Why does a dropped pen fall but the Moon never crashes into Earth? The answer is gravitation — a universal attractive force between any two masses. For the NDA exam, this is a scoring chapter: a handful of formulas (Newton's law, g, escape velocity, orbital velocity) reappear in PYQs almost every year. Master them once and bank the marks.
Why gravitation matters for NDA
Gravitation is the force that holds the entire universe together. It keeps your feet on the ground, gives weight to objects, drives the tides, and binds planets to the Sun. In the NDA General Ability Test (Physics section), questions from this topic are direct and formula-based, so they are easy marks if you have memorised a few relations.
The story begins with Sir Isaac Newton, who realised that the same force pulling an apple to the ground also keeps the Moon in its orbit. This single insight unified events on Earth with events in the heavens.
Gravitation is always attractive — there is no "repulsive gravity." It is also the weakest of the four fundamental forces, yet it dominates at astronomical scales because it acts over huge distances and never cancels out.
Newton's universal law of gravitation
Every particle of matter 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 them.
F = G × (m1 m2) ÷ r2
where F = force of attraction, m1, m2 = the two masses, r = distance between their centres, and G = universal gravitational constant.
The value of the gravitational constant is G = 6.67 × 10−11 N·m2/kg2. It is the same everywhere in the universe, which is why the law is called "universal."
Key features
- The force acts along the line joining the two centres.
- It obeys an inverse-square law: double the distance, and the force becomes one-quarter.
- The two forces form an action–reaction pair (Newton's third law): the Earth pulls you down with the same force you pull the Earth up.
Do not confuse G (universal gravitational constant, fixed everywhere) with g (acceleration due to gravity, which changes from place to place). They are different quantities with different units.
Acceleration due to gravity (g)
When an object falls freely, gravity gives it an acceleration called g. Near Earth's surface, its standard value is g = 9.8 m/s2 (often rounded to 9.81 m/s2).
We can connect g to Newton's law. The force on a mass m at Earth's surface equals its weight, mg, and also equals the gravitational pull from Earth's mass M:
g = G M ÷ R2
where M = mass of Earth and R = radius of Earth. Notice that g does not depend on the mass of the falling object — a feather and a stone fall with the same g in a vacuum.
Weight vs mass
- Mass is the amount of matter in a body. It is constant everywhere and measured in kilograms.
- Weight is the gravitational force on the body, W = mg. It changes with location because g changes.
On the Moon, g is about 1/6 of Earth's value, so your weight on the Moon is one-sixth your weight on Earth — but your mass stays exactly the same.
How g varies with height, depth and shape
The value of g is not truly constant over the whole planet. The NDA loves to test these variations.
With height (altitude)
As you go up above the surface, r increases, so g decreases. At a height h above the surface, g falls roughly in proportion to 1/(R + h)2. That is why g is slightly less at the top of a mountain than at sea level.
With depth
As you go down into the Earth, g also decreases, because only the mass below you contributes to the pull. At the centre of the Earth, g = 0, since mass surrounds you symmetrically and the net pull cancels.
With latitude and Earth's shape
The Earth is not a perfect sphere — it bulges at the equator and is flattened at the poles. So:
- g is maximum at the poles (you are closer to the centre).
- g is minimum at the equator (you are farther out, and rotation reduces effective gravity).
A handy memory line: g is greatest at the poles, least at the equator, decreases going up, decreases going down, and is zero at the centre. Many one-mark NDA questions are answered just by this single sentence.
Gravitational field and potential energy
The region around a mass where another mass feels a gravitational pull is called its gravitational field. The field strength at a point equals the force per unit mass placed there, and it is numerically equal to g.
Gravitational potential energy
Lifting an object higher stores energy in it. Near the surface, the gravitational potential energy is:
P.E. = m g h
where h is the height above the reference level. This energy converts to kinetic energy as the object falls.
For motion far from Earth (in space problems), physicists use a more general form, U = −G M m ÷ r, where the negative sign shows that energy must be supplied to move an object away from Earth against gravity. This negative potential energy idea leads directly to the concept of escape velocity.
Free fall feels like "weightlessness" not because gravity vanishes, but because the object and everything around it fall together with the same acceleration g. Astronauts in orbit are in continuous free fall.
Escape velocity
Escape velocity is the minimum speed an object must be given at the surface so that it leaves the planet's gravity forever and never falls back.
ve = √(2 g R) = √(2 G M ÷ R)
For Earth, escape velocity is about 11.2 km/s (roughly 11,200 m/s). The Moon's escape velocity is only about 2.4 km/s.
Important properties
- It does not depend on the mass of the escaping object — a pebble and a rocket need the same escape speed.
- It does not depend on the direction of launch.
- A larger or denser planet has a higher escape velocity.
Escape velocity from Earth ≈ 11.2 km/s is one of the most frequently asked single values in NDA Physics. Memorise it as a fact, and also note ve = √2 × orbital velocity.
Satellites and orbital velocity
A satellite is any body that revolves around a larger body under gravity. The Moon is Earth's natural satellite; communication and weather satellites are artificial ones.
For a satellite to stay in a circular orbit, the gravitational pull provides exactly the centripetal force needed to keep it turning. This gives the orbital velocity:
vo = √(G M ÷ r) = √(g R2 ÷ r)
For a satellite orbiting close to Earth's surface, vo ≈ 7.9 km/s (about 8 km/s).
Geostationary satellites
A geostationary satellite orbits in the equatorial plane with a period of exactly 24 hours, matching Earth's rotation. So it appears fixed over one point on the ground — ideal for TV broadcasting and communication. It sits at a height of about 36,000 km above the equator.
A satellite stays up not because gravity is absent, but because its high sideways speed makes it "fall around" the Earth. If you removed gravity, the satellite would fly off in a straight line, not orbit.
Kepler's three laws of planetary motion
Before Newton, Johannes Kepler described how planets move around the Sun using three laws based on careful observation.
1. Law of orbits
Every planet moves in an ellipse with the Sun at one focus. Orbits are not perfect circles.
2. Law of areas
The line joining a planet to the Sun sweeps out equal areas in equal times. This means a planet moves faster when nearer the Sun and slower when farther away. (It is really a statement of conservation of angular momentum.)
3. Law of periods
T2 ∝ r3
The square of a planet's orbital period is proportional to the cube of the average radius of its orbit. So planets farther from the Sun take much longer to complete one revolution.
Kepler observed the pattern; Newton later explained it using his law of gravitation. The two together form the foundation of classical astronomy.
Worked example
Let us see how a typical numerical works out step by step.
The mass of a planet is the same as Earth's, but its radius is half of Earth's. If g on Earth is 9.8 m/s2, find g on this planet.
So halving the radius (with the same mass) makes gravity four times stronger. This is the inverse-square law in action — a favourite NDA trick.
Common mistakes to avoid
These slips cost students easy marks every year. Read them twice.
- G vs g: G is constant everywhere; g changes with location.
- Mass vs weight: mass is constant; weight = mg changes with g.
- g at the centre: it is zero, not maximum.
- g at poles vs equator: maximum at the poles, minimum at the equator — do not swap these.
- Escape velocity: independent of the object's mass; for Earth it is 11.2 km/s, not the orbital 7.9 km/s.
- Weightlessness: astronauts are not beyond gravity; they are in free fall.
Many students write that gravity "increases as you dig towards the centre." It actually decreases with depth and becomes zero at the centre. Going up also decreases g. Only one direction (towards the surface from inside, or down towards sea level from a mountain) increases it.
Previous-year style question
Here is the kind of question you can expect in the exam.
Q. The escape velocity of a body from the surface of the Earth is approximately:
Answer: 11.2 km/s. This is the minimum speed needed for a body to overcome Earth's gravity completely and not return. It is independent of the mass of the body and is equal to √2 times the orbital velocity (about 7.9 km/s) near the surface.
When a question gives "same mass, different radius" or "same density, different size," go straight to g = GM/R2 and substitute. These appear almost every year and are quick to solve.
Quick revision
- Newton's law: F = G m1m2 / r2; G = 6.67 × 10−11 N·m2/kg2.
- Acceleration due to gravity: g = GM/R2 ≈ 9.8 m/s2; independent of the falling object's mass.
- g is maximum at the poles, minimum at the equator, decreases with height and depth, and is zero at the centre.
- Escape velocity from Earth ≈ 11.2 km/s; orbital velocity near surface ≈ 7.9 km/s.
- Geostationary satellite: period 24 h, height ≈ 36,000 km, appears fixed over the equator.
- Kepler: ellipse orbits, equal areas in equal times, T2 ∝ r3.
Revise these six lines the night before the exam and you can confidently attempt almost any gravitation MCQ. Pair the formulas with the worked example above, and this becomes one of your safest scoring chapters.
Frequently asked questions
What is the difference between G and g?
G is the universal gravitational constant (6.67 × 10−11 N·m2/kg2) and is the same everywhere in the universe. g is the acceleration due to gravity (about 9.8 m/s2 on Earth) and changes with location.
What is the escape velocity from Earth?
About 11.2 km/s. It is the minimum speed a body needs to leave Earth's gravity permanently, and it does not depend on the mass of the body.
Why do astronauts feel weightless in orbit?
Not because gravity is absent, but because they and their spacecraft are in continuous free fall around the Earth, falling together with the same acceleration, so there is no supporting force pressing on them.
Where is the value of g greatest on Earth?
g is maximum at the poles and minimum at the equator, because Earth bulges at the equator. It also decreases as you go up in height or down towards the centre, where it becomes zero.
What is a geostationary satellite?
A satellite that orbits in the equatorial plane with a period of 24 hours, matching Earth's rotation, so it appears fixed over one point. It sits about 36,000 km above the equator and is used for communication and broadcasting.
Related NDA Physics topics
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