Two racing cars of masses m, and m2 are moving in circles of radii r, and r2 respectively. Their speeds are such that each car makes a complete circle in the same time ‘t The ratio of angular speed of the first to that of the second car is :

1. Basics of Motion: Speed and Velocity (basic)

Welcome to your first step in mastering mechanics! To understand how the world moves, we must first distinguish between how fast something is going and the direction it is moving in. At its simplest, motion is a change in the position of an object over time. When an object moves along a straight path, like a train traveling between two stations, we call this linear motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116. However, objects rarely move at a constant rate; a train might start slow, speed up, and then slow down again as it reaches its destination.

The most fundamental concept here is Speed, which is the distance covered by an object in a unit of time. It tells us how 'fast' an object is, but it doesn't care about the direction. If we add direction to speed, we get Velocity. For example, '50 km/h' is a speed, but '50 km/h towards the North' is a velocity. This is a crucial distinction in physics: speed is a scalar quantity (magnitude only), while velocity is a vector quantity (magnitude plus direction).

Feature	Speed	Velocity
Definition	Distance covered per unit time.	Displacement (distance in a specific direction) per unit time.
Type	Scalar (only magnitude).	Vector (magnitude + direction).
Change	Changes only if the rate of motion changes.	Changes if either the speed or the direction changes.

In our daily lives, we mostly encounter non-uniform motion, where the speed of an object keeps changing as it moves Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119. To simplify this, we often calculate the Average Speed, which is the total distance traveled divided by the total time taken. Conversely, if an object covers equal distances in equal intervals of time, we say it is in uniform motion Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119

2. Introduction to Uniform Circular Motion (UCM) (basic)

To understand circular motion, let’s first contrast it with what we know. In linear motion, an object moves along a straight path Science-Class VII, Measurement of Time and Motion, p.116. However, when an object moves along a circular path at a constant speed, we call it Uniform Circular Motion (UCM). While its speed remains constant, its direction is constantly changing at every point on the circle.

In UCM, we often look at Angular Displacement (θ)—the angle swept by the object at the center of the circle. One complete revolution equals an angular displacement of 2π radians (or 360°). The time it takes to complete this one full revolution is called the Time Period (T). This leads us to a vital concept: Angular Speed (ω). Defined as the rate of change of angular displacement, it is calculated as:

ω = 2π / T

A fascinating aspect of angular speed is that it depends only on the time taken to complete a circle (the period), not on the size of the circle (radius) or the mass of the object. For instance, whether we are talking about a proton moving in a magnetic field Science, class X, Magnetic Effects of Electric Current, p.203 or the Earth rotating on its axis Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309, the angular velocity is a measure of how fast the "angle" changes, regardless of the physical distance covered.

Feature	Linear Motion	Uniform Circular Motion
Path	Straight line	Circular path
Constant Value	Linear Speed (v)	Angular Speed (ω) and Linear Speed (v)
Direction	Fixed	Changing continuously

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.116-117; Science , class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.203; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

3. Forces in Circular Motion: Centripetal Force (intermediate)

When we think of motion, we often imagine a straight line. However, in nature and engineering—from the swirling winds of a cyclone to a car navigating a sharp bend on the Mumbai-Pune Express Highway—circular motion is everywhere. To keep an object moving in a circle, a specific kind of influence is required. As we know, a force is a push or pull resulting from an interaction Science Class VIII NCERT, Exploring Forces, p.77. In the case of circular motion, this force must constantly pull the object toward the center of its path; we call this the Centripetal Force.

Why is this force necessary? Even if an object moves at a constant speed, its direction is changing every millisecond. In physics, velocity is a vector (it has both speed and direction). Therefore, a change in direction is a change in velocity, which means the object is accelerating. According to Newton’s laws, acceleration can only happen if a net force is applied. This centripetal acceleration acts at right angles to the motion, directed inwards toward the center of rotation Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309. Without this inward tug, the object would simply fly off in a straight line (tangent to the circle) due to its inertia.

To understand the "rate" of this rotation, we look at Angular Speed (ω). While linear speed measures distance over time, angular speed measures how quickly the angle changes (Δθ/Δt). For any object completing one full revolution, the angular displacement is 2π radians. If the time taken for one full circle is the Period (T), then the angular speed is calculated as:

ω = 2π / T

This reveals a fascinating insight: the angular speed depends only on the time it takes to complete a lap, not on how wide the circle is or how heavy the object is. In a cyclonic vortex, for example, the intense low pressure at the center provides the centripetal pull that holds the rotating winds in place against the outward "centrifugal" tendency Physical Geography by PMF IAS, Tropical Cyclones, p.365.

Concept	Centripetal Force	Centrifugal "Force"
Direction	Inward (toward the center)	Outward (away from the center)
Nature	A real force (e.g., gravity, friction, tension)	An apparent force (felt due to inertia)
Example	Low pressure pulling air in a cyclone	Feeling pushed against a car door during a turn

Sources: Science Class VIII NCERT, Exploring Forces, p.77; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309; Physical Geography by PMF IAS, Tropical Cyclones, p.365

4. S&T Connection: Satellite Orbits and Periods (exam-level)

To understand how satellites move, we must first master the concept of Angular Speed (ω). Imagine two satellites orbiting the Earth. While linear speed measures how many kilometers a satellite travels per second, angular speed measures how many degrees or radians it sweeps through per unit of time. In physics, we define angular speed as the rate of change of angular displacement (Δθ/Δt). For any object completing a full circle, the total angular displacement is 2π radians (which is equivalent to 360°).

The time it takes to complete one full revolution is called the Period (T). This leads us to a fundamental relationship: ω = 2π/T. As noted in Science, Class VIII, NCERT, p.185, artificial satellites orbiting about 800 km above Earth take roughly 100 minutes to complete one orbit. This "100 minutes" is their Period. Because the value of 2π is a constant, the angular speed depends entirely on the time period. If two objects have the same period, they must have the exact same angular speed, regardless of any other factors.

A common point of confusion in competitive exams is whether mass or the size of the orbit changes this specific ratio. From a purely mechanical perspective, if the time taken (T) is held constant, the mass of the object and the radius of the circle do not affect the angular speed. Whether it is a massive natural satellite like the Moon or a small man-made ISRO satellite Science, Class VIII, NCERT, p.188, if they were to complete a revolution in the same amount of time, their angular speeds would be identical (a 1:1 ratio).

However, in actual orbital mechanics, the period itself is often determined by the distance from the center. As described by Kepler’s Laws, the square of the orbital period is proportional to the cube of the orbital radius Physical Geography by PMF IAS, The Solar System, p.21. But once that period is fixed, the angular speed is locked in. For example, all Geostationary satellites have a period of 24 hours to match Earth's rotation; therefore, every geostationary satellite, regardless of its size or weight, shares the exact same angular speed.

Sources: Science, Class VIII NCERT, Keeping Time with the Skies, p.185; Science, Class VIII NCERT, Keeping Time with the Skies, p.188; Physical Geography by PMF IAS, The Solar System, p.21

5. Angular Displacement and Time Period (T) (intermediate)

In our study of motion, we often focus on how far something travels in a straight line. However, in the realm of circular motion—whether it's a car on a racing track or the Earth orbiting the Sun—we must look at Angular Displacement. This is the angle (usually measured in radians) through which an object moves around a central point. While linear displacement measures distance in meters, angular displacement measures the "sweep" of the motion. For a complete circle, the angular displacement is always 2π radians (or 360°).

Just as we see in geography, where latitudes and longitudes are defined as angular distances measured from the center of the Earth Physical Geography by PMF IAS, Latitudes and Longitudes, p.250, mechanics uses these angles to describe position. Interestingly, even in geomorphology, the concept of shear stress is linked to angular displacement or "slippage" of earth materials FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geomorphic Processes, p.39. Whenever a body rotates or moves in a curve, we are tracking how its angular position changes over time.

The time it takes to complete one full revolution is called the Time Period (T). For instance, the Earth's time period for one revolution around the Sun is approximately 365.25 days Certificate Physical and Human Geography, The Earth's Crust, p.6. The relationship between this time and the "speed" of rotation is known as Angular Speed (ω). Mathematically, it is expressed as:

ω = 2π / T

The most critical takeaway for UPSC aspirants is that Angular Speed (ω) depends only on the Time Period (T). It does not depend on the mass of the object or the radius of the circle. If two objects complete a circle in the same amount of time, their angular speeds are identical, regardless of how heavy they are or how large their respective circles are.

Feature	Linear Motion	Circular Motion
Displacement	Distance in a straight line (meters)	Angle swept (radians)
Speed	Rate of change of distance (v)	Rate of change of angle (ω)
Full Cycle	N/A	2π radians

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.250; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geomorphic Processes, p.39; Certificate Physical and Human Geography, The Earth's Crust, p.6

6. Angular Velocity (ω) and its Independence (exam-level)

To master mechanics, we must distinguish between how fast an object moves along a path and how fast it rotates. Angular velocity (ω) is defined as the rate of change of angular displacement. While linear velocity measures the distance covered per unit of time (meters per second), angular velocity measures the angle covered per unit of time (usually radians per second). If an object completes one full revolution, it has traveled an angle of 2π radians (360°).

The time taken to complete one full revolution is known as the Time Period (T). This gives us the foundational formula: ω = 2π/T. This relationship reveals a crucial insight: angular velocity is independent of the radius (r) of the path and the mass (m) of the object. For example, even though the Earth's linear rotational velocity is approximately 1675 km/hr at the equator Physical Geography by PMF IAS, The Solar System, p.23, every point on a rigid rotating body (like the Earth) shares the exact same angular velocity because they all complete one rotation in the same 24-hour period.

Feature	Linear Velocity (v)	Angular Velocity (ω)
Definition	Rate of change of distance.	Rate of change of angle.
Formula	v = distance / time	ω = 2π / T
Dependency	Depends on the radius (v = rω).	Independent of radius and mass.

This independence is a common "trap" in exam questions. If two objects—regardless of their weight or how wide their circular tracks are—complete a lap in the same amount of time, their angular velocities are identical. This concept is vital when calculating the Coriolis Force, which depends on the Earth's constant angular velocity (ω) rather than varying linear speeds Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309.

Sources: Physical Geography by PMF IAS, The Solar System, p.23; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

7. Solving the Original PYQ (exam-level)

This question is a masterclass in applying the fundamental definitions of Circular Motion you have just studied. To arrive at the solution, you must connect the concept of angular displacement to the time period (T). The building blocks here are simple: angular speed (ω) measures how quickly an object rotates through an angle, defined by the formula ω = 2π/T. Since the problem explicitly states that both cars complete a full circle in the same time t, their time periods are identical, regardless of the path they follow or their physical characteristics.

As your coach, I want you to walk through the logic step-by-step: first, identify that the angular displacement for any complete circle is always 2π radians. Second, recognize that the rate at which this angle is covered depends solely on the time t. When you set up the ratio of the first car's speed (2π/t) to the second car's speed (2π/t), the terms cancel out completely. This leads us directly to the correct answer (B) 1:1. The beauty of this problem lies in its simplicity once you strip away the unnecessary variables.

UPSC frequently uses distractors to test whether you can identify which physical laws actually apply. Option (A) m1:m2 is a trap for students who might overthink the role of centripetal force or mass, which are irrelevant to the rate of rotation here. Option (C) r1:r2 is a common trap designed to confuse angular speed with linear velocity (v). While it is true that the car on the larger radius (r2) must travel at a higher linear speed to finish the lap in the same time, their rotational rates remain the same. This distinction is a recurring theme in NCERT Class 11 Physics and is vital for your success in the Prelims.