A solid sphere, a disc and a ring of the same mass and radius are allowed to roll down a frictionless inclined plane simultaneously from the same height. In this context, which one of the following statements is correct?

1. Newton’s Laws and Gravitational Acceleration (basic)

To understand how objects move, we must first look at Newton’s Second Law of Motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration (F = ma). In simpler terms, force is what causes an object to change its state of motion. When an object moves along a straight path, such as a train moving between two stations, it is undergoing linear motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116. The standard unit we use to measure this push or pull is the newton (N) Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.65.

One of the most universal forces we experience is gravity. While we often think of gravity as a constant force, it actually varies slightly depending on where you are on Earth. Because the Earth is not a perfect sphere—it bulges at the equator—the distance from the center of the Earth to the surface is greater at the equator than at the poles. Consequently, the gravitational force is stronger at the poles and slightly weaker at the equator FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19. Geologists even use these variations, known as gravity anomalies, to map out the uneven distribution of mass within the Earth's crust Physical Geography by PMF IAS, Earths Interior, p.58.

When gravity is the only force acting on a falling object (in a vacuum), it produces a specific acceleration called gravitational acceleration (g). On Earth, this value is approximately 9.8 m/s². It is crucial to distinguish between an object's mass (the amount of matter it contains) and its weight (the force exerted on it by gravity). While your mass stays the same whether you are on the Moon or the Earth, your weight changes because the gravitational pull is different.

Feature	Mass (m)	Weight (W)
Definition	The quantity of matter in an object.	The force of gravity acting on an object.
SI Unit	Kilogram (kg)	Newton (N)
Variability	Constant everywhere.	Changes based on the local value of gravity (g).

Sources: Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.65; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116; 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

2. Motion on an Inclined Plane (basic)

To understand motion on an inclined plane, we must first look at how gravity behaves on a slope. Unlike a free-falling object that drops straight down, an object on an incline is restricted by the surface. Gravity still pulls the object vertically toward the center of the Earth, but the surface of the incline pushes back. We resolve the force of gravity into two components: one that presses the object against the surface and another that pulls it down along the slope. On a frictionless surface, the acceleration down the plane is determined solely by the angle of the incline (θ) and the acceleration due to gravity (g), expressed as a = g sinθ.

A critical distinction in mechanics is between sliding and rolling. For an object like a ball or a cylinder to 'roll' without slipping, there must be a force of friction acting at the point of contact to provide 'torque' (the twisting force that causes rotation). In a perfectly frictionless environment, there is no torque. Therefore, whether you place a block, a sphere, or a ring on a frictionless incline, they will all simply slide down like a block of ice. Because the mass (m) appears on both sides of the force equation (F = ma and F = mg sinθ), it cancels out, meaning the acceleration is identical for all objects regardless of their mass or shape.

While we often treat gravity as a constant 9.8 m/s², it is worth noting that the actual value of g can vary slightly depending on your location on Earth. For instance, gravity is stronger at the poles than at the equator because the poles are closer to the Earth's center FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19. Additionally, the uneven distribution of mass within the Earth's crust creates 'gravity anomalies' Physical Geography by PMF IAS, Earths Interior, p.58. However, for most basic mechanics problems, we assume a uniform gravitational pull acting on our inclined plane.

Scenario	Primary Force	Resulting Motion
Frictionless Incline	Component of Gravity (g sinθ)	Pure Sliding (All shapes move together)
Incline with Friction	Gravity vs. Friction	Rolling or Sliding (Shape/Mass distribution matters)

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; Science Class VIII NCERT (Revised ed 2025), Exploring Forces, p.78

3. Types of Friction: Static and Kinetic (basic)

Imagine trying to push a heavy wooden crate across a concrete floor. Initially, no matter how hard you push, the crate doesn't budge. This resistance is due to friction, a contact force that arises because no surface is perfectly smooth. At a microscopic level, every surface has tiny peaks and valleys called irregularities. When two surfaces touch, these irregularities interlock like the teeth of two combs, opposing any effort to slide one over the other Science, Class VIII, Exploring Forces, p.68. This force always acts in the direction opposite to the intended or actual motion Science, Class VIII, Exploring Forces, p.68.

We categorize this resistance into two primary types based on whether the object is moving or still. Static Friction is the force that acts on an object when it is stationary. It is a "smart" force—it matches the strength of your push exactly to keep the object at rest, up to a certain maximum limit. Once your push exceeds this limit, the interlocking "teeth" break free, and the object begins to move. At this point, Kinetic (or Sliding) Friction takes over. Because the surfaces are already in relative motion, the irregularities do not have enough time to lock back into each other as deeply as they did when the object was still. Consequently, kinetic friction is usually slightly weaker than the maximum static friction.

Feature	Static Friction	Kinetic Friction
State of Object	Stationary (trying to move)	In motion (sliding)
Magnitude	Variable (self-adjusting)	Constant (for a given pair of surfaces)
Interlocking	Deep and firm	Surface bumps "glide" over each other

This concept is not just limited to boxes on floors; it applies to nature as well. For instance, the irregularities of the Earth's surface resist wind movement, creating friction that slows down winds and changes their direction. This effect is strongest near the ground (up to 1-3 km) and is much lower over smooth surfaces like the sea Physical Geography by PMF IAS, Pressure Systems and Wind System, p.307.

Sources: Science, Class VIII, Exploring Forces, p.68; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.307

4. Law of Conservation of Energy (intermediate)

The Law of Conservation of Energy is a fundamental pillar of physics stating that energy can neither be created nor destroyed; it can only be transformed from one form to another. In any isolated system, the total amount of energy remains constant over time. For a UPSC aspirant, understanding this principle is crucial because it governs everything from the movement of tectonic plates to the efficiency of renewable energy sources. When we look at the Earth's surface, exogenic processes (like weathering and erosion) derive their energy from the sun and the gravitational gradients created by tectonic factors FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Chapter 5, p.39.

In classical mechanics, we often focus on Mechanical Energy, which is the sum of Kinetic Energy (KE)—the energy of motion—and Potential Energy (PE)—the energy stored due to position. As an object moves, these two forms are constantly interchanging. For instance, a rock perched on a steep slope possesses high gravitational potential energy. If it begins to slide downslope as a landslide or earthflow, that potential energy is converted into kinetic energy Physical Geography by PMF IAS, Geomorphic Movements, p.87. Similarly, human technology harnesses these transformations; for example, wind turbines convert the kinetic energy of blowing wind into electrical energy INDIA PEOPLE AND ECONOMY, Geography Class XII (NCERT 2025 ed.), Chapter 4, p.61.

In an idealized environment without friction (non-dissipative forces), the conversion between PE and KE is perfectly efficient. However, in the real world, some energy is often 'lost' to the environment as heat or sound due to friction or air resistance. Even then, the law holds true: the energy isn't gone; it has simply transitioned into a less useful thermal form. This is why energy conservation is a policy priority; while the total energy in the universe is constant, the 'available' or 'useful' energy (like fossil fuels) is exhaustible and needs careful management Geography of India, Majid Husain, Chapter 30, p.31.

Type of Energy	Definition	Example in Geography
Potential	Energy due to position/height	Water stored behind a dam or a rock on a cliff.
Kinetic	Energy due to motion	Flowing river water or blowing wind.
Thermal	Energy due to friction/heat	Heat generated during the shearing of rocks.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Chapter 5: Geomorphic Processes, p.39; Physical Geography by PMF IAS, Geomorphic Movements, p.87; INDIA PEOPLE AND ECONOMY, Geography Class XII (NCERT 2025 ed.), Chapter 4: Mineral and Energy Resources, p.61; Geography of India, Majid Husain, Chapter 30: Energy Resources, p.31

5. Rigid Body Dynamics: Moment of Inertia (intermediate)

In our previous steps, we discussed how mass is a measure of inertia—the resistance an object offers to any change in its state of linear motion. However, when we talk about rotation, mass alone isn't enough. We need a new concept: the Moment of Inertia (I). While mass tells us how hard it is to push an object, the Moment of Inertia tells us how hard it is to rotate it. It depends not just on how much matter an object has, but crucially on how that mass is distributed relative to the axis of rotation.

Mathematically, for a collection of particles, the Moment of Inertia is the sum of the mass of each particle multiplied by the square of its distance from the axis (I = Σ mr²). This means that mass located further from the center has a much greater impact on rotational resistance than mass located near the center. We see this principle even in planetary science; for instance, the distribution of mass within the Earth's crust influences its gravitational pull, a phenomenon known as a gravity anomaly Physical Geography by PMF IAS, Earths Interior, p.58. Similarly, while the Sun contains 99.8% of the Solar System's mass, it holds very little of its angular momentum because of the way its mass and rotation are concentrated Physical Geography by PMF IAS, The Solar System, p.23.

Feature	Linear Motion	Rotational Motion
Measure of Inertia	Mass (m)	Moment of Inertia (I)
Cause of Motion	Force (F)	Torque (τ)
Basic Law	F = ma	τ = Iα (Alpha is angular acceleration)

A fascinating application of this concept occurs on inclined planes. If you release a solid sphere, a disc, and a ring down a ramp with friction, they will reach the bottom at different times because their different Moments of Inertia cause them to resist rotation differently. However, if the surface is frictionless, there is no torque to start the rotation in the first place. In such a case, the objects do not roll; they simply slide down the incline like a block of ice Physical Geography by PMF IAS, Geomorphic Movements, p.89. Without rotation, the Moment of Inertia becomes irrelevant to the speed of the descent, and all objects would accelerate at the same rate (g sinθ).

Sources: Physical Geography by PMF IAS, Earths Interior, p.58; Physical Geography by PMF IAS, The Solar System, p.23; Physical Geography by PMF IAS, Geomorphic Movements, p.89

6. Torque and the Mechanics of Rolling (exam-level)

To understand why objects move the way they do on a slope, we must first distinguish between translation (moving from point A to point B) and rotation (spinning around an axis). In the context of a round object on an incline, 'rolling' is actually a combination of both. However, for an object to start spinning, it requires Torque (τ). Torque is the rotational equivalent of force; while a force causes linear acceleration, torque causes angular acceleration.

On an inclined plane, the force of gravity pulls an object downward through its center of mass. But gravity alone cannot make an object rotate because it doesn't provide torque relative to the center. For rotation to occur, there must be a tangential force acting at the point of contact between the object and the surface. This force is Static Friction. As explained in Science, Class VIII . NCERT, Exploring Forces, p.68, friction arises from the interlocking of microscopic irregularities between two surfaces. In a 'frictionless' scenario, these irregularities are absent, meaning there is no force to 'grip' the surface and create the necessary torque. Without torque, the object cannot rotate; it simply slides down the incline.

When objects slide without rotation, their internal mass distribution (whether they are hollow like a ring or solid like a sphere) becomes irrelevant to their speed. In a frictionless environment, the only force acting along the slope is the component of gravity (mg sinθ). Therefore, every object—regardless of its shape—undergoes the same linear acceleration: a = g sinθ. This is why in geomorphic processes like a 'debris slide,' materials can move rapidly down a slope without backward rotation if the conditions favor sliding over rolling FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Geomorphic Processes, p.42.

Condition	Primary Motion	Cause of Rotation	Acceleration (a)
With Friction	Rolling without slipping	Static friction provides torque	Depends on shape (Moment of Inertia)
Frictionless	Pure Sliding	None (Zero Torque)	Identical for all shapes (g sinθ)

Sources: Science, Class VIII . NCERT, Exploring Forces, p.68; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Geomorphic Processes, p.42

7. Sliding vs. Rolling: The Frictionless Constraint (exam-level)

In mechanics, the movement of an object down an incline is usually a contest between sliding and rolling. While we often imagine a ball rolling down a hill, physics tells us that rolling is not a 'default' state—it is a result of a specific interaction called friction. As noted in Science Class VIII, Exploring Forces, p.68, friction is the force that acts between surfaces in contact, opposing relative motion. In the context of a slope, static friction at the point of contact provides the torque (turning force) necessary to make an object rotate. Without this 'grip,' the object cannot initiate rotation around its center of mass; it simply skids.

When we introduce the frictionless constraint, the rules of the race change entirely. On a perfectly smooth (frictionless) inclined plane, there is no tangential force to create torque. Gravity pulls the object down the slope with a force of mg sinθ, but because this force acts through the center of mass, it cannot cause the object to spin. Consequently, every object—regardless of whether it is a solid sphere, a hollow ring, or a flat disc—will fail to roll. Instead, they will all slide down the incline like a block of ice. This is similar to 'debris slides' in geomorphology, where material moves rapidly down a slope without significant backward rotation Physical Geography by PMF IAS, Geomorphic Movements, p.89.

Since there is no energy being 'spent' on rotational kinetic energy, all the potential energy is converted solely into translational kinetic energy. Because the acceleration for a sliding object on a frictionless plane is always a = g sinθ (independent of mass or shape), every object will reach the bottom at the exact same time. The moment of inertia, which usually decides the winner in a rolling race, becomes irrelevant when the 'frictionless' condition is applied.

Feature	With Friction (Rolling)	Frictionless (Sliding)
Torque	Provided by static friction.	Zero (no friction to provide grip).
Motion Type	Rotation + Translation.	Pure Translation (Sliding).
Acceleration	Depends on shape (Moment of Inertia).	Same for all (a = g sinθ).

Sources: Science Class VIII, Exploring Forces, p.68; Physical Geography by PMF IAS, Geomorphic Movements, p.89

8. Solving the Original PYQ (exam-level)

To solve this, we must synthesize two fundamental building blocks you recently mastered: Torque and Frictional Force. You’ve learned that for an object to truly "roll," it requires an external force to act at a distance from the center of mass to create torque. On a frictionless inclined plane, however, there is no static friction to provide this necessary torque. This means that despite the shapes involved, the solid sphere, disc, and ring do not actually rotate; they slide. According to the principles of classical mechanics, all objects sliding down a frictionless slope experience the same constant acceleration (a = g sinθ), regardless of their mass or internal distribution.

The correct answer is (D) All of them reach down simultaneously. The reasoning follows a precise logical chain: because the surface is frictionless, no gravitational potential energy is diverted into rotational kinetic energy. Instead, 100% of the energy is converted into translational kinetic energy. Since the acceleration is identical for all three bodies in a pure sliding scenario, they will cover the same displacement in the exact same time. This concept of gravitational force as the sole driver of motion is fundamental to understanding geomorphic processes and landslides, as noted in FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.) and Physical Geography by PMF IAS.

UPSC often uses distractor options like (B) to catch students who over-rely on memorized formulas. If the plane had friction, the solid sphere would reach first because its lower moment of inertia allows it to accelerate faster while rolling. Options (A) and (C) are similarly traps based on varying mass distributions (the ring being the slowest in a friction scenario). By using the word "roll" but specifying a "frictionless" surface, the examiner is testing your ability to identify contradictory constraints. In the world of UPSC, the absence of friction is the ultimate "game-changer" that resets the rules of the race.