Which one of the following is dimensionless quantity ?

1. Systems of Measurement and SI Units (basic)

Measurement is the cornerstone of physics and engineering. It involves comparing an unknown physical quantity with a known, fixed standard called a unit. Historically, different regions used different systems (like the FPS or CGS systems), which led to confusion in global trade and science. To solve this, the world adopted the International System of Units (SI). This system is built upon seven Base Units, such as the metre (m) for length, the kilogram (kg) for mass, and the second (s) for time. As noted in Science-Class VII . NCERT, Measurement of Time and Motion, p.111, symbols for these units are always written in lowercase (unless they are at the start of a sentence) and are never pluralized—for instance, we write '10 kg', not '10 kgs'.

While base units measure fundamental properties, Derived Units are formed by mathematically combining these base units. For example, density is defined as mass per unit volume. Since mass is measured in kilograms and volume in cubic metres, the SI unit of density is kg/m³ Science, Class VIII . NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.141. Similarly, pressure is defined as force per unit area, measured in Pascals (Pa), which is equivalent to Newtons per square metre (N/m²).

An interesting category in measurement is the dimensionless quantity. These are numbers that have no physical unit because they are a ratio of two identical physical quantities. For instance, if you divide a length by another length, the units cancel out completely, leaving a pure number. Strain is a classic example: it measures the fractional change in an object's dimensions (like change in length divided by original length). Because it is "length/length," it has no units, unlike force (Newtons) or stress (Pascals), which always carry physical dimensions.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.111; Science ,Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.141

2. Fundamental vs. Derived Physical Quantities (basic)

In the world of physics, we measure the universe using Physical Quantities. To keep things organized, scientists have identified a small set of Fundamental Quantities (also called base quantities) that are entirely independent of one another. Think of these as the 'alphabet' of physics—they cannot be broken down further or defined in terms of other quantities. In the standard International System (SI), there are seven fundamental quantities, but for basic mechanics, we primarily focus on three: Mass (measured in kilograms), Length (meters), and Time (seconds). As you explore forces, you'll find that mass is a core property used to describe how much matter is in an object Science, Class VIII, Chapter 5, p.72.

Once we have our 'alphabet,' we can combine these base units to form 'words' known as Derived Quantities. These are physical quantities that are expressed as a combination of fundamental quantities through multiplication or division. For example, Speed is derived by dividing Length by Time (m/s). Similarly, Pressure is a derived quantity defined as the force acting per unit area Science, Class VIII, Chapter 6, p.82. Another fascinating derived concept is Relative Density, which is a ratio of two densities—since it compares the same type of quantity, the units cancel out, leaving us with a dimensionless number Science, Class VIII, p.141.

Feature	Fundamental Quantities	Derived Quantities
Definition	Independent and basic; cannot be simplified further.	Dependent on fundamental quantities for their definition.
Examples	Mass, Length, Time, Temperature.	Area, Volume, Density, Force, Pressure.
Units	Base units (e.g., kg, m, s).	Derived units (e.g., kg/m³, m/s²).

Sources: Science, Class VIII, Chapter 5: Exploring Forces, p.72; Science, Class VIII, Chapter 6: Pressure, Winds, Storms, and Cyclones, p.82; Science, Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.141

3. Dynamics: Understanding Force and Motion (basic)

In our journey through mechanics, understanding Force is our cornerstone. At its simplest level, a force is a push or a pull resulting from an interaction between two or more objects Science, Class VIII, Chapter 5, p. 65. However, for a civil services aspirant, it is crucial to look deeper: forces are the agents of change. They can alter an object's speed, change its direction of motion, or even transform its physical shape Science, Class VIII, Chapter 5, p. 77. Whether it is the muscular force you use to lift a book or the invisible gravitational force the Earth exerts to keep us grounded, every force is measured in the SI unit called the newton (N).

It is important to distinguish between force and its related concepts, Weight and Pressure. Many students mistakenly think weight and mass are the same; however, Weight is actually a force—specifically, the pull of Earth's gravity on an object. Because it is a force, weight is also measured in newtons (N) Science, Class VIII, Chapter 5, p. 72. Pressure, on the other hand, is the force applied per unit area (Force/Area). Its SI unit is the pascal (Pa), which is equivalent to one newton per square metre (N/m²) Science, Class VIII, Chapter 6, p. 82.

Finally, let's explore a sophisticated concept: dimensionless quantities. Most physical measurements have units (like meters for length or seconds for time). However, when we calculate a ratio of two identical physical quantities, the units cancel out, leaving us with a pure number. A prime example in mechanics is Strain. Strain measures how much an object deforms; for instance, tensile strain is the change in length divided by the original length (Length / Length). Since the units of length in the numerator and denominator cancel each other out, strain has no units—it is dimensionless. This stands in stark contrast to force or pressure, which always carry their respective physical dimensions.

Sources: Science, Class VIII, Exploring Forces, p.65; Science, Class VIII, Exploring Forces, p.72; Science, Class VIII, Exploring Forces, p.77; Science, Class VIII, Pressure, Winds, Storms, and Cyclones, p.82

4. Pressure and Its Applications in Geography (intermediate)

In our journey through basic mechanics, pressure is a fundamental concept that bridges the gap between pure physics and the complex systems of our planet. Simply put, pressure is defined as the force acting perpendicularly per unit area of a surface. While force tells us how hard an object is being pushed, pressure tells us how concentrated that force is. The formula is written as Pressure = Force / Area. Because force is measured in Newtons (N) and area in square metres (m²), the standard SI unit of pressure is N/m², more commonly known as the pascal (Pa) Science, Class VIII . NCERT(Revised ed 2025), Chapter 6, p. 82.

In Geography, we focus heavily on atmospheric pressure—the weight of the column of air above a given point. Unlike solid objects, gases and liquids exert pressure in all directions. Because air is compressible, it is densest at sea level and becomes thinner as we go up. Consequently, pressure decreases rapidly with altitude. On average, pressure drops by about 1 millibar (mb) for every 10 metres of ascent in the lower atmosphere FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Chapter 9, p. 76. By the time you reach the summit of Mt. Everest, the air pressure is nearly two-thirds lower than at sea level, which explains why breathing becomes so difficult Physical Geography by PMF IAS, Pressure Systems and Wind System, p. 305.

It is also crucial to distinguish pressure from other mechanical terms like strain. While pressure and stress have physical dimensions (Force/Area), strain is a dimensionless ratio of change in length to original length. In a geographical context, horizontal pressure differences (pressure gradients) are the primary drivers of wind, as air naturally moves from high-pressure areas to low-pressure areas. However, we don't feel a constant upward "wind" from the massive vertical pressure gradient because it is perfectly balanced by the downward pull of gravity—a state known as hydrostatic equilibrium FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Chapter 9, p. 76.

Unit Type	Unit Name	Conversion / Context
SI Unit	Pascal (Pa)	1 N/m²
Meteorological	Millibar (mb)	100 Pa (equivalent to 1 hPa)
Meteorological	Hectopascal (hPa)	100 Pa

Sources: Science, Class VIII . NCERT(Revised ed 2025), Chapter 6: Pressure, Winds, Storms, and Cyclones, p.82, 87; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.305; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Chapter 9: Atmospheric Circulation and Weather Systems, p.76

5. Fluid Mechanics: Viscosity and Surface Tension (intermediate)

In our journey through mechanics, we now move from solids to fluids—substances like liquids and gases that flow because they cannot withstand even a small amount of shear stress without deforming. The two defining characteristics of how fluids move and interact with surfaces are Viscosity and Surface Tension. Think of Viscosity as 'internal friction.' Just as friction resists the motion of a block on a floor, viscosity resists the relative motion between layers of a fluid. A fluid with high viscosity, like honey or motor oil, flows slowly and feels 'thick,' whereas a low-viscosity fluid like water flows easily. Unlike the pressure we studied earlier, which acts normally (perpendicularly) to a surface with units of Pascal (N/m²) Science, Class VIII, Chapter 6, p.87, viscosity is about the tangential drag between fluid layers.

Surface Tension is a property unique to the interface of a liquid. It is caused by cohesion—the attractive force between identical molecules. Inside a liquid, a molecule is pulled in all directions by its neighbors, but a molecule at the surface is only pulled inward and sideways. This creates a state of tension, making the surface behave like a stretched elastic membrane or 'skin.' This is why small insects can walk on water and why raindrops naturally pull themselves into a spherical shape to minimize surface area. When we observe water in a container, this tension, combined with adhesion (attraction to the container walls), causes the surface to curve. This curve is known as the meniscus Science, Class VIII, Chapter 9, p.144.

To compare these two vital fluid properties, let’s look at their core drivers:

Feature	Viscosity	Surface Tension
Physical Cause	Internal friction/resistance between fluid layers.	Unbalanced cohesive forces at the liquid's surface.
Primary Effect	Determines how fast or slow a fluid flows.	Determines how a liquid beads up or wets a surface.
Temperature Effect	Generally decreases for liquids as temperature rises (they get 'thinner').	Decreases as temperature rises (molecular attractions weaken).

Sources: Science, Class VIII (NCERT 2025), Chapter 6: Pressure, Winds, Storms, and Cyclones, p.87; Science, Class VIII (NCERT 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.144

6. Elasticity and Hooke's Law (intermediate)

When we study mechanics, we often assume objects are perfectly rigid. However, in the real world, materials deform when subjected to external forces. Elasticity is the property of a body by virtue of which it tends to regain its original size and shape when the applied force is removed. This happens because the constituent particles in solids are held together by strong interparticle interactions Science, Class VIII, Particulate Nature of Matter, p.102. When you stretch a spring, you are fighting these internal bonds; when you let go, those bonds pull the particles back to their fixed positions Science, Class VIII, Particulate Nature of Matter, p.113.

To quantify these changes, we use two critical terms: Stress and Strain. Stress is the internal restoring force acting per unit area of a deformed body. It is mathematically similar to pressure, which is also defined as force per unit area and measured in Pascals (N/m²) Science, Class VIII, Pressure, Winds, Storms, and Cyclones, p.94. However, while pressure is usually an external force acting on a surface, stress represents the internal resistance within the material. For example, repeated temperature changes in deserts cause rocks to expand and contract, setting up internal stresses that eventually cause the rock to crack Certificate Physical and Human Geography, Weathering, Mass Movement and Groundwater, p.38.

Strain, on the other hand, describes the deformation itself. It is the ratio of the change in a dimension (like length or volume) to the original dimension. Because it is a ratio of two identical physical quantities (e.g., meters divided by meters), strain is a dimensionless quantity—it has no units at all. This brings us to Hooke’s Law, which states that within the elastic limit of a material, the stress applied is directly proportional to the strain produced. In simple terms: the harder you pull, the more it stretches, in a predictable, linear fashion.

Concept	Definition	SI Unit
Stress	Internal restoring force per unit area.	Pascal (N/m²)
Strain	Fractional change in dimension (deformation).	None (Dimensionless)
Elasticity	Ability to return to original shape.	Property, not a unit.

Sources: Science, Class VIII, Particulate Nature of Matter, p.102; Science, Class VIII, Particulate Nature of Matter, p.113; Science, Class VIII, Pressure, Winds, Storms, and Cyclones, p.94; Certificate Physical and Human Geography, Weathering, Mass Movement and Groundwater, p.38

7. Defining Strain and Dimensionless Ratios (exam-level)

In the study of mechanics, when we apply an external force to a body, it tends to undergo deformation. While stress represents the internal restoring force per unit area, strain is the mathematical measure of how much the object actually deforms. Specifically, strain is defined as the fractional change in the dimensions of an object. For instance, tensile strain is the ratio of the change in length (ΔL) to the original length (L) of the object.

The most unique characteristic of strain is that it is a dimensionless quantity. This occurs because strain is a ratio of two identical physical quantities. If you are measuring the change in length of a metal rod in meters and dividing it by the original length also in meters, the units (m/m) cancel each other out. Unlike force (measured in Newtons) or pressure and stress (measured in Pascals or N/m²), strain does not have a unit of measurement. It is simply a pure number that describes the intensity of deformation. This concept of ratios is common in science; for example, the refractive index is also dimensionless because it is a ratio of the speed of light in different media Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148.

Understanding dimensionless ratios is vital because they allow scientists to compare different materials or systems regardless of their size. Whether you are stretching a 1-meter wire or a 100-meter cable, a strain of 0.01 tells you that both have stretched by 1% of their original length. While quantities like mass density depend on the material's mass and volume Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.72, strain focus purely on the proportion of change, making it a universal tool for analyzing material strength.

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.72

8. Solving the Original PYQ (exam-level)

Having mastered the fundamental definitions of force and its effects, you can now see how UPSC tests your understanding of physical dimensions. This question requires you to look beyond just the formula and identify whether a quantity carries a unit or represents a pure ratio. As you recall from your foundational concepts in Science, Class VIII. NCERT (Revised ed 2025), dimensions are lost when we compare two identical physical properties, resulting in a unitless value.

The correct answer is (B) Strain. To arrive at this, walk through the definition: Strain is the fractional change in an object’s dimensions (like length or volume) relative to its original state. Because it is a ratio of length divided by length, the units cancel out entirely, leaving a dimensionless number. This is a classic conceptual pivot; while strain describes a physical change, it does not require a unit to express its magnitude.

UPSC often includes Pressure and Stress as distractors because they are frequently confused. While both represent a ratio of force over area, they result in the unit of Pascals (N/m²), meaning they are not dimensionless. Similarly, Force is a primary interaction measured in Newtons. As highlighted in Science, Class VIII. NCERT (Revised ed 2025), only a quantity that compares two identical units—like strain—can be truly dimensionless.