Four wires of same material and of dimensions as under are stretched by a load of same magnitude separately. Which one of them will be elongated maximum?

1. Introduction to Mechanical Properties of Solids (basic)

Concept: Introduction to Mechanical Properties of Solids

2. Elasticity, Plasticity, and the Elastic Limit (basic)

In mechanics, every solid material reacts to external forces by changing its shape or size. Elasticity is the property of a body to regain its original dimensions once the deforming force is removed. Think of a high-quality steel wire or a rubber band; when you stop pulling, they return to their initial state. This happens because of internal restoring forces that develop within the material to resist change.

On the flip side, we have Plasticity. If a material does not return to its original shape and stays deformed even after the force is gone, it is called plastic. Common examples include putty, mud, or lead. While we often use the word "elastic" in economics to describe how demand responds to price changes NCERT Microeconomics Class XII 2025, Theory of Consumer Behaviour, p.29, in physics, it specifically refers to this ability to "snap back" to a starting physical configuration.

The transition between these two behaviors is defined by the Elastic Limit. This is the maximum stress (force per unit area) a material can withstand while still being able to return to its original shape. If you apply a force that exceeds this limit, the material enters the plastic region, meaning it will suffer a "permanent set" or lasting deformation. Understanding these limits is crucial for engineers and civil servants involved in infrastructure, as it ensures that bridges and buildings are designed to stay well within their elastic zones to prevent structural failure.

Property	Reaction to Force Removal	Example
Elasticity	Regains original shape and size perfectly.	Steel springs, Rubber
Plasticity	Retains the deformed shape permanently.	Modeling clay, Plasticine

Sources: NCERT Microeconomics Class XII 2025, Theory of Consumer Behaviour, p.29, 34

3. Stress, Strain, and Hooke's Law (intermediate)

In mechanics, when we apply an external force to a body, it tends to deform. To understand this, we use two fundamental concepts: Stress and Strain. Stress is the internal restoring force acting per unit area of a body. Think of it as the internal 'push-back' a material gives when you try to change its shape. On the other hand, Strain is the measure of that deformation—specifically, it is the fractional change in the dimensions (like length or volume) of the body. Even in our own bodies, we experience this; for instance, holding a book too close to your eyes causes the ciliary muscles to work harder, leading to eye strain Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162.

Hooke’s Law states that for relatively small deformations, the stress is directly proportional to the strain. The constant of proportionality for tensile or compressive stress is called Young’s Modulus (Y). Mathematically, Stress = Y × Strain. By expanding this, we find that the change in length (ΔL) of a wire or rod is given by the formula:

ΔL = (F · L) / (A · Y)

Here, F is the force, L is the original length, A is the cross-sectional area, and Y is the material's Young's Modulus. This principle is not just for laboratories; it explains physical weathering in geography. When rocks undergo temperature changes, the molecular stresses caused by thermal expansion and contraction eventually lead to fatigue and rock fracture FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.41.

When comparing different wires of the same material under the same load, the elongation depends entirely on their geometry. Since the area (A) of a circular wire is proportional to the square of its diameter (d²), we can say that ΔL is proportional to L/d². This means a longer wire will stretch more, but a thicker wire (larger diameter) will stretch significantly less because the diameter's effect is squared.

Term	Definition	Formula/Relationship
Stress	Restoring force per unit area	Force / Area
Strain	Fractional deformation	Change in Length / Original Length
Elongation (ΔL)	The total stretch produced	ΔL ∝ L / d² (for same material/load)

Sources: Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.162; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Geomorphic Processes, p.41

4. Fluid Mechanics: Viscosity and Drag (intermediate)

In our previous discussions, we looked at how solids interact. Now, imagine trying to run through waist-deep water versus running through air. You feel a distinct resistance, don't you? In fluid mechanics, this internal 'stickiness' or resistance to flow is called Viscosity. Just as solid surfaces have irregularities that cause friction when they slide over each other Science, Class VIII, Exploring Forces, p.68, fluids (liquids and gases) possess internal friction between their layers. When one layer of fluid moves over another, these layers exert a force that tries to oppose the relative motion. This is why honey flows much more slowly than water—it has a higher viscosity.

When an object moves through a fluid, it doesn't just face internal fluid resistance; it experiences an external frictional force known as Drag. You might have noticed this while sticking your hand out of a moving car window. The magnitude of this drag depends on three primary factors: the speed of the object relative to the fluid, the nature of the fluid (its viscosity), and most importantly, the shape of the object. To minimize this energy-wasting drag, nature and engineers use streamlining—giving objects a sleek shape (like birds, fish, or airplanes) to allow fluid to flow past them smoothly.

Interestingly, environmental conditions also play a role. For instance, temperature significantly affects how fluids behave; generally, the density of a substance decreases as temperature increases Science, Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.150, and for most liquids, heating them makes them less viscous and easier to pour. Furthermore, the way fluids move and exert pressure is governed by Bernoulli's principle, which states that within a horizontal flow, points of higher fluid speed will have less pressure than points of slower fluid speed Physical Geography by PMF IAS, Tropical Cyclones, p.358. This relationship between speed, pressure, and resistance is fundamental to everything from the flight of an aircraft to the circulation of blood in our veins.

Feature	Viscosity	Drag (Fluid Friction)
Definition	Internal resistance of a fluid to flow.	Opposing force exerted by a fluid on an object moving through it.
Analogy	The "thickness" of the liquid (Honey vs. Water).	The "air resistance" felt by a cyclist.
Dependency	Depends on molecular makeup and temperature.	Depends on shape, speed, and fluid viscosity.

Sources: Science, Class VIII, Exploring Forces, p.68; Science, Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.150; Physical Geography by PMF IAS, Tropical Cyclones, p.358

5. Surface Tension and Capillarity (intermediate)

Imagine the surface of a liquid not as a simple boundary, but as a stretched elastic membrane. This phenomenon is Surface Tension. At a molecular level, a molecule inside a liquid is pulled in every direction by its neighbors, resulting in a net force of zero. However, a molecule at the surface has no liquid molecules above it; it is pulled only sideways and inwards. This creates an internal pressure that forces the liquid to occupy the least possible surface area, which is why raindrops are spherical.

This molecular "tug-of-war" involves two critical forces: Cohesion (attraction between similar molecules, like water-to-water) and Adhesion (attraction between different molecules, like water-to-glass). We see the practical application of altering these forces when using soap. Soap molecules act as mediators; one end attaches to oil and the other to water, effectively lowering the surface tension of water so it can "wet" the fabric more effectively and lift away dirt Science Class VIII, Particulate Nature of Matter, p.111.

Capillarity (or capillary action) is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity. This happens because the adhesion between the liquid and the solid walls is stronger than the cohesion within the liquid. In biology, we use the term "capillaries" to describe our smallest blood vessels because they are so narrow Science Class X, Life Processes, p.93. While blood flow is driven by the heart, in physical systems like soil or a lamp wick, the liquid "climbs" through these tiny pores due to this surface tension effect.

Force Type	Description	Result in Capillarity
Cohesion	Attraction between same molecules.	Tries to keep the liquid together (pulls surface down).
Adhesion	Attraction between liquid and tube wall.	Tries to spread the liquid out (pulls liquid up the wall).

Sources: Science Class VIII, Particulate Nature of Matter, p.111; Science Class X, Life Processes, p.93

6. Modulus of Elasticity: Young’s Modulus (exam-level)

To understand how materials behave under tension, we look at the Young’s Modulus (Y), which is a measure of a material’s stiffness. It tells us how much a solid object will stretch or compress when a force is applied. From first principles, Young’s Modulus is defined as the ratio of tensile stress (force per unit area) to tensile strain (fractional change in length). This relationship is expressed by the formula:

ΔL = (F · L) / (A · Y)

Where ΔL is the elongation, F is the applied force, L is the original length, A is the cross-sectional area, and Y is the Young's Modulus of the material. In the context of earth sciences, these molecular stresses are what eventually lead to rock fracture and weathering when the material can no longer handle the strain Fundamentals of Physical Geography, Geomorphic Processes, p.39.

When comparing wires made of the same material and subjected to the same force, the Young’s Modulus (Y) and Force (F) become constants. In this scenario, the elongation (ΔL) depends entirely on the geometry of the wire. Specifically, ΔL is directly proportional to the length (L) and inversely proportional to the cross-sectional area (A). Since most wires are circular, the area (A) is proportional to the square of the diameter (d²). This gives us a highly useful proportionality for exams:

ΔL ∝ L / d²

This means if you want to find out which wire will stretch the most under the same load, you don't need the exact value of Y; you simply need to calculate the ratio of Length to the square of the Diameter. A long, thin wire will always elongate much more than a short, thick one because the stress is concentrated over a smaller area and distributed over a greater length. This concept of repeated expansion and contraction leading to material fatigue is a fundamental driver of physical weathering in nature Fundamentals of Physical Geography, Geomorphic Processes, p.41.

Change in Variable	Effect on Elongation (ΔL)	Reasoning
Increasing Length (L)	Increases ΔL	More material to distribute the strain.
Increasing Diameter (d)	Decreases ΔL	Area increases by square of d, reducing stress.
Higher Young’s Modulus (Y)	Decreases ΔL	The material is stiffer (e.g., steel vs. lead).

Sources: Fundamentals of Physical Geography, Geomorphic Processes, p.39; Fundamentals of Physical Geography, Geomorphic Processes, p.41

7. Factors Governing Wire Elongation (ΔL) (exam-level)

When we apply a pulling force (tensile load) to a wire, it undergoes a physical change in length known as elongation (ΔL). This property is not just a matter of chance; it is governed by the Elastic Modulus (specifically Young's Modulus, Y) of the material. Think of it as a tug-of-war between the force trying to stretch the wire and the internal atomic bonds trying to hold it together. The fundamental relationship is expressed as ΔL = (F · L) / (A · Y), where F is the force applied, L is the original length, A is the cross-sectional area, and Y is the Young's Modulus of the material.

To master this concept for competitive exams, you must look at the geometric factors. For a given material and a constant load, the elongation depends entirely on the ratio of the wire's length to its thickness. Just as the resistance of a wire increases with length and decreases with thickness Science, Class X (NCERT 2025 ed.), Electricity, p.178, the mechanical elongation follows a similar logic. Specifically, because most wires are circular, the area (A) is proportional to the square of the diameter (d²). Therefore, if you keep the material and force the same, ΔL is proportional to L / d².

This means that a longer wire will stretch more than a shorter one, but a thicker wire (larger diameter) will stretch significantly less because the force is distributed over a larger area. In practical calculations, if you are comparing different wires, you should calculate the value of L / d² for each. The wire with the highest value will experience the greatest elongation. Note that even a small change in diameter has a massive impact because it is squared in the denominator Science, Class X (NCERT 2025 ed.), Electricity, p.180.

Sources: Science, Class X (NCERT 2025 ed.), Electricity, p.178; Science, Class X (NCERT 2025 ed.), Electricity, p.180

8. Solving the Original PYQ (exam-level)

This question brings together your understanding of Young’s Modulus and the geometric properties of materials. As you learned in the NCERT Class 11 Physics chapter on Mechanical Properties of Solids, the elongation ($ΔL$) of a wire is governed by the formula ΔL = (F · L) / (A · Y). Since the problem states the material (Young’s Modulus, $Y$) and the load ($F$) are identical for all wires, the elongation depends purely on the ratio of length ($L$) to cross-sectional area ($A$). Because the area of a circular wire is proportional to the square of its diameter ($d^2$), you can simplify your search: the maximum elongation occurs where the ratio $L/d^2$ is highest.

Walking through the options, we calculate this ratio to compare them directly. For (A), the ratio is $1/2^2 = 0.25$; for (B), it is $2/2^2 = 0.5$; and for (D), it is $1/1^2 = 1$. However, for Option (C), the ratio is $3 / 1.5^2$, which simplifies to $3 / 2.25 ≈ 1.33$. This confirms that Option (C) is the correct answer because it yields the highest value. By systematically applying the formula, you can see that the significant 3m length combined with a relatively small diameter makes this wire the most susceptible to stretching.

A common trap in UPSC Prelims is to rely on singular intuitions—such as assuming the longest wire (Option B) or the thinnest wire (Option D) must be the answer. UPSC tests your ability to see how variables interact. While a thinner wire increases elongation, that effect is squared compared to the length. Option (C) is the winner because it balances a high numerator (length) with a sufficiently small denominator (diameter squared), overcoming the simpler dimensions of the other choices. Always calculate the combined ratio rather than guessing based on one dimension alone.