If radius of the earth were to shrink by 1%, its mass remaining the same, g would decrease by nearly

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

At its heart, Newton’s Universal Law of Gravitation explains the invisible 'glue' that holds our universe together. Sir Isaac Newton proposed that every particle of matter 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 centers. This theory represented the climax of the scientific revolution, moving humanity from mere observation to mathematical precision Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119. Unlike magnetic or electrostatic forces, which can push objects away, gravitational force is always attractive and acts as a non-contact force—meaning it pulls on you even without touching you Science, Class VIII, Exploring Forces, p.72. To understand how this force works in practice, we use the formula F = G(m₁m₂)/r². Here, m₁ and m₂ are the masses of the two objects, r is the distance between them, and G is the Universal Gravitational Constant. The force is measured in Newtons (N) Science, Class VIII, Exploring Forces, p.65. A crucial takeaway from this formula is the Inverse Square Law: if you double the distance between two objects, the gravitational pull doesn't just halve—it drops to one-fourth (1/2²) of its original strength. This relationship is why even a small change in Earth's radius or your distance from its center can significantly impact the weight you feel. While we often think of gravity as just 'what makes things fall,' it is also a vital tool for modern cosmology. Scientists now use gravitational waves—ripples in the fabric of space-time—to calculate the Hubble constant, which tells us how fast the universe is expanding Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.5. In our daily lives, we experience this force as gravity, the specific pull the Earth exerts on objects toward its center Science, Class VIII, Exploring Forces, p.72.

Feature	Gravitational Constant (G)	Acceleration due to gravity (g)
Nature	A scalar constant; same everywhere in the universe.	A vector quantity; changes based on location (like altitude or planet).
Role in Formula	The proportionality link in F = G(m₁m₂)/r².	The result of Earth's pull on a mass (g = GM/R²).

Sources: Themes in world history, History Class XI (NCERT 2025 ed.), Changing Cultural Traditions, p.119; Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.72; Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.65; Physical Geography by PMF IAS, The Universe, The Big Bang Theory, Galaxies & Stellar Evolution, p.5

2. Understanding Acceleration due to Gravity (g) (basic)

At its heart, acceleration due to gravity (g) is the rate at which an object speeds up as it falls toward a massive body like Earth. While we often treat it as a constant (9.8 m/s²), it is actually a variable determined by the physical characteristics of the planet. This relationship is defined by the formula: g = GM/R², where G is the universal gravitational constant, M is the mass of the body, and R is the distance from its center to the surface. This formula reveals a critical inverse square law: if you move further from the center of the Earth, the pull of gravity weakens significantly.

Because Earth is not a perfect sphere but an oblate spheroid (bulging at the middle due to its rotation), the distance from the center to the surface is not uniform. The radius at the equator is greater than the radius at the poles. Consequently, the value of g is greater near the poles and less at the equator because the poles are closer to the Earth's center of mass. FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19.

Beyond just the distance from the center, the distribution of mass within the Earth's crust also plays a role. If a specific region has higher density materials underground, the local gravitational pull might be slightly higher than expected. This difference between the actual measured gravity and the theoretical expected value is known as a gravity anomaly. Physical Geography by PMF IAS, Earths Interior, p.58. To put Earth's gravity in perspective, compare it with other celestial bodies below:

Body	Surface Gravity (m/s²)	Comparison to Earth
Sun	274	~28 times stronger
Earth	9.8	Baseline (1g)
Moon	1.62	~1/6th of Earth

Physical Geography by PMF IAS, The Solar System, p.23

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19; Physical Geography by PMF IAS, Earths Interior, p.58; Physical Geography by PMF IAS, The Solar System, p.23

3. Mass vs. Weight: Conceptual Distinction (basic)

To master mechanics, we must first clear a very common hurdle: the difference between mass and weight. In our daily lives, we often use these terms interchangeably—for instance, saying a bag of wheat "weighs" 10 kg—but in the world of physics, they represent two fundamentally different concepts. Understanding this distinction is crucial because it explains why your physical body remains the same everywhere in the universe, even if a weighing scale says otherwise.

Mass is the measure of the quantity of matter present in an object Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.141. Think of it as the total number of atoms and molecules that make you, "you." Because the amount of matter doesn't change based on your location, your mass remains constant whether you are on Earth, the Moon, or floating in deep space. The standard unit for mass is the kilogram (kg).

Weight, on the other hand, is not an inherent property of the object itself; it is a force. Specifically, it is the gravitational force with which a planet or celestial body pulls an object toward its center Science Class VIII, Exploring Forces, p.75. Because gravity varies depending on the mass and size of the planet you are standing on, your weight changes depending on your location. Since weight is a force, its proper scientific unit is the Newton (N), though many commercial scales are calibrated to display kilograms for convenience Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.142.

Feature	Mass	Weight
Definition	Quantity of matter in an object.	Gravitational pull exerted on an object.
Nature	Intrinsic property (does not change).	Extrinsic property (changes with gravity).
SI Unit	Kilogram (kg)	Newton (N)
Measurement	Measured using a physical balance.	Measured using a spring balance or digital scale.

An easy way to visualize this is by looking at different worlds. An object with a mass of 1 kg will always have a mass of 1 kg. However, its weight will be approximately 10 N on Earth, but only 1.6 N on the Moon because the Moon's gravity is much weaker Science Class VIII, Exploring Forces, p.75. This demonstrates that weight is a product of mass and the local acceleration due to gravity (W = m × g).

Sources: Science Class VIII (NCERT 2025), The Amazing World of Solutes, Solvents, and Solutions, p.141; Science Class VIII (NCERT 2025), Exploring Forces, p.75; Science Class VIII (NCERT 2025), The Amazing World of Solutes, Solvents, and Solutions, p.142; Science Class VIII (NCERT 2025), Exploring Forces, p.77

4. Variation of 'g' with Altitude and Depth (intermediate)

To understand why the weight of an object changes depending on where it is on Earth, we must first look at the formula for acceleration due to gravity (g). At the Earth's surface, this is expressed as g = GM/R², where G is the universal gravitational constant, M is the mass of the Earth, and R is the distance from the center of the Earth to the surface. This relationship follows the inverse square law: as the distance from the center increases, the pull of gravity decreases sharply.

When we move away from the Earth's surface (altitude), the distance from the center increases, which causes g to decrease. Conversely, as we go deep into the Earth (depth), you might think gravity would increase because we are getting closer to the center. However, the mass above you no longer contributes to the downward pull; only the mass of the smaller sphere "below" you matters. Consequently, gravity also decreases with depth, reaching zero at the Earth's center.

An interesting application of this concept occurs when we consider Earth's shape. Because the Earth is an oblate spheroid (bulging at the equator and flattened at the poles), the radius at the poles is smaller than at the equator. This variation in radius directly impacts gravitational force Physical Geography by PMF IAS, Latitudes and Longitudes, p. 241. Using a mathematical approximation for small changes (differentials), we find that if the radius R decreases by 1% while the mass remains constant, the value of g actually increases by approximately 2% (calculated as Δg/g = -2 × ΔR/R). This sensitivity shows that even a small "shrinking" of the Earth would make every object on the surface feel significantly heavier.

Scenario	Change in 'g'	Reason
Increasing Altitude	Decreases	Increase in distance from the center (R + h).
Increasing Depth	Decreases	Reduction in the "effective" mass of Earth pulling you.
Shrinking Radius (Mass constant)	Increases	Inverse relationship; smaller R leads to higher g.

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.241

5. Impact of Earth's Shape on Gravity (intermediate)

To understand how Earth's shape affects gravity, we must first move past the idea that Earth is a perfect sphere. In reality, our planet is an oblate spheroid or, more accurately, a geoid. This shape is a direct result of Earth's rotation. As the Earth spins, centrifugal force (which is strongest at the equator) pushes the mass outward, creating an equatorial bulge and flattening the areas around the North and South Poles Physical Geography by PMF IAS, Latitudes and Longitudes, p.241. Consequently, the distance from the center of the Earth to the surface (the radius, R) is greater at the equator than it is at the poles.

This physical variation in radius has a direct impact on the acceleration due to gravity (g). According to Newton's Law of Universal Gravitation, the formula for gravity at the surface is g = GM/R², where G is the gravitational constant and M is the mass of the Earth. Notice that R is in the denominator and squared; this follows the inverse square law. This means that gravity is inversely proportional to the square of the radius. Therefore, as the distance from the center (R) increases, the pull of gravity (g) decreases significantly.

Region	Radius (R)	Gravitational Pull (g)	Reasoning
Equator	Larger (Bulged)	Lower	Further from the Earth's center of mass.
Poles	Smaller (Flattened)	Higher	Closer to the Earth's center of mass.

Interestingly, because of the square in the formula (R²), gravity is very sensitive to even small changes in the Earth's radius. For instance, using a mathematical approximation (differentiation), we find that a 1% decrease in the radius leads to approximately a 2% increase in gravitational acceleration. This sensitivity explains why weight varies slightly depending on your latitude: you would actually weigh slightly more at the North Pole than you would at the Equator, even though your mass remains exactly the same Physical Geography by PMF IAS, Latitudes and Longitudes, p.241.

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.241

6. Effect of Earth's Rotation on 'g' (intermediate)

To understand how the Earth's rotation affects the acceleration due to gravity (g), we must look at two distinct physical phenomena: the shape of the Earth and the centrifugal force generated by its spin. While we often treat Earth as a perfect sphere, it is actually an oblate spheroid — it bulges at the equator and flattens at the poles because of its rotation Physical Geography by PMF IAS, Chapter 18, p.241. This means that if you are standing at the North Pole, you are physically closer to the Earth's center of mass than if you were standing at the Equator. Since gravity follows the inverse square law (g = GM/R²), a smaller radius (R) at the poles results in a stronger gravitational pull.

The second factor is the Centrifugal Force. As the Earth rotates from west to east Physical Geography by PMF IAS, Chapter 18, p.251, every object on its surface experiences an outward-acting force. This force is most intense where the rotational speed is highest — at the Equator — and vanishes entirely at the Poles where the rotational radius is zero. Because this centrifugal force acts in the opposite direction to gravity, it effectively 'cancels out' a small portion of the gravitational pull Physical Geography by PMF IAS, Chapter 18, p.241.

The combined result of these two factors is that the effective value of g is maximum at the poles and minimum at the equator. This variation is significant enough that objects actually weigh slightly less at the equator than they do at the poles. This principle is also why space agencies prefer to launch rockets from sites closer to the equator, as the lower effective gravity and the 'boost' from the Earth's rotational speed help save fuel.

Factor	Effect at Equator	Effect at Poles
Distance from Center	Greater (Lower gravity)	Smaller (Higher gravity)
Centrifugal Force	Maximum (Reduces weight)	Zero (No effect)
Value of 'g'	Minimum (~9.78 m/s²)	Maximum (~9.83 m/s²)

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.241; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.251

7. Proportionality and Inverse Square Relationships (exam-level)

In physics and economics, proportionality defines how two variables change in relation to one another. A direct proportionality means that as one factor increases, the other increases at a constant rate. For instance, in production, Constant Returns to Scale (CRS) occurs when doubling all inputs leads to exactly double the output Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.42. However, when variables move in opposite directions—one goes up while the other goes down—we call it an inverse relationship. A classic example is found in macroeconomics, where the ratio of money balance to the value of transactions represents an inverse relationship Macroeconomics (NCERT class XII 2025 ed.), Money and Banking, p.44.

A more powerful version of this is the Inverse Square Law, which is fundamental to understanding gravity. The acceleration due to gravity (g) is determined by the formula g = GM/R². Here, g is inversely proportional to the square of the radius (R). This means gravity doesn't just get weaker as you move away; it gets weaker much faster than the distance increases. Because Earth is not a perfect sphere but an oblate spheroid, the radius is shorter at the poles than at the equator, which causes variations in gravitational pull across the planet's surface Physical Geography by PMF IAS, Chapter 18: Latitudes and Longitudes, p.241.

For the UPSC exam, you should master the Differential Approximation for small percentage changes. If a variable follows an inverse square relationship (y ∝ 1/x²), a small percentage change in x results in a change in y that is approximately negative two times that percentage. For example, if the radius (R) of a planet were to shrink by 1%, gravity (g) would not just increase by 1%—it would increase by approximately 2%. This "double the change" rule is a vital shortcut for solving mechanics problems quickly without complex calculators.

Sources: Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.42; Macroeconomics (NCERT class XII 2025 ed.), Money and Banking, p.44; Physical Geography by PMF IAS, Chapter 18: Latitudes and Longitudes, p.241

8. Approximation Method for Small Percentage Changes (exam-level)

In physics and economics, we often need to calculate how a small change in one variable affects another. When these changes are very small (typically less than 5%), we can skip complex calculations and use a powerful differential approximation. This method is rooted in the power rule of calculus: if a variable y depends on x raised to some power n (y = xⁿ), then the percentage change in y is approximately n times the percentage change in x.

Consider the acceleration due to gravity (g), which follows the formula g = GM/R². Here, G is the gravitational constant and M is the Earth's mass. If we assume mass is constant, gravity is inversely proportional to the square of the radius (g ∝ R⁻²). Applying our approximation rule, the exponent is -2. Therefore, the percentage change in g is approximately -2 times the percentage change in R. This mathematical shortcut is incredibly useful for understanding how Earth's slightly elliptical shape—being flatter at the poles and bulging at the equator—creates variations in gravitational pull Physical Geography by PMF IAS, Chapter 18, p. 241.

To visualize how this works in practice, look at the table below. Notice how the "power" or exponent dictates the multiplier for the change:

Relationship	Change in Independent Variable	Approximate Change in Result
Area of a square (A = s²)	Side increases by 1%	Area increases by 2%
Gravity (g ∝ R⁻²)	Radius decreases by 1%	Gravity increases by 2%
Volume of a sphere (V ∝ r³)	Radius increases by 1%	Volume increases by 3%

This principle of proportional shifts is also mirrored in social sciences. For instance, in economics, the relationship between price and quantity demanded determines whether total expenditure increases or decreases. If the percentage change in quantity exactly offsets the percentage change in price, the total expenditure remains unchanged Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p. 32. Mastering these small-change approximations allows you to make quick, accurate estimations during your exams without getting bogged down in heavy arithmetic.

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.241; Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.32

9. Solving the Original PYQ (exam-level)

You have just mastered the fundamental relationship between mass, distance, and gravity. This question tests your ability to apply the inverse square law derived from the formula g = GM/R². In geography and physics, understanding how Earth's dimensions influence gravity is crucial, as discussed in Physical Geography by PMF IAS. Here, the building blocks of gravitational constants and proportionality converge: if mass (M) is constant, then gravity (g) depends solely on the square of the radius (R).

To solve this, think like a coach using differential approximation for small changes. Since g ∝ R⁻², a small percentage change in radius results in a change in gravity equal to the exponent multiplied by that percentage change. If the radius shrinks by 1% (a change of -1%), the calculation becomes -2 × (-1%) = +2%. While the question uses the word "decrease," the numerical magnitude of the shift is (B) 2%. Note that scientifically, gravity actually increases as the radius shrinks because you are closer to the center of mass, but UPSC often focuses on the absolute magnitude of the change.

UPSC designs distractors to catch specific errors in your logic. Option (A) 1% is a "linear trap" for students who forget that the radius is squared in the denominator; they see a 1% change in R and assume a 1% change in g. Options (C) 3% and (D) 4% target students who might confuse the formula with volume calculations (R³) or simply guess higher magnitudes without applying the power rule. Always remember: in an inverse square relationship, the percentage change doubles for small variations.