity still acts on it.
• Astronauts in an orbiting satellite feel weightless because both the astronaut and the satellite fall freely towards the Earth at the same rate.
• Weightlessness does not mean gravity is absent — gravity is still acting; only the apparent weight is zero.
• The weight of a body is also zero at the centre of the Earth because g = 0 there.

The Moon is much smaller and less massive than the Earth, so its surface gravity is far weaker — only about one-sixth of the Earth's.

• Gravity on the Moon ≈ 1.6 m/s² (Earth's 9.8 m/s² ÷ 6).
• A body weighs only one-sixth as much on the Moon as on the Earth, but its mass stays the same.
• The Moon has weaker gravity because it has much smaller mass and radius than the Earth.
• The Moon has no atmosphere partly because its weak gravity cannot hold gas molecules.
• This is why astronauts can take long, high jumps on the Moon — they weigh six times less there.

Before Newton, Johannes Kepler discovered three laws that describe exactly how the planets move around the Sun, based on careful observations of their orbits.

• Second law (Law of Areas): a planet moves faster when nearer the Sun and slower when farther away — equal areas are swept in equal times.
• Third law (Law of Periods): T² ∝ r³, so planets farther from the Sun take much longer to complete one revolution.
• Newton later explained Kepler's laws using his law of gravitation.

For a satellite to revolve around the Earth in a stable orbit, it must move with a particular speed called the orbital velocity, which balances gravity against its circular motion.

• v_o = √(GM/r) — where M = mass of the Earth, r = radius of the orbit (distance from Earth's centre).
• For a satellite close to the Earth's surface, orbital velocity ≈ 7.9 km/s (about 8 km/s).
• The orbital velocity decreases as the height of the orbit increases (larger r means smaller v_o).
• Orbital velocity does not depend on the mass of the satellite.
• At this speed, gravity provides exactly the centripetal force needed to keep the satellite in orbit.

If a body is thrown upward fast enough, it can overcome the Earth's gravity completely and never fall back — the minimum speed needed for this is the escape velocity.

• v_e = √(2GM/R) — where M = mass of the planet, R = radius of the planet.
• The escape velocity from the Earth's surface is 11.2 km/s (exam-correct value).
• Escape velocity does not depend on the mass of the escaping body — it is the same for a small or a large object.
• Escape velocity = √2 × orbital velocity, so it is about 1.414 times the orbital velocity.
• The escape velocity from the Moon is about 2.4 km/s — much smaller because the Moon's gravity is weak.
• A body thrown at or above 11.2 km/s will never return to the Earth.

A satellite is any body that revolves around a larger body in space; some occur naturally while others are built and launched by humans.

• The Moon is the natural satellite of the Earth.
• Planets themselves are natural satellites of the Sun as they revolve around it.

• An artificial satellite is a man-made object placed into orbit around the Earth (or another body).
• Artificial satellites are launched by rockets and kept in orbit by gravity.
• Aryabhata was India's first artificial satellite, launched in 1975.
• Artificial satellites are used for communication, weather forecasting, navigation and research.

A special kind of satellite revolves around the Earth in exactly the same time the Earth takes to spin once, so it appears to stay fixed over one spot — these are vital for communication.

• Its time period is 24 hours (exactly one day), the same as the Earth's rotation period.
• It is placed at a height of about 36,000 km above the Earth's surface (equatorial orbit).
• A geostationary satellite appears stationary (fixed) from the Earth because it moves with the Earth.
• It revolves in the equatorial plane from west to east, the same way the Earth spins.
• A geosynchronous satellite also has a period of 24 hours; a geostationary satellite is a special geosynchronous satellite placed directly above the equator.
• Geostationary satellites are mainly used for communication and television broadcasting because they always face the same region.

Artificial satellites have become essential to modern life, serving in communication, weather prediction, navigation, mapping and scientific study.

• Weather forecasting — to observe clouds, storms and cyclones and predict the weather.
• Navigation — GPS satellites help locate positions and guide ships, aircraft and vehicles.
• Remote sensing — for studying land, oceans, forests, crops and natural resources.
• Military and surveillance — for keeping watch on borders and gathering intelligence.
• Scientific research — for studying space, the atmosphere and other planets.
• Disaster management — to monitor floods, earthquakes and other natural calamities.