The radius of the Moon is about one-fourth that of the Earth and acceleration due to gravity on the Moon is about one-sixth that on the Earth. From this, we can conclude that the ratio of the mass of Earth to the mass of the Moon is about

1. Newton's Law of Universal Gravitation (basic)

Hello! Let’s begin our journey into the stars by understanding the invisible thread that holds everything together: Gravity. At its most fundamental level, Isaac Newton's law of gravitation represented a climax in the scientific revolution Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119. He proposed that every object in the universe pulls on every other object with a force that depends on just two factors: how heavy they are (mass) and how far apart they are (distance).

Newton formulated this as an equation: F = G (m₁m₂ / r²). In this formula, F represents the gravitational force, measured in newtons (N) Science, Class VIII NCERT (Revised ed 2025), Exploring Forces, p.65. The terms m₁ and m₂ are the masses of the two objects, while r is the distance between their centers. The letter G stands for the Universal Gravitational Constant—a value that never changes, regardless of where you are in the universe.

This law is famously known as an inverse-square law. This means that distance has a much more dramatic effect on gravity than mass does. For example, if you double the distance between two planets, the gravitational pull between them doesn't just drop by half; it drops to one-fourth (2²) of its original strength. Today, scientists even use gravitational waves to measure the Hubble constant, which describes how fast our universe is expanding Physical Geography by PMF IAS, The Universe, p.5. However, Newton’s basic law remains the bedrock for understanding celestial mechanics.

Factor	Relationship to Force	Effect of Doubling
Mass	Directly Proportional	Force Doubles
Distance	Inversely Proportional to Square	Force drops to 1/4th

Sources: Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119; Science, Class VIII NCERT (Revised ed 2025), Exploring Forces, p.65; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.5

2. Mass vs. Weight and Acceleration due to Gravity (g) (basic)

To understand the universe, we must first distinguish between two terms often confused in daily life: mass and weight. Mass is the fundamental property of an object representing the actual quantity of matter it contains (Science, Class VIII, Chapter 9, p.142). It is constant; whether you are on Earth, the Moon, or floating in deep space, your mass remains the same. Weight, however, is a measure of the gravitational force exerted on that mass by a celestial body (Science, Class VIII, Chapter 5, p.75). While mass is measured in kilograms (kg), weight is a force measured in Newtons (N).

The bridge between mass and weight is acceleration due to gravity (g). The weight of an object is calculated by the formula Weight = mass × g. On any celestial body, the value of 'g' is determined by its own mass (M) and its radius (R) using the formula g = GM/R², where G is the Universal Gravitational Constant. This explains why your weight changes depending on where you are: the Moon is less massive than Earth, so its 'g' is only about 1/6th of Earth's, making you feel much lighter there even though your body's matter (mass) hasn't changed.

Feature	Mass	Weight
Definition	Quantity of matter in an object.	Gravitational pull exerted on an object.
Nature	Intrinsic property (Constant everywhere).	Variable (Depends on the local gravity).
SI Unit	Kilogram (kg)	Newton (N)
Measurement	Two-pan balance.	Spring balance or digital scale.

Even on Earth, 'g' is not perfectly uniform. Because the Earth is an oblate spheroid—slightly flattened at the poles and bulging at the equator—the distance from the center (R) is smaller at the poles. Consequently, gravity is greater at the poles and less at the equator (Fundamentals of Physical Geography, Class XI, Chapter 2, p.19). Furthermore, the uneven distribution of mass within the Earth's crust causes slight local variations known as gravity anomalies. These variations are crucial for geographers because gravity is the primary force driving geomorphic processes like erosion and the movement of surface materials (Fundamentals of Physical Geography, Class XI, Chapter 6, p.38).

Sources: Science, Class VIII (NCERT 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.142; Science, Class VIII (NCERT 2025), Chapter 5: Exploring Forces, p.75; Fundamentals of Physical Geography, Class XI (NCERT 2025), Chapter 2: The Origin and Evolution of the Earth, p.19; Fundamentals of Physical Geography, Class XI (NCERT 2025), Chapter 6: Geomorphic Processes, p.38

3. Variations in Gravity on Earth (intermediate)

To understand why gravity isn't the same everywhere on Earth, we must start with the fundamental formula for acceleration due to gravity: g = GM/R². In this equation, G is a universal constant, M is the mass of the planet, and R is the distance from the center. This tells us two critical things: gravity is directly proportional to the mass of the body and inversely proportional to the square of the distance from its center. On a perfectly uniform, static sphere, gravity would be identical everywhere. However, Earth is neither perfectly spherical nor static. There are three primary reasons for variations in gravity on Earth:

• Shape (Latitudinal Variation): Earth is an oblate spheroid—it bulges at the Equator and is flattened at the Poles. Because the polar radius is shorter than the equatorial radius, a person at the North Pole is closer to Earth's center of mass than someone at the Equator. Consequently, gravity is strongest at the Poles and weakest at the Equator.
• Rotation: As the Earth rotates on its axis Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.251, it generates an outward centrifugal force. This force is maximum at the Equator and zero at the Poles. This outward push partially counteracts the inward pull of gravity, further reducing the effective value of 'g' at the Equator.
• Altitude and Depth: As you climb a mountain or fly in an airplane, your distance (R) from the Earth's center increases, causing gravity to decrease Science, Class VIII, NCERT, Chapter 5, p.78. Conversely, as you go deep into a mine, although you are getting closer to the center, the mass of the Earth "above" you starts pulling you upward, which actually reduces the net gravitational pull toward the center.

When we compare Earth to other celestial bodies, these principles explain the dramatic differences we see. For instance, the Moon’s gravity is only about 1/6th of Earth's because the Moon has significantly less mass (approximately 1/100th of Earth's mass) and a smaller radius Physical Geography by PMF IAS, The Solar System, p.28.

Factor	Change in Condition	Effect on Gravity (g)
Latitude	Moving from Equator to Pole	Increases (Radius decreases)
Altitude	Increasing height above sea level	Decreases (Distance increases)
Rotation	Faster rotation speed	Decreases (at Equator)

Sources: Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.251; Science, Class VIII, NCERT, Chapter 5: Exploring Forces, p.78; Physical Geography by PMF IAS, The Solar System, p.28

4. Escape Velocity and Planetary Atmospheres (intermediate)

To understand why some planets are lush with air while others are barren rocks, we must look at the Escape Velocity. This is the minimum speed an object (like a rocket or a gas molecule) must reach to break free from a planet's gravitational pull and never fall back. If an object's speed is lower than this threshold, gravity eventually pulls it back; if it is higher, the object escapes into deep space Physical Geography by PMF IAS, The Solar System, p.39.

Whether a planet can hold onto an atmosphere depends on a delicate 'tug-of-war' between two factors:

• Gravitational Grip: Larger, denser planets have higher escape velocities. For example, Earth's escape velocity is about 11.2 km/s, whereas the Moon's is only about 2.4 km/s.
• Thermal Agitation: Gas molecules are constantly moving. The hotter the gas, the faster the molecules zip around. If the average speed of these molecules (determined by temperature and molecular mass) exceeds the planet's escape velocity, the atmosphere literally leaks away into space.

This is why the Moon has no atmosphere. Because of its low mass and small radius, its escape velocity is so low that even at moderate temperatures, gas molecules gain enough energy to fly away. On Earth, while we have a strong grip, we still lose light gases like Hydrogen and Helium from the outermost layer (the exosphere) because they move fast enough to reach escape velocity Physical Geography by PMF IAS, Earths Atmosphere, p.280. Furthermore, external factors like Solar Winds can strip away atmospheric particles unless a planet has a strong Magnetic Field to act as a shield Physical Geography by PMF IAS, Earths Atmosphere, p.280.

Body	Escape Velocity	Atmospheric Status
Earth	~11.2 km/s	Thick, life-sustaining atmosphere.
Moon	~2.4 km/s	Virtually no atmosphere (gas escapes easily).
Jupiter	~59.5 km/s	Massive gravity; retains even the lightest gases.

Sources: Physical Geography by PMF IAS, Earths Atmosphere, p.280; Physical Geography by PMF IAS, The Solar System, p.39

5. Kepler's Laws and Orbital Motion (intermediate)

To understand how celestial bodies move, we must look at the groundwork laid by Johannes Kepler. Before Kepler, it was widely believed that planets moved in perfect circles. However, his First Law (The Law of Orbits) revealed that planets move in elliptical orbits, with the Sun situated at one of the two "foci" of the ellipse Physical Geography by PMF IAS, The Solar System, p.21. This means the distance between a planet and the Sun is constantly changing throughout its "year."

His Second Law (The Law of Areas) is perhaps the most fascinating for geography students because it explains why our seasons aren't equal in length. It states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. For this to happen, the planet must move faster when it is closer to the Sun (perihelion) and slower when it is further away (aphelion) Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.257. This is why, in the Northern Hemisphere, summer is actually about 92 days long while winter is only about 89 days; the Earth is further from the Sun during the northern summer, moving slower in its orbit and taking more time to cover that arc of the ellipse Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.256.

Finally, the Third Law (The Law of Periods) establishes a mathematical harmony between a planet's distance from the Sun and its orbital period. It states that the square of the orbital period (T) is proportional to the cube of the semi-major axis (a) of its orbit (T² ∝ a³) Physical Geography by PMF IAS, The Solar System, p.21. Simply put, the further a planet is from the Sun, the significantly longer its year will be, not just because it has a longer path to travel, but because its orbital velocity is fundamentally slower.

Kepler's Law	Common Name	Key Takeaway
First Law	Law of Orbits	Orbits are ellipses, not circles; the Sun is at one focus.
Second Law	Law of Areas	Planets speed up when close to the Sun and slow down when far away.
Third Law	Law of Periods	The further a planet is from the Sun, the longer its orbital period.

Sources: Physical Geography by PMF IAS, The Solar System, p.21; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.256-257

6. Comparing Gravity and Mass across Celestial Bodies (intermediate)

To understand how celestial bodies exert force, we must look at Surface Gravity (g). This isn't just a fixed number; it is a derived value based on two fundamental properties of a planet: its mass (M) and its radius (R). The relationship is governed by the formula g = GM/R², where G is the Universal Gravitational Constant. This formula tells us that gravity is directly proportional to mass (more mass means a stronger pull) but inversely proportional to the square of the radius. This means if you move twice as far from the center of a planet, the gravity doesn't just halve; it drops to one-fourth Physical Geography by PMF IAS, Earths Interior, p.58. When we compare celestial bodies, we see massive variations. For instance, the Sun’s surface gravity is a staggering 274 m/s², which is about 28 times stronger than Earth's, primarily because its mass is so immense Physical Geography by PMF IAS, The Solar System, p.23. Conversely, the Moon is much smaller and less dense, resulting in a surface gravity of about 1.62 m/s², roughly 1/6th of Earth’s 9.8 m/s². It is also important to remember that gravity is not perfectly uniform across a planet's surface. Variations in the density of materials within the crust lead to what scientists call gravity anomalies, which help us map the internal distribution of mass Physical Geography by PMF IAS, Earths Interior, p.58. By rearranging the gravity formula, we can compare the total mass of two bodies if we know their gravity and size. If we want to find the ratio of Earth's mass (Me) to the Moon's mass (Mm), we use the ratio of their gravitational pulls and the square of their radii: Me/Mm = (ge / gm) × (Re / Rm)². Since Earth's gravity is ~6 times that of the Moon and its radius is ~4 times larger, the mass ratio works out to 6 × 4², or roughly 96. This explains why Earth is nearly 100 times more massive than its satellite despite being only 4 times wider.

Celestial Body	Surface Gravity (m/s²)	Comparison to Earth
Sun	274	~28x Earth
Earth	9.8	1x (Baseline)
Moon	1.62	~1/6th Earth

Sources: Physical Geography by PMF IAS, Earths Interior, p.58; Physical Geography by PMF IAS, The Solar System, p.23

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental relationship between mass, radius, and gravity, you can see how the formula g = GM/R² acts as the mathematical bridge for this question. This PYQ tests your ability to manipulate variables within a proportional relationship rather than just memorizing facts. As highlighted in Science, Class VIII, NCERT, gravity is not a fixed constant but a result of a celestial body's physical properties. By rearranging the formula to isolate Mass (M), we find that M ∝ g × R², meaning the mass is directly proportional to the gravity and the square of the radius.

To arrive at the correct answer, we must compare the ratios. Since Earth's gravity (ge) is 6 times that of the Moon (gm) and its radius (Re) is 4 times that of the Moon (Rm), we set up the ratio as 6 × (4)². Reasoning through the calculation, 4 squared is 16, and 6 multiplied by 16 gives us 96. In the context of UPSC's approximation-style questions, 96 is closest to (B) 100. This logical deduction is a core skill for tackling science-based geography questions, as discussed in Physical Geography by PMF IAS.

UPSC often includes "trap" options to catch students who rush their reasoning. Option (A) might attract someone who forgets to square the radius (simply calculating 6 × 4 = 24), while options (C) and (D) are designed to catch order-of-magnitude errors or students who mistakenly use the formula for volume (R³) instead of the gravitational formula. By carefully applying the inverse-square law logic inherent in the gravity formula, you can confidently identify (B) 100 as the only mathematically sound conclusion.