A particle oscillates in one dimension about the equilibrium position subject to a force Fx(x) that has an associated potential energy U(x). If k is the force constant, which one of the following relations is true?

1. Understanding Periodic and Oscillatory Motion (basic)

Welcome to your first step in mastering mechanics! To understand how the physical world moves, we must first distinguish between motion that happens once and motion that repeats. Periodic motion is any movement that repeats itself at regular intervals of time. For example, the Earth orbiting the Sun or the rotation of the hands on a clock are periodic because they repeat their path consistently over a fixed duration.

However, when an object moves back and forth or to and fro about a central point, we call this oscillatory motion. A classic example is the simple pendulum, which consists of a metallic ball (the bob) suspended by a thread. When the bob is moved and released, it moves away from its mean position (the resting point) and returns to it repeatedly Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109. The time taken to complete one full back-and-forth movement is known as the time period, which remains constant for a pendulum of a given length Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118.

What drives this motion? It is the restoring force. For an object to oscillate, there must be a force that always tries to pull it back toward the center. In most basic systems, this is governed by Hooke’s Law, expressed as F = -kx. Here, the force (F) is directly proportional to the displacement (x) but acts in the opposite direction. This mathematical relationship explains why the further you pull a spring or a pendulum, the stronger it pulls back toward the equilibrium.

To keep these terms clear, remember this distinction:

Feature	Periodic Motion	Oscillatory Motion
Definition	Motion that repeats at regular intervals.	To and fro motion about a mean position.
Path	Can be circular, elliptical, or linear.	Always involves moving back and forth over the same path.
Example	Earth revolving around the Sun.	A child on a swing or a vibrating guitar string.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118

2. Newton’s Laws and the Concept of Restoring Force (basic)

In our first hop, we looked at how forces interact. Now, let’s go deeper into Newton’s Second Law (F = ma) to understand a specific, vital type of force: the Restoring Force. Imagine a system at its happy place, or equilibrium. When you disturb this system—like pushing a box or stretching a spring—nature often tries to fight back to bring it to its original state. As we observe in everyday objects, a force is required to bring a moving object to rest or to change its state Science, Class VIII, Exploring Forces, p.67.

The most classic example of this is a spring. When you hang an object from a spring, it stretches Science, Class VIII, Exploring Forces, p.73. The further you pull it from its resting position (the displacement, represented by x), the harder the spring pulls back. This "pull back" is the Restoring Force. It is mathematically defined by Hooke’s Law: Fₓ(x) = -kx. The negative sign is the most important part—it tells us that the force always acts in the opposite direction of the displacement. If you pull the spring right, the force pulls left.

In this relationship, k is the force constant (or spring constant), which measures the stiffness of the system. Because this force is conservative, the work done against it is stored as Elastic Potential Energy. This energy is expressed as U(x) = ½ kx². This quadratic relationship means that even a small increase in displacement results in a much larger increase in stored energy. Whether it is the tension in a plucked guitar string or the internal forces in a building swaying during an earthquake, the restoring force is what provides stability to our physical world.

Sources: Science, Class VIII, Exploring Forces, p.67; Science, Class VIII, Exploring Forces, p.73

3. Work, Energy, and Conservative Forces (basic)

In the study of mechanics, energy is defined as the capacity to perform work. When we analyze systems like a vibrating spring or a pendulum, we encounter Conservative Forces. A force is considered conservative if the work it performs depends only on the starting and ending points of the motion, rather than the specific path taken. As a fundamental principle, when work is performed, energy is transformed from one form to another Environment and Ecology, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14.

A classic example of a conservative system is the Simple Harmonic Oscillator, which follows Hooke’s Law. This law describes a "restoring force"—a force that always tries to bring the system back to its neutral equilibrium position. Mathematically, the force (Fx) is proportional to the displacement (x) from equilibrium: Fx(x) = -kx. The negative sign is essential; it signifies that the force acts in the opposite direction of the displacement. The term 'k' is known as the force constant, which measures the stiffness of the system.

Because these forces are conservative, we can associate them with Potential Energy (U), which is energy stored due to the position or configuration of the object. While the restoring force is linear (directly proportional to x), the potential energy is quadratic. By calculating the work done against this force, we find the relationship: U(x) = ½kx². This means that if you stretch a spring twice as far, the force you feel doubles, but the energy stored in the spring actually quadruples.

Feature	Restoring Force (F)	Potential Energy (U)
Relationship	Linear (proportional to x)	Quadratic (proportional to x²)
Formula	-kx	½kx²
Direction	Always towards equilibrium	Scalar (magnitude only)

Sources: Environment and Ecology, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14"

4. The Mathematical Link: Force and Potential Energy (intermediate)

In mechanics, the relationship between force and potential energy is one of the most elegant mathematical links in physics. To understand this from first principles, imagine a ball on a hilly landscape. The ball naturally rolls 'downhill' toward the lowest possible energy state. In physical geography, we see this when gravitational force acts on earth materials on a sloping surface, producing a downslope movement FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.39. Mathematically, we say that Force (F) is the negative gradient of Potential Energy (U). This means that force always points in the direction where potential energy decreases most rapidly. For a system like a Simple Harmonic Oscillator (like a spring), this relationship is expressed through Hooke’s Law. The restoring force is directly proportional to the displacement from the equilibrium position but acts in the opposite direction. We write this as Fₓ(x) = -kx, where k is the force constant. The negative sign is critical; it signifies that the force is 'restoring' the object back to its starting point. This is similar to how a pressure gradient creates a force that moves air from high to low pressure areas to reach equilibrium FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Atmospheric Circulation and Weather Systems, p.78. To find the potential energy associated with this force, we calculate the work done against it. Because work involves multiplying force by displacement (or integrating it over a distance), a linear force relationship (proportional to x) leads to a quadratic energy relationship (proportional to x²). Specifically, the potential energy is given by U(x) = ½kx². While the force is a vector that tells us where to push, the potential energy is a scalar that tells us how much 'stored' energy the system has at a specific point.

Feature	Force (F)	Potential Energy (U)
Mathematical Nature	Vector (has direction)	Scalar (magnitude only)
SHM Relationship	Linear: F = -kx	Quadratic: U = ½kx²
Direction	Points toward equilibrium	Increases as you move away from equilibrium

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.39; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Atmospheric Circulation and Weather Systems, p.78

5. Elasticity and Hooke's Law (intermediate)

When we apply force to an object like a rubber band or a metal coil, it deforms. Elasticity is the internal property of a material that allows it to resist this deformation and return to its original shape once the external force is removed. At the heart of this behavior is a principle known as Hooke’s Law. This law states that for relatively small deformations, the restoring force (the internal force trying to snap the object back) is directly proportional to the displacement (the distance it has been stretched or compressed).

Mathematically, we express Hooke’s Law as F = -kx. In this equation, F represents the restoring force, x is the displacement from the equilibrium position, and k is the force constant (or spring constant), which measures the stiffness of the material. The negative sign is crucial—it indicates that the restoring force always acts in the opposite direction to the displacement. You can see this principle in action with a spring balance, where the amount of stretching directly indicates the weight or force applied Science, Class VIII, Exploring Forces, p.73.

Because you must do work against this restoring force to stretch a spring, that work is stored within the system as Elastic Potential Energy (U). Unlike the force, which varies linearly with displacement, the potential energy follows a quadratic relationship: U = ½kx². This means that if you double the distance you stretch a spring, you actually quadruple the energy stored within it. This balance of forces is a fundamental concept in physics, similar to how atmospheric pressure and internal body pressure must balance to prevent us from being crushed Science, Class VIII, Pressure, Winds, Storms, and Cyclones, p.87.

Sources: Science, Class VIII (NCERT), Exploring Forces, p.73; Science, Class VIII (NCERT), Pressure, Winds, Storms, and Cyclones, p.87

6. Dynamics of Simple Harmonic Motion (SHM) (exam-level)

To understand the **Dynamics of Simple Harmonic Motion (SHM)**, we must look at what happens 'under the hood' to make an object oscillate. As we see in the behavior of a simple pendulum, an object at rest stays in its **mean position** until an external force is applied Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.109. Once moved and released, the object doesn't just sit there; a specific type of force, called a **Restoring Force**, takes over. In SHM, this force is unique because it is always proportional to how far the object has been moved from the center. This is governed by **Hooke's Law**, mathematically expressed as:

Fₓ(x) = -kx

In this equation, k is the force constant (a measure of stiffness), and x is the displacement. The **negative sign** is the most critical part of the dynamics—it tells us that the force always acts in the opposite direction of the displacement. If you pull a pendulum to the right, the restoring force pulls it back to the left. This constant 'tug' toward the center is what creates the repetitive, periodic motion. From a perspective of energy, as the object moves away from the equilibrium, work is done against this restoring force. This work is stored as **Elastic Potential Energy (U)**. Because the force varies linearly with distance, the potential energy grows quadratically: **U(x) = ½kx²**. This means that even a small increase in displacement leads to a significant increase in the energy stored in the system. Furthermore, because a force causes a change in the speed or direction of motion Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.64, the object accelerates most rapidly when it is furthest from the center (where the force is strongest) and has maximum speed when it passes through the mean position (where the force is zero but its momentum is highest).

Sources: Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.109; Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.64

7. Energy Quantities in a Harmonic Oscillator (exam-level)

In basic mechanics, when we look at a harmonic oscillator, such as a mass on a spring, we are essentially looking at how a system stores and releases energy through movement. The foundation of this system is the restoring force. As observed in simple experiments with hanging masses, the stretch (displacement) of a spring varies depending on the force applied Science, Class VIII NCERT (Revised ed 2025), Exploring Forces, p.73. This is formally defined by Hooke’s Law, which states that the force (F) exerted by the spring is directly proportional to the displacement (x) but acts in the opposite direction to bring the system back to equilibrium. Mathematically, this is expressed as F = -kx, where k is the force constant. While the force follows a linear relationship with displacement, the potential energy (U) stored in the system follows a quadratic relationship. This is because potential energy is the accumulation of work done against the restoring force as you pull the object further away. Since the force itself keeps increasing as you pull, the energy grows much faster than the force does. The standard physical relation for the potential energy in a one-dimensional harmonic oscillator is U(x) = ½kx². This creates a parabolic energy curve, meaning that if you double the displacement, you actually quadruple the stored energy. Understanding these quantities is similar to understanding the relationship between current and potential difference in a circuit, where a constant ratio defines the behavior of the system Science, Class X NCERT (2025 ed.), Electricity, p.175. In our oscillator, the constant k (the force constant) serves as the defining characteristic of the system's "stiffness," determining both how much force is required to move it and how much energy it can store at a given distance.

Sources: Science, Class VIII NCERT (Revised ed 2025), Exploring Forces, p.73; Science, Class X NCERT (2025 ed.), Electricity, p.175

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamentals of simple harmonic motion (SHM) and the relationship between conservative forces and potential energy, this question serves as the perfect synthesis of those building blocks. In any one-dimensional oscillatory system, the movement is governed by a restoring force that attempts to bring the particle back to its equilibrium position. As you learned in ScienceDirect: Oscillation, this force must be proportional to the displacement but opposite in direction, a relationship fundamentally defined by Hooke’s Law.

To arrive at the correct answer, we must identify the precise mathematical expression of this physical reality. If we define k as the force constant and x as the displacement, the restoring force F_x(x) is expressed as -kx. This negative sign is critical; it signifies that the force is always directed toward the origin. While the potential energy (U) is indeed the integral of this force—resulting in 1/2 kx²—we must look for the most accurate and properly formatted representation provided. Thus, Option (B) Fx(x) = —kx is the definitive correct choice.

Understanding UPSC's "trap" mechanics is just as important as knowing the physics. Option (C) suggests a linear potential energy, which would result in a constant force rather than an oscillation. Option (D) is a classic typographical distractor; while it aims for the correct energy formula (1/2 kx²), the inclusion of extra characters and incorrect notation makes it technically invalid. Always verify the notation carefully, as UPSC often includes options that are "conceptually close" but mathematically garbled to test your precision under exam pressure.