A man is sitting on a rotating stool with his arms outstretched. If suddenly he folds his arms the angular velocity of the man would

1. Basics of Rotational Motion (basic)

To understand mechanics, we must distinguish between how objects move in a straight line versus how they spin. Rotational motion is the movement of an object where all its parts move in circles around an imaginary line called the axis of rotation Science-Class VII, Earth, Moon, and the Sun, p.171. For example, the Earth rotates on a tilted axis that passes through its North and South poles, spinning from West to East Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.251. It is crucial to distinguish this from revolution, which is the motion of one object around another Science-Class VII, Earth, Moon, and the Sun, p.175.

The physics of rotation introduces two vital concepts: Angular Velocity (ω) and the Moment of Inertia (I). While linear velocity measures distance over time, angular velocity measures the rate of rotation (how fast the angle changes). The Moment of Inertia represents an object's resistance to rotational motion. Unlike mass in linear motion, which is constant, the Moment of Inertia depends on how mass is distributed relative to the axis. If mass is concentrated far from the axis, the Moment of Inertia is high; if mass is pulled closer to the axis, it becomes low.

The relationship between these is governed by the Law of Conservation of Angular Momentum (L). Angular momentum is defined by the formula:

L = Iω
In an isolated system where no external twisting force (torque) is applied, the total angular momentum (L) remains constant. This leads to a fascinating trade-off: if you decrease your Moment of Inertia (I) by pulling your mass closer to the center, your Angular Velocity (ω) must increase to keep L the same. This is exactly why a figure skater or a person on a rotating stool spins much faster when they fold their arms inward.

Concept	Linear Motion Equivalent	Rotational Motion Concept
Speed	Velocity (v)	Angular Velocity (ω)
Resistance	Mass (m)	Moment of Inertia (I)
Quantity of Motion	Linear Momentum (p = mv)	Angular Momentum (L = Iω)

Sources: Science-Class VII, Earth, Moon, and the Sun, p.171; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.251; Science-Class VII, Earth, Moon, and the Sun, p.175

2. Moment of Inertia: Mass in Rotation (basic)

In our previous hop, we understood that Inertia is the inherent tendency of an object to resist changes in its state of motion. When we move from linear motion to rotational motion, this concept evolves into the Moment of Inertia (I). While mass measures how much an object resists being pushed in a straight line, the Moment of Inertia measures how much an object resists being rotated or having its spinning speed changed.

What makes the Moment of Inertia fascinating is that it doesn't just depend on how much mass an object has, but crucially on how that mass is distributed relative to the axis of rotation. Just as the distribution of mass within the Earth's crust creates "gravity anomalies" and influences gravitational pull Fundamentals of Physical Geography, Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19, the distribution of mass in a spinning body determines its rotational behavior. If you concentrate mass further away from the center (the axis), the Moment of Inertia increases, making it harder to start or stop the spin.

This principle leads us to the Conservation of Angular Momentum (L). In physics, angular momentum is the product of the Moment of Inertia (I) and the angular velocity (ω), expressed as L = Iω. In a system where no external twisting force (torque) is applied, this total value L must remain constant. This creates a fascinating trade-off: if you decrease your Moment of Inertia by pulling your mass closer to the center, your rotational speed must increase to keep the total momentum balanced. This is why a figure skater or a man on a rotating stool spins much faster the moment they pull their arms inward.

Action	Mass Distribution	Moment of Inertia (I)	Spinning Speed (ω)
Arms Extended	Mass is far from axis	High	Slow
Arms Folded In	Mass is close to axis	Low	Fast

Sources: Fundamentals of Physical Geography, Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19; Environment and Ecology, Majid Hussain, Locational Factors of Economic Activities, p.32

3. Torque: The Cause of Rotational Change (intermediate)

In our study of mechanics, we often focus on linear motion—moving from point A to point B. However, most machines and natural systems involve rotation. Torque (τ) is the rotational equivalent of force. Just as a force causes an object to accelerate in a straight line, torque is the turning effect that causes an object to acquire angular acceleration. If you want to change how fast something is spinning, or start it spinning from a standstill, you must apply torque.

The effectiveness of torque depends on more than just the amount of strength you use. It is determined by three variables: the magnitude of the force, the lever arm (the perpendicular distance from the axis of rotation to the point where force is applied), and the angle of application. Mathematically, this is expressed as τ = rF sin θ. This explains why door handles are always placed far from the hinges; the greater the distance (r), the more torque you generate for the same amount of effort.

We see this principle applied in technology and nature. For example, in renewable energy, vertical-axis wind turbines are designed specifically to yield a high torque even at low speeds, which is essential for overcoming the initial resistance to rotation Environment, Shankar IAS Academy, Renewable Energy, p.290. Similarly, while we often discuss the Coriolis force in terms of atmospheric deflection Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309, it is helpful to remember that any change in the rotational state of a mass requires an underlying torque to be present.

Finally, it is vital to understand the relationship between torque and the object's resistance to spinning. This is captured in the rotational version of Newton’s Second Law: τ = Iα, where I is the Moment of Inertia and α is the angular acceleration. If an object has a very high moment of inertia (meaning its mass is spread far from the center), you will need a significantly higher torque to get it spinning at the same rate as a more compact object.

Sources: Environment, Shankar IAS Academy, Renewable Energy, p.290; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

4. Circular Motion: Centripetal and Centrifugal Forces (intermediate)

In our study of mechanics, Circular Motion introduces us to a fascinating dynamic where an object is constantly changing its direction. Even if the speed remains constant, the velocity is changing because the direction of motion is shifting at every point. To achieve this, a force must be applied to change the object's path from a straight line to a curve NCERT Science Class VIII, Exploring Forces, p.65. This brings us to two critical concepts: Centripetal and Centrifugal forces.

Centripetal Force is the "center-seeking" force that pulls an object toward the axis of rotation. Without it, the object would fly off in a straight tangent. In physical geography, we see this in cyclones, where intense low pressure at the center acts like an invisible string, pulling the wind inward to create a vortex PMF IAS, Tropical Cyclones, p.365. Conversely, Centrifugal Force is the outward "apparent" force felt by an object moving in a circle. It is why the Earth bulges at the equator—the rotation speed is higher there, creating a greater outward force that slightly counteracts gravity PMF IAS, Latitudes and Longitudes, p.241. These two forces are central to understanding tides, as the moon's gravity pulls water on one side, while centrifugal force causes a secondary bulge on the opposite side of the Earth NCERT Geography Class XI, Movements of Ocean Water, p.109.

To master the mechanics of these rotations, we must understand the Law of Conservation of Angular Momentum (L). Angular momentum is defined by the formula: L = Iω, where I is the Moment of Inertia (how mass is distributed relative to the center) and ω (omega) is the Angular Velocity (speed of rotation). In an isolated system, L remains constant. If you decrease your Moment of Inertia—for example, by pulling your arms inward while spinning—you are bringing your mass closer to the axis. To keep the total momentum (L) the same, your rotational speed (ω) must automatically increase. This is why a figure skater or a man on a rotating stool spins much faster the moment they tuck their limbs in.

Force Type	Direction	Real-World Example
Centripetal	Inward (toward center)	Low-pressure center in a cyclone pulling air inward.
Centrifugal	Outward (away from center)	The equatorial bulge of the Earth due to rotation.

Sources: NCERT Science Class VIII, Exploring Forces, p.65; PMF IAS Physical Geography, Tropical Cyclones, p.365; PMF IAS Physical Geography, Latitudes and Longitudes, p.241; NCERT Geography Class XI, Movements of Ocean Water, p.109

5. Kepler's Laws and Planetary Speed (intermediate)

Hello! Now that we have a solid grasp of basic mechanics, let’s apply those principles to the vast scale of our solar system. For centuries, astronomers believed planetary orbits were perfect circles. However, Johannes Kepler revolutionized our understanding by proving that orbits are actually ellipses, with the Sun sitting at one of the two "foci" (focal points) Physical Geography by PMF IAS, The Solar System, p.21. This means the distance between a planet and the Sun is constantly changing as it travels through space.

Kepler’s Second Law (The Law of Equal Areas) is the most fascinating for understanding planetary speed. It states that an imaginary line connecting a planet to the Sun sweeps out equal areas in equal intervals of time Physical Geography by PMF IAS, The Solar System, p.21. Imagine a triangular slice of a pizza: if the slice is short (planet is close to the Sun), the crust must be very wide to get the same amount of cheese (area) as a long, skinny slice (planet is far away). Therefore, to "sweep" that wide crust in the same amount of time, the planet must move much faster when it is near the Sun.

Position	Distance from Sun	Orbital Speed
Perihelion (or Perigee for Earth-Moon)	Closest	Fastest
Aphelion (or Apogee for Earth-Moon)	Farthest	Slowest

This variance in speed has a tangible impact on our lives. For instance, in the Northern Hemisphere, Earth reaches its aphelion (farthest point) during the summer. Because Earth is moving at its slowest orbital velocity at this point, it takes longer to travel through that part of its orbit Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.256. This is why summer in the Northern Hemisphere is roughly 92 days long, while winter is only about 89 days—the Earth is literally "speeding" through the winter months!

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; Certificate Physical and Human Geography, The Earth's Crust, p.3

6. Law of Conservation of Angular Momentum (exam-level)

To understand the Law of Conservation of Angular Momentum, we must first define what angular momentum is. In the world of rotation, Angular Momentum (L) is the product of two factors: the Moment of Inertia (I), which describes how mass is distributed around an axis, and the Angular Velocity (ω), which is the speed of rotation. Mathematically, this is expressed as L = Iω.

The principle of conservation states that if no external torque (a twisting force) acts on a system, the total angular momentum remains constant. This leads to a fascinating inverse relationship: if the Moment of Inertia decreases (by moving mass closer to the center), the Angular Velocity must increase to keep the total momentum the same. This is why a figure skater spins much faster when they pull their arms inward—they are reducing their moment of inertia, forcing their body to rotate at a higher speed to conserve momentum.

In the context of our planetary system, this law explains why the distribution of motion is so uneven. For instance, while the Sun contains approximately 99.8% of the solar system's mass, it accounts for only about 2% of its angular momentum Physical Geography by PMF IAS, The Solar System, p.23. The rest is held by the planets, particularly the gas giants. Similarly, the Earth’s constant 24-hour rotation Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.251 is a manifestation of this momentum, which would only change if an immense external torque were applied to the planet.

To visualize how these variables interact, consider the following table:

Action	Moment of Inertia (I)	Angular Velocity (ω)	Total Momentum (L)
Mass moved inward	Decreases	Increases (Spins faster)	Constant
Mass moved outward	Increases	Decreases (Spins slower)	Constant

Sources: Physical Geography by PMF IAS, The Solar System, p.23; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.251

7. Real-world Applications of Iω Relationship (exam-level)

To understand the dynamics of rotating systems, we look at the fundamental relationship L = Iω, where L is angular momentum, I is the moment of inertia (the distribution of mass relative to the axis), and ω (omega) is the angular velocity (rotational speed). According to the law of conservation of angular momentum, if no external torque acts on a system, its total angular momentum remains constant. This creates an inverse relationship: if the moment of inertia decreases, the angular velocity must increase to compensate, and vice versa.

One of the most elegant demonstrations of this principle is seen in figure skating. When a skater starts a spin with their arms and one leg extended, their mass is far from the axis of rotation, resulting in a high moment of inertia. As they pull their limbs inward toward their body, they are effectively redistributing their mass closer to the axis. This reduces their moment of inertia significantly, causing their rotational speed (ω) to accelerate dramatically to keep L constant Geography of India, Climate of India, p.30. A similar effect occurs when a person sits on a rotating friction-less stool; folding one's arms inward leads to a noticeable increase in spin speed.

This principle isn't just limited to the sports arena; it governs massive atmospheric phenomena as well. In the formation of a tornado, a large, rotating air mass known as a mesocyclone may span several kilometers. As the vortex extends vertically and contracts horizontally, the air mass is brought closer to the central axis. Just like the ice skater, this horizontal contraction reduces the moment of inertia, causing the wind speeds to accelerate into the violent, high-velocity rotations characteristic of a destructive tornado Geography of India, Climate of India, p.30.

Sources: Geography of India, Climate of India, p.30

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental building blocks of rotational mechanics, this question serves as the perfect bridge from theory to application. You have learned that Angular Momentum (L) is defined by the relationship L = Iω, where I represents the Moment of Inertia and ω represents Angular Velocity. According to the NCERT Physics principle of conservation, if no external torque acts on a system, its total angular momentum remains constant. This question tests your ability to recognize that folding one's arms is not just a physical movement, but a strategic redistribution of mass that alters the rotational characteristics of the system.

To arrive at the correct answer, follow this logic: When the man folds his arms inward, he moves his mass closer to the axis of rotation. As you’ve learned, the Moment of Inertia is a measure of how mass is distributed relative to that axis; therefore, pulling his arms in causes his Moment of Inertia (I) to decrease significantly. Since the product I × ω must stay the same to satisfy the Law of Conservation of Angular Momentum, a decrease in I must be compensated by a proportional rise in ω. Consequently, the man’s rotational speed must increase, making (A) the correct choice.

UPSC often includes distractors to test the depth of your conceptual clarity. Option (B) is a common trap; the velocity would only decrease if he extended his arms, thereby increasing his inertia. Option (D) is the most frequent mistake made by students who confuse the conservation of momentum with the conservation of velocity; velocity only remains constant if the body's shape and mass distribution do not change. Finally, option (C) is physically impossible in this frictionless idealization because an object in motion will stay in motion unless acted upon by an external torque. Always remember: in rotational dynamics, speed and inertia share an inverse dance to keep the total momentum steady.