When a stone tied to a string is whirled in a circle, the work done on it by the string :

1. Understanding Work and Energy (basic)

Welcome to the first step of your journey into basic mechanics! To understand how the physical world moves, we must first master the concepts of Work and Energy. In everyday language, we use the word "work" to describe any effort—like studying for the UPSC or standing for hours at a bus stop. However, in physics, work has a very precise definition: it is the process of a force causing the displacement of an object. If there is no displacement, or if the force is applied in a specific way, no "work" is done in the eyes of physics.

The mathematical heart of this concept is the formula: W = F · d · cosθ. Here, F is the force applied, d is the displacement, and θ (theta) is the angle between the force and the direction of motion. For work to be non-zero, the force must have a component acting in the direction of the displacement. We see energy utilized in everything from propelling vehicles to driving industrial machinery NCERT, Contemporary India II, p.113. Essentially, Energy is defined as the capacity or ability to do this work. Whether it is the chemical energy in the food we eat Science, Class X (NCERT 2025 ed.), Our Environment, p.210 or the electrical energy moving a charge between two points Science, Class X (NCERT 2025 ed.), Electricity, p.173, energy is the "currency" spent to perform physical work.

One of the most fascinating applications of this is Circular Motion. Imagine you are whirling a stone tied to a string in a perfect circle. The tension in the string provides a centripetal force that pulls the stone inward toward the center. However, at any single moment, the stone wants to fly off in a straight line—its instantaneous displacement is tangential to the circle. This means the force (inward) and the displacement (tangential) are perfectly perpendicular to each other (θ = 90°). Since the cosine of 90° is zero, the work done by the string on the stone is actually zero! The force is change the direction of the stone, but it isn't doing "work" to change its speed in a uniform circular path.

Scenario	Is Work Done?	Reason
Pushing a wall (no movement)	No	Displacement (d) is zero.
Carrying a box while walking horizontally	No (by gravity)	Force (upward) is perpendicular to displacement (forward).
Lifting a book off the floor	Yes	Force and displacement are in the same direction.

Sources: Science, Class X (NCERT 2025 ed.), Our Environment, p.210; NCERT, Contemporary India II, Energy Resources, p.113; Science, Class X (NCERT 2025 ed.), Electricity, p.173

2. The Mathematical Definition of Work (basic)

In everyday language, 'work' usually refers to any mental or physical effort. However, in the realm of physics and mechanics, **Work** has a very precise mathematical definition. For work to be done, two things are essential: a force must be applied, and that force must cause a displacement of the object. If you push against a wall with all your might but it doesn't move, mathematically, you have done zero work because the displacement is zero.

The mathematical formula for work is W = F × d × cosθ. Here, F is the magnitude of the force, d is the displacement, and θ (theta) is the angle between the force vector and the displacement vector. This formula tells us that work is essentially the dot product of force and displacement. It isn't just about how much force you apply, but how much of that force is acting in the same direction as the movement. This concept of directionality is critical; for instance, when studying magnetic effects, we often see that the force acting on a conductor is most effective when the directions are at right angles Science, class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.203.

The role of the angle θ leads to three fascinating scenarios:

• Positive Work: When the force and displacement are in the same direction (θ = 0°), work is maximum.
• Negative Work: When the force opposes the motion (like friction), the angle is 180°, and the work done is negative.
• Zero Work: This is the most counter-intuitive part. If the force applied is perpendicular (θ = 90°) to the direction of motion, no work is done by that specific force. This is because cos(90°) = 0. We see this principle when considering forces acting perpendicular to a surface Science, Class VIII, NCERT(Revised ed 2025), Pressure, Winds, Storms, and Cyclones, p.81. This explains why a satellite orbiting Earth in a perfect circle has zero work done on it by gravity — the gravitational pull is always at a right angle to its instantaneous path.

Sources: Science, class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.203; Science, Class VIII, NCERT(Revised ed 2025), Pressure, Winds, Storms, and Cyclones, p.81

3. Positive, Negative, and Zero Work (intermediate)

In physics, "work" is not just a measure of effort, but a precise calculation of how much a force contributes to the movement of an object. It is defined by the formula W = F·d cosθ, where 'F' is the magnitude of the force, 'd' is the displacement, and 'θ' is the angle between them. Essentially, a force only does work if it has a component acting in the direction of the object's motion Science, Class VIII, Exploring Forces, p.77. Understanding the angle (θ) is the key to distinguishing between the three types of work. Positive Work occurs when the force and the displacement are in the same direction (θ is acute). In this case, the force is actively helping the object move, thereby increasing its energy. For example, when you push a cart and it moves forward, you are doing positive work. Conversely, Negative Work occurs when the force opposes the direction of motion (θ is obtuse). A classic example is friction; because it arises from the irregularities of surfaces locking together to resist movement, it always acts in the opposite direction of the displacement Science, Class VIII, Exploring Forces, p.68. Similarly, the friction exerted by the Earth's surface on moving wind acts to resist its movement, performing negative work that slows the wind down Physical Geography by PMF IAS, Pressure Systems and Wind System, p.307. Zero Work occurs under three specific conditions: if the force is zero, if the displacement is zero, or if the force and displacement are perpendicular (θ = 90°). Because the cosine of 90° is zero, the total work done becomes zero. This is a vital concept in circular motion. For instance, when a stone is whirled in a circle, the tension in the string (centripetal force) pulls inward toward the center, while the stone moves tangentially. Since the inward force and the tangential movement are always at a right angle, the string does no work on the stone.

Type of Work	Angle (θ)	Impact on Object	Example
Positive	0° ≤ θ < 90°	Adds energy/speed	A horse pulling a cart forward
Negative	90° < θ ≤ 180°	Removes energy/slows down	Frictional force on a sliding book
Zero	θ = 90°	No energy change via that force	Carrying a load horizontally on your head

Sources: Science, Class VIII, Exploring Forces, p.77; Science, Class VIII, Exploring Forces, p.68; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.307

4. Dynamics of Uniform Circular Motion (intermediate)

In Uniform Circular Motion (UCM), an object moves along a circular path at a constant speed. However, even though the speed is constant, the velocity is continuously changing because the direction of motion is constantly shifting. To maintain this path, a centripetal force (center-seeking force) must act on the object, pulling it toward the center of the circle. This force acts at right angles to the direction of motion. For example, when air flows around centers of atmospheric circulation, centripetal acceleration creates a force directed inwards, producing the circular patterns we see in cyclones and anticyclones Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309.

One of the most fascinating aspects of UCM is the Work-Energy relationship. In physics, Work (W) is calculated as the product of force (F), displacement (d), and the cosine of the angle (θ) between them: W = Fd cosθ. In circular motion, the centripetal force is always directed radially inward, while the instantaneous displacement of the object is tangential to the circle. Because a radius and a tangent always meet at 90°, the angle θ is always 90°. Since cos 90° = 0, the work done by the centripetal force is zero. This means the force changes the direction of the object, but it does not change its kinetic energy or speed.

This principle is visible in nature beyond just stones on strings. In oceanography, the actual motion of water particles beneath surface waves is circular. As a wave approaches, the water is carried up and forward, then down and back as it passes, completing a loop FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109. Similarly, in a tropical cyclone, intense tangential forces acting on high-speed winds contribute to the formation of the eye—a region of relative calmness at the center of the vortex Physical Geography by PMF IAS, Tropical Cyclones, p.364.

Sources: Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109; Physical Geography by PMF IAS, Tropical Cyclones, p.364

5. Centripetal Force in Action (intermediate)

When we observe an object moving in a circle—whether it is a stone whirled on a string or a planet orbiting a star—there is a hidden geometry at play. For any object to maintain a circular path, it must be constantly pulled toward the center. This "center-seeking" requirement is met by Centripetal Force. It is important to remember that centripetal force isn't a new kind of force; rather, it is a role played by existing forces. For a stone on a string, the tension provides the force; for a planet, it is gravitational force Science, Class VIII NCERT (Revised ed 2025), Exploring Forces, p.72; and for air circling a low-pressure system, it is centripetal acceleration acting on the wind Physical Geography by PMF IAS, Chapter 23, p.309.

One of the most fascinating aspects of this force is its relationship with Work Done. In physics, work is defined by the formula W = F · d cosθ, where θ is the angle between the force and the direction of displacement. In uniform circular motion, the centripetal force always acts radially inward (toward the center), while the object’s instantaneous displacement is tangential (along the edge of the circle). Because a radius and a tangent are always perpendicular (90°), and the cosine of 90° is zero, the work done by the centripetal force is always zero. This explains why the force changes the direction of the object but does not change its kinetic energy or speed.

In the context of physical geography, this concept helps us understand global phenomena like cyclones and tides. While centripetal acceleration maintains the circular flow of winds around pressure centers, its counterpart, the centrifugal force (an apparent force acting outward), plays a massive role in shaping our oceans. As explained in Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109, the interaction between the moon's gravitational pull and the Earth's centrifugal force is responsible for the dual tidal bulges we see on opposite sides of the planet.

System	Force Providing Centripetal Action	Resulting Motion
Stone on String	Tension in the string	Circular path of the stone
Planetary Orbit	Gravity	Elliptical/Circular orbit
Atmospheric Cyclone	Pressure Gradient & Centripetal Acceleration	Vortex (Circular flow)

Sources: Science, Class VIII NCERT (Revised ed 2025), Exploring Forces, p.72; Physical Geography by PMF IAS, Chapter 23: Pressure Systems and Wind System, p.309; Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109

6. Work and Energy in Planetary Orbits (exam-level)

To understand how planets and satellites maintain their paths, we must look at the relationship between Force, Displacement, and Work. In physics, Work (W) is defined as the product of the force applied and the displacement in the direction of that force. Mathematically, this is expressed as W = F · d cosθ, where θ is the angle between the force vector and the displacement vector. For an object in a nearly circular orbit, such as the Earth around the Sun Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.176, the gravitational pull acts as a centripetal force, constantly pulling the object toward the center of the orbit.

In uniform circular motion, a fascinating geometric reality emerges: the gravitational force is always directed radially inward (toward the Sun or Earth), while the instantaneous displacement of the planet is always tangential to the path. This means the force and the displacement are always perpendicular to each other (θ = 90°). Since the cosine of 90° is zero, the work done by gravity on the planet is zero. This explains why a planet can orbit for billions of years without "using up" its orbital energy or slowing down due to gravity itself; the force changes the direction of motion, but it does not change the speed or perform work in the thermodynamic sense.

However, space is rarely a perfect vacuum with zero resistance. In the exosphere, where high and mid-earth orbit satellites reside, the air is extremely thin, allowing them to move with very little atmospheric drag Physical Geography by PMF IAS, Earths Atmosphere, p.280. If drag were significant, it would perform negative work, slowing the satellite down and causing it to lose altitude. Furthermore, energy can be transferred through other means, such as tidal friction. For instance, as Earth's rotation slows due to tidal interaction with the Moon, energy is transferred, causing the Moon to slowly move further away by about four centimeters per year Physical Geography by PMF IAS, The Solar System, p.29. This highlights that while gravity in a perfect circle does no work, complex interactions in our solar system involve subtle energy shifts over millions of years.

Sources: Science-Class VII . NCERT(Revised ed 2025), Earth, Moon, and the Sun, p.176; Physical Geography by PMF IAS, Earths Atmosphere, p.280; Physical Geography by PMF IAS, The Solar System, p.29

7. Angle Analysis: Centripetal Force vs Displacement (exam-level)

In mechanics, understanding the spatial relationship between force and movement is crucial for determining energy transfer. When an object undergoes uniform circular motion, such as a stone being whirled on a string or air circulating around a low-pressure center, it is acted upon by a centripetal force. This force is always directed radially inward, toward the center of rotation. In meteorology, for instance, this force is what produces the circular vortex of wind around atmospheric pressure systems Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309.

To analyze the work being done, we must look at the instantaneous displacement of the object. While the force pulls the object toward the center, the object itself is trying to move forward. At any given micro-moment, its displacement is tangential to the circle. This creates a geometric constant: the inward-pointing force and the forward-pointing displacement are perpendicular to each other, forming a 90° angle. This is a fundamental principle seen in other areas of physics too, such as when the displacement of a conductor is maximized when the current and magnetic field are at right angles Science, class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.203.

The mathematical implication of this 90° angle is profound for energy calculations. Work (W) is defined by the formula W = F · d cosθ, where θ is the angle between the force and displacement. Because the centripetal force and tangential displacement are always at 90°, we use cos(90°), which equals zero. Consequently, the work done by a centripetal force is always zero. This explains why the force only changes the direction of the object's velocity, not its speed or kinetic energy.

Vector	Direction in Circular Motion	Role
Centripetal Force	Radially Inward (toward center)	Maintains the curved path
Instantaneous Displacement	Tangential (along the curve)	The direction of motion

Sources: Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309; Science, class X (NCERT 2025 ed.), Magnetic Effects of Electric Current, p.203

8. Solving the Original PYQ (exam-level)

Now that you've mastered the individual components of Work and Circular Motion, this question brings them together in a classic application of physics geometry. To solve this, you must bridge the gap between the effort exerted by the string and the mathematical definition of Work. As you learned in the module on Physical Geography by PMF IAS, any object moving in a curve experiences a centripetal force. In this specific case, the tension in the string acts as that force, pulling the stone radially inward toward the center of the circle to maintain its path.

To arrive at the correct answer, apply the Work-Energy formula: W = F · d cosθ. Think about the direction of motion: at any micro-second, the stone’s instantaneous displacement is tangential to the circle. Because the force (tension) is pulling inward and the displacement is moving sideways along the tangent, they are always perpendicular to each other. Since the angle (θ) is 90°, and the cosine of 90° is zero, the total work done is zero. It is a vital coaching tip to remember: if the force doesn't have a component in the direction of motion, it can't do work!

UPSC often includes distractors like "positive" or "negative" to catch students who confuse the intensity of the pull with the result of the work. You might feel the tension, but because the stone never moves closer to your hand, no work is performed in the physics sense. Furthermore, saying the work "depends on the mass" is a common trap designed to make you think of the formula for centripetal force ($F = mv^2/r$); however, while mass changes the amount of force, it does not change the angle, which is what ultimately dictates that the work done is zero.