If the radius of the earth were shrink by one per cent , its mass remaining the same, the value of ‘g’ on the earth’s surface would

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

Imagine that every single object in the universe—from the smartphone in your hand to the distant stars—is pulling on every other object. This invisible tug-of-war is what we call Gravity. While thinkers had wondered about this for centuries, the Scientific Revolution reached its peak when Isaac Newton formulated the Universal Law of Gravitation Themes in world history, History Class XI, p.119. He realized that the same force that makes an apple fall to the ground is responsible for keeping the Moon in its orbit around the Earth.

Newton’s law states that the force of attraction between two objects depends on two primary factors: their masses and the distance between them. We express this with the formula: F = G(m₁m₂)/r². In this equation, F represents the gravitational force, measured in Newtons (N) Science, Class VIII, p.65. The symbols m₁ and m₂ are the masses of the two objects, while r is the distance between their centers. The G is the Universal Gravitational Constant, a fixed value that applies everywhere in the cosmos.

The beauty of this law lies in its proportions. If you increase the mass of the objects, the force increases (direct proportionality). However, the relationship with distance is an "inverse-square" law. This means that if you double the distance (r becomes 2r), the force doesn't just halve—it becomes four times weaker (1/2² = 1/4). This sensitivity to distance is why we feel Earth's gravity so strongly, but the gravity of a much larger, distant star doesn't pull us off the ground.

Change in Variable	Effect on Force (F)
Mass of one object doubles	Force doubles (2F)
Distance (r) doubles	Force becomes 1/4th (F/4)
Distance (r) is halved	Force becomes 4 times stronger (4F)

Sources: Themes in world history, History Class XI, Changing Cultural Traditions, p.119; Science, Class VIII, Exploring Forces, p.65

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

When we talk about acceleration due to gravity (g), we are describing the rate at which an object speeds up as it falls toward a celestial body, like Earth. This isn't just a random number; it is determined by the fundamental physical properties of the planet. According to the law of universal gravitation, the formula for surface gravity is g = GM/R², where G is the universal gravitational constant, M is the mass of the body, and R is the distance from the center (the radius).

This formula tells us two vital things that are frequently tested in UPSC:

• Mass (M) is directly proportional: If a planet is more massive, its gravity is stronger. This is why the Sun's surface gravity is a staggering 274 m/s², which is 28 times that of Earth Physical Geography by PMF IAS, The Solar System, p.23.
• Radius (R) follows an inverse-square law: Because R is in the denominator and squared, gravity is extremely sensitive to distance. If you move further from the center, gravity drops significantly; if you get closer, it increases sharply.

On Earth, this value is not uniform everywhere. Our planet is an oblate spheroid—it bulges at the equator and is flattened at the poles. Because the distance from the center to the equator is greater than the distance to the poles, the value of g is lower at the equator and higher at the poles FUNDAMENTALS OF PHYSICAL GEOGRAPHY, The Origin and Evolution of the Earth, p.19. Furthermore, the mass inside the Earth isn't distributed perfectly; variations in local rock density can cause small differences in gravity readings, known as gravity anomalies Physical Geography by PMF IAS, Earths Interior, p.58.

Understanding the sensitivity of the inverse-square relationship is key. For example, if the mass of a planet stays the same but it shrinks (radius decreases), the value of 'g' will increase. Because the radius is squared in the denominator, a small percentage change in the radius results in roughly double that percentage change in gravity (in the opposite direction). This mathematical relationship is a favorite for conceptual questions regarding planetary physics.

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

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

To understand how gravity changes, we must first look at the force that keeps us grounded. The acceleration due to gravity (g) is determined by the formula g = GM/R². Here, 'G' is the universal gravitational constant, 'M' is the Earth's mass, and 'R' is the distance from the Earth's center. This formula reveals a critical principle: g is not a universal constant; it is a variable that depends entirely on your location. Because of the inverse-square law (the 1/R² part), even small changes in your distance from the center have a magnified effect on how much gravity you feel.

When we move away from the surface into the atmosphere—reaching the heights of peaks like Mt. Everest (8,848m) or Kanchenjunga (Contemporary India-I, Physical Features of India, p.8)—the value of g decreases. This is because your altitude increases your total distance from the Earth's center. Interestingly, gravity also decreases if you go in the opposite direction: depth. As you descend into a deep mine, the mass of the Earth "above" you no longer pulls you downward; only the shell of Earth beneath your feet exerts a net downward pull. Therefore, g decreases as you go deeper, eventually reaching zero at the Earth's center.

Finally, we must consider the Earth's actual shape. Our planet is not a perfect sphere but an oblate spheroid, bulging at the equator and flattened at the poles. As noted in Fundamentals of Physical Geography, The Origin and Evolution of the Earth, p.19, the distance from the center is greater at the equator than at the poles. This leads to a distinct pattern of variation:

Factor	Change in Location	Effect on 'g'
Altitude	Moving to higher elevations (e.g., Himalayas)	Decreases
Depth	Moving toward the Earth's core	Decreases
Latitude	Moving from Equator to Poles	Increases

Geologists also track gravity anomalies—differences between observed and expected gravity—which provide clues about the uneven distribution of mass (like dense mineral deposits) within the Earth's crust (Physical Geography by PMF IAS, Earths Interior, p.58).

Sources: Contemporary India-I NCERT, Physical Features of India, p.8; Fundamentals of Physical Geography NCERT, The Origin and Evolution of the Earth, p.19; Physical Geography by PMF IAS, Earths Interior, p.58

4. Shape of Earth and its Impact on Gravity (intermediate)

To understand how gravity varies across our planet, we must first look at the Earth's true shape. While we often imagine Earth as a perfect sphere, it is actually an oblate spheroid or Geoid. Because the Earth rotates on its axis, the centrifugal force generated is strongest at the Equator, causing the planet to bulge outward at its middle and flatten slightly at the poles Physical Geography by PMF IAS, Latitudes and Longitudes, p.241. This means that if you stand at the North Pole, you are actually closer to the Earth's center of mass than if you were standing on the Equator Exploring Society: India and Beyond, Locating Places on the Earth, p.14.

The strength of surface gravity (g) is governed by the formula g = GM/R², where G is the gravitational constant, M is the Earth's mass, and R is the radius (the distance from the center). Because the radius (R) is in the denominator and squared, gravity is inversely proportional to the square of the distance from the center. Therefore, at the poles, where the radius is smaller, the gravitational pull is stronger. Conversely, at the Equator, where the radius is larger, the gravitational pull is weaker FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.19.

The mathematical relationship 1/R² makes gravity very sensitive to changes in distance. For instance, if the Earth were to shrink such that its radius decreased by only 1% while keeping its mass constant, the surface gravity would not just increase by 1%—it would increase by approximately 2%. This is because the square of the smaller radius (0.99R)² results in a divisor of roughly 0.98, making the final value of g about 1.02 times its original strength. Additionally, gravity is influenced by the mass distribution within the crust; variations from the expected gravitational value at a specific location are known as gravity anomalies Physical Geography by PMF IAS, Earths Interior, p.58.

Location	Distance from Center (Radius)	Gravitational Force (g)
Poles	Shorter (Flattened)	Higher
Equator	Longer (Bulged)	Lower

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.241; Exploring Society: India and Beyond, Locating Places on the Earth, p.14; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, The Origin and Evolution of the Earth, p.19; Physical Geography by PMF IAS, Earths Interior, p.58

5. Escape Velocity and Orbital Mechanics (exam-level)

To understand how objects move in space or how a planet retains its atmosphere, we must look at the tug-of-war between gravity and velocity. Escape velocity is the minimum speed an object must reach to break free from a planet's gravitational pull and travel into space without ever falling back. This velocity depends on the mass of the planet and the distance from its center. For Earth, this speed is approximately 11.2 km/s. If an object (like a gas molecule or a rocket) moves slower than this but fast enough to counteract gravity's downward pull, it enters an orbit. This balance is known as orbital velocity.

This concept explains why Earth has an atmosphere while smaller bodies like the Moon do not. In the exosphere, gas molecules move very fast due to heat from the sun. Light gases such as hydrogen and helium frequently reach escape velocity and are lost to space—a process known as atmospheric escape or stripping Physical Geography by PMF IAS, Earths Atmosphere, p.280. On Earth, our magnetic field acts as a shield, preventing solar winds from accelerating these particles even further, which helps us retain our atmosphere Physical Geography by PMF IAS, Earths Atmosphere, p.280.

In terms of technology, orbital mechanics dictates where we place our satellites. Satellites in High Earth Orbit (HEO) and Mid Earth Orbit (MEO) are strategically placed within the exosphere. Why? Because the air density there is extremely low, meaning there is almost no atmospheric drag to slow the satellite down, allowing it to maintain its orbit for long periods with minimal fuel Physical Geography by PMF IAS, Earths Atmosphere, p.280. Understanding the relationship between height, gravity, and speed is the foundation of all space exploration missions, including those by ISRO.

Sources: Physical Geography by PMF IAS, Earths Atmosphere, p.280

6. Mass vs Weight: Conceptual Differences (basic)

In our daily lives, we often use the terms mass and weight as if they mean the same thing. However, in the realm of physics and for your UPSC preparation, it is vital to distinguish between them. Mass is an intrinsic property of an object; it represents the quantity of matter contained within it Science, Class VIII, NCERT, p.142. Whether you are standing on the peak of Mt. Everest, floating in the rarefied air of the thermosphere Physical Geography, PMF IAS, p.277, or sitting on the Moon, your mass remains exactly the same because the amount of "stuff" you are made of hasn't changed.

Weight, on the other hand, is not a property of the object alone—it is a force. Specifically, it is the gravitational force with which a planet (like Earth) pulls an object toward its center Science, Class VIII, NCERT, p.75. Because weight depends on gravity, it can change depending on where you are. For example, if you move to a planet with stronger gravity, you would weigh more, even though your mass is unchanged. We typically measure mass in kilograms (kg) and weight in Newtons (N) using instruments like a spring balance Science, Class VIII, NCERT, p.74.

The relationship between the two is defined by the formula: Weight = Mass × Gravity (W = mg). This means your weight is directly proportional to the local acceleration due to gravity (g). A fascinating consequence of this is that if Earth were to magically shrink in size (radius decreases) while keeping its mass constant, you would find yourself closer to the Earth's center of gravity. According to the inverse-square law, the gravitational pull (g) would actually increase. Specifically, if the Earth's radius were to decrease by 1%, the surface gravity would increase by approximately 2%, making you feel 2% heavier even though your actual mass remains identical!

Feature	Mass	Weight
Definition	Quantity of matter in an object.	Force of gravitational attraction.
Nature	Constant everywhere in the universe.	Variable; changes with location/gravity.
SI Unit	Kilogram (kg)	Newton (N)
Measurement	Measured using a beam balance.	Measured using a spring balance.

Sources: Science, Class VIII, NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.142; Science, Class VIII, NCERT, Exploring Forces, p.74-75; Physical Geography, PMF IAS, Earth's Atmosphere, p.277

7. Percentage Changes and Inverse Square Relations (intermediate)

In physics and economics alike, understanding how one variable responds to a change in another is crucial. An inverse square relation exists when one quantity is inversely proportional to the square of another, expressed as y ∝ 1/x². This is a fundamental concept in mechanics, most famously seen in Newton’s Law of Universal Gravitation and the calculation of surface gravity (g = GM/R²). While some relationships are linear, such as the relationship between a mirror's radius of curvature and its focal length (R = 2f) Science, Class X, Light – Reflection and Refraction, p.137, inverse square relationships are much more sensitive to change.

When we deal with small percentage changes in these relationships, a very useful mathematical shortcut emerges. If a variable y depends on x raised to a power n (y = xⁿ), then the percentage change in y is approximately n times the percentage change in x. In an inverse square relation, the power of the denominator is 2, which mathematically acts as an exponent of -2. Therefore, if the radius (R) of a planet changes by a small amount, the surface gravity (g) will change by approximately twice that percentage, but in the opposite direction. This is similar to how price elasticity dictates that expenditure changes based on the percentage shift in quantity versus price Microeconomics, Class XII, Theory of Consumer Behaviour, p.32.

To visualize this in a mechanics context, consider a scenario where Earth’s mass remains constant but its radius shrinks. Because gravity is governed by the inverse square of the radius, the "pull" becomes stronger as you get closer to the center of mass. If the radius decreases by 1%, the value of g does not just increase by 1%; it increases by approximately 2%. This occurs because (1/0.99)² is roughly 1.0204. Understanding these small fluctuations is vital when considering planetary orbits or even the slight variations in distance between the Earth and Sun in its oval-shaped path Science-Class VII, Earth, Moon, and the Sun, p.186.

Change in Radius (R)	Effect on Gravity (g)	Approximate Result
Decrease by 1%	g ∝ 1/(0.99R)²	Increase by ~2%
Increase by 1%	g ∝ 1/(1.01R)²	Decrease by ~2%
Decrease by 2%	g ∝ 1/(0.98R)²	Increase by ~4%

Sources: Science, Class X, Light – Reflection and Refraction, p.137; Microeconomics, Class XII, Theory of Consumer Behaviour, p.32; Science-Class VII, Earth, Moon, and the Sun, p.186

8. Solving the Original PYQ (exam-level)

This question is a perfect application of the Universal Law of Gravitation and the specific formula for acceleration due to gravity (g = GM/R²) that you just mastered. By stating that the mass remains constant, the examiner focuses your attention entirely on the inverse-square relationship between gravity and the radius. This highlights a fundamental building block in physics: when the denominator of a fraction decreases (the radius shrinks), the overall value of the fraction (gravity) must increase. Understanding this proportional logic is the first step to eliminating half of the choices immediately.

To arrive at the correct answer, (B) increase by 2%, you can use a quick mental shortcut used by top rankers: for small percentage changes, the change in the result is approximately the exponent multiplied by the percentage change of the variable. Since g is proportional to R⁻², a 1% decrease in R results in a (-2) × (-1%) = +2% change in g. Mathematically, as shown in NCERT Class 11 Physics, if the new radius is 0.99R, the new gravity becomes g/(0.99)², which calculates to approximately 1.02g, confirming a 2% rise.

The other options are classic UPSC traps designed to catch students who rush their logic. Options (C) and (D) are incorrect because they suggest a decrease; however, because of the inverse relationship, a smaller Earth would actually pull objects more strongly at its surface. Option (A) is a "calculation trap" for those who might mistakenly apply a square root or linear relationship instead of the inverse-square law. In the exam hall, always remember: if the radius shrinks, the gravity must increase, and the square power ensures that the change in gravity is double the percentage change of the radius.