Which one of the following pairs does not have the same dimension?

1. Fundamental and Derived Physical Quantities (basic)

To understand the physical world, we must first learn how to measure it. A physical quantity is any property of a material or system that can be quantified by measurement. In physics, we categorize these into two main types: Fundamental and Derived quantities. Think of fundamental quantities as the 'alphabet' of science—they are independent and cannot be defined in terms of anything else. The most common examples are Mass [M], Length [L], and Time [T]. Every other measurement we use is built by combining these basic building blocks.

Derived quantities are created when we multiply or divide fundamental quantities. For instance, speed is calculated by dividing distance (length) by time. Therefore, speed is a derived quantity with the unit metre/second (m/s) Science-Class VII, Measurement of Time and Motion, p.113. Similarly, Density—which helps us understand why Earth is the densest planet in our solar system Physical Geography by PMF IAS, The Solar System, p.26—is derived by dividing Mass by Volume (Length³), giving it the dimensional formula [ML⁻³].

Interestingly, some quantities are dimensionless. This usually happens when we compare two identical types of quantities as a ratio. A classic example is Relative Density (or Specific Gravity). Since it is the ratio of the density of a substance to the density of water, the units cancel each other out, leaving us with a pure number that has no dimensions Science, Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.141. Understanding this distinction is vital for dimensional analysis, where we check if both sides of an equation represent the same physical 'nature'.

Type	Definition	Examples
Fundamental	Independent quantities that serve as the foundation.	Mass, Length, Time, Temperature
Derived	Quantities expressed as a combination of fundamental ones.	Speed, Force, Density, Acceleration

Sources: Science-Class VII NCERT, Measurement of Time and Motion, p.113; Physical Geography by PMF IAS, The Solar System, p.26; Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.141

2. Introduction to Dimensional Analysis (MLT System) (basic)

At the heart of physics lies the ability to measure the world around us. However, whether we measure a distance in centimeters, meters, or kilometers, we are fundamentally measuring the same thing: Length. Dimensional Analysis is a powerful tool that allows us to look past specific units and see the 'fundamental nature' of a physical quantity. In mechanics, we express almost everything using three base dimensions: Mass [M], Length [L], and Time [T]. For instance, when you calculate the volume of a notebook by multiplying its length, width, and height, you are multiplying three length measurements together, giving volume the dimensional formula of [L³] Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.145.

Derived quantities are formed by combining these base dimensions. Consider Speed: it is defined as distance divided by time. Therefore, its dimensions are Length divided by Time, written as [LT⁻¹] Science Class VII, Measurement of Time and Motion, p.113. This system helps us understand that seemingly different concepts often share the same 'DNA.' For example, Potential Energy and Kinetic Energy might look different in their formulas, but they both represent energy and thus share the exact same dimensions: [ML²T⁻²]. Similarly, whether it is the pull of gravity or the drag of friction, every Force has the dimensions [MLT⁻²].

One of the most important lessons in dimensional analysis is recognizing dimensionless quantities. These occur when we take a ratio of two similar quantities, causing the units and dimensions to cancel out completely. A classic example is Specific Gravity. While Density measures mass per unit volume (dimensions [ML⁻³]), Specific Gravity is the ratio of a substance's density to the density of water. Because it is a 'density divided by a density,' all dimensions cancel out, leaving it as a pure number without any MLT components. Understanding this distinction is crucial for scientific accuracy.

Sources: Science Class VIII, The Amazing World of Solutes, Solvents, and Solutions, p.145; Science Class VII, Measurement of Time and Motion, p.113

3. Work and Energy: The [ML²T⁻²] Dimension (intermediate)

In physics, we use dimensional analysis to understand the fundamental nature of physical quantities. Every derived unit is built from three core building blocks: Mass [M], Length [L], and Time [T]. To find the dimension of Work, we look at its definition: Work = Force × Displacement. Since Force is Mass × Acceleration ([MLT⁻²]), multiplying it by Displacement ([L]) gives us the dimensional formula [ML²T⁻²].

It is a profound rule in science that Energy and Work are two sides of the same coin. Energy is essentially the capacity to do work. Whether we look at Kinetic Energy (½mv²) or Potential Energy (mgh), the dimensional result remains identical. For instance, in Kinetic Energy, mass [M] multiplied by the square of velocity [LT⁻¹]² also yields [ML²T⁻²]. This consistency confirms that energy, in all its forms, is dimensionally equivalent to work, which is why they share the common SI unit, the Joule (J) Science, class X (NCERT 2025 ed.), Electricity, p.191.

Quantity	Defining Formula	Dimensional Derivation	Final Dimension
Work	Force × Distance	[MLT⁻²] × [L]	[ML²T⁻²]
Kinetic Energy	½ × Mass × Velocity²	[M] × [LT⁻¹]²	[ML²T⁻²]
Potential Energy	Mass × Gravity × Height	[M] × [LT⁻²] × [L]	[ML²T⁻²]

Understanding this equivalence is vital for the Law of Conservation of Energy. Because they share the same dimensions, energy can be transformed into work and vice-versa without losing its fundamental physical nature Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14. This is why a battery's stored chemical energy or a moving car's kinetic energy can both be measured and compared using the same unit: the Joule.

Sources: Science, class X (NCERT 2025 ed.), Electricity, p.191; Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14

4. Nature of Forces: Gravitational vs. Frictional (intermediate)

In our journey through mechanics, we must understand that while all forces share a common identity, they differ fundamentally in how they interact with objects. At its simplest, a force is a push or a pull resulting from an object's interaction with another Science, Class VIII, Exploring Forces, p.77. Whether it is the Earth pulling on a falling apple or a brake pad slowing down a bicycle, both are measured in Newtons (N) and share the same dimensional formula (Mass × Acceleration). However, their "nature of contact" sets them poles apart.

Frictional force is a classic example of a contact force. It comes into play only when two surfaces are in physical contact and move (or attempt to move) relative to each other Science, Class VIII, Exploring Forces, p.66. Friction always acts in a direction opposite to the motion, effectively resisting it. Without physical contact, friction simply cannot exist. In contrast, gravitational force is a non-contact force (also known as an "action-at-a-distance" force). It is the attractive pull that exists between any two masses. For instance, the Earth pulls an object toward its center even if the object is mid-air and not touching the ground Science, Class VIII, Exploring Forces, p.77. This pull is what we commonly refer to as the weight of the object Science, Class VIII, Exploring Forces, p.72.

Understanding these distinctions is crucial for solving complex mechanics problems. While both can change an object's speed or direction, they arise from different physical principles. Here is a quick comparison to keep these clear in your mind:

Feature	Frictional Force	Gravitational Force
Category	Contact Force	Non-contact Force
Requirement	Physical contact between surfaces	Masses present (no contact needed)
Direction	Opposite to the direction of motion	Always attractive (toward the center of mass)
SI Unit	Newton (N)	Newton (N)

Sources: Science, Class VIII, Exploring Forces, p.77; Science, Class VIII, Exploring Forces, p.66; Science, Class VIII, Exploring Forces, p.72

5. Basics of Optics: Focal Length and Distance (intermediate)

In the study of optics, focal length (f) is a fundamental measurement that determines how strongly a lens converges or diverges light. For any lens, the distance from its optical centre to its principal focus is defined as the focal length. Interestingly, a lens has two principal foci (F₁ and F₂), one on each side, but the focal length refers to the distance to these points from the center. From a dimensional perspective, focal length is essentially a measure of distance, meaning its physical dimension is simply Length [L]. Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.151

To use these measurements in calculations, we follow a strict New Cartesian Sign Convention. This is crucial for solving numerical problems correctly. All distances are measured from the optical centre. Distances measured in the direction of incident light are positive, while those against it are negative. This leads to a standard rule: the focal length of a convex (converging) lens is always positive, whereas that of a concave (diverging) lens is always negative. Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.155

The Power of a lens (P) is mathematically defined as the reciprocal of its focal length in metres (P = 1/f). The SI unit of power is the dioptre (D), where 1 D = 1 m⁻¹. A lens with a short focal length bends light rays more sharply, thus possessing higher power. If a doctor prescribes a lens with a power of +2.0 D, you can immediately identify it as a convex lens with a focal length of +0.50 m. Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158

Lens Type	Nature	Focal Length (f)	Power (P)
Convex	Converging	Positive (+)	Positive (+)
Concave	Diverging	Negative (–)	Negative (–)

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.151; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.155; Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158

6. Density vs. Specific Gravity (Relative Density) (exam-level)

To understand how objects interact with their environment—like why an iron nail sinks while a massive wooden log floats—we must distinguish between Density and Specific Gravity (also known as Relative Density). While they are related, they represent fundamentally different ways of measuring matter.

Density is an intrinsic physical property defined as the mass present in a unit volume of a substance (Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.140). Mathematically, it is expressed as Density = Mass / Volume. In terms of dimensional analysis, density has the formula [ML⁻³], meaning it carries the units of mass divided by length cubed (such as kg/m³ or g/cm³). It is important to note that while density is independent of an object's shape or size, it is sensitive to external factors; for instance, temperature and pressure can alter density, particularly in gases (Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.140). In our oceans, we see this in action as cold, saline water becomes denser and sinks, driving global currents (Physical Geography by PMF IAS, Ocean Movements Ocean Currents And Tides, p.487).

Specific Gravity, on the other hand, is a relative measure. It is the ratio of the density of a substance to the density of a reference substance (usually pure water at 4°C). Because it is a ratio of two similar quantities (Density / Density), the units cancel out completely. This makes Specific Gravity a dimensionless quantity. For example, if a rock has a density of 3000 kg/m³ and water is 1000 kg/m³, its specific gravity is simply 3. This number tells us exactly how many times heavier the substance is compared to an equal volume of water, without needing to worry about whether we are using metric or imperial units.

Feature	Density	Specific Gravity
Definition	Mass per unit volume.	Ratio of substance density to reference density.
Dimensions	[ML⁻³]	Dimensionless (No dimensions).
Units	kg/m³, g/cm³, etc.	None (Pure number).
Dependency	Changes based on the unit system used.	Remains the same regardless of units.

Sources: Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.140; Physical Geography by PMF IAS, Ocean Movements Ocean Currents And Tides, p.487

7. Dimensionless Quantities in Physics (exam-level)

In physics, most physical quantities are expressed in terms of fundamental dimensions like Mass [M], Length [L], and Time [T]. For instance, both kinetic and potential energy share the dimensional formula [ML²T⁻²], while any type of force, whether gravitational or frictional, is expressed as [MLT⁻²]. However, there is a special class of quantities known as dimensionless quantities. These are pure numbers that do not possess any physical dimensions because they usually represent a ratio of two identical physical quantities where the units cancel out.

A classic example is the Refractive Index (n). It is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in a specific medium (v). Since both the numerator and denominator are speeds (measured in m/s), the dimensions [LT⁻¹] cancel each other out, leaving a pure number. This number tells us how much the light slows down in a medium; for example, the refractive index of water is approximately 1.33 Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148. It is important to note that while a substance might be "optically denser" (higher refractive index), this does not necessarily mean it has a higher mass density Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.149.

Another critical distinction often tested in exams is between Density and Specific Gravity. While they sound similar, their dimensional nature is entirely different. Density is mass per unit volume and carries the dimensions [ML⁻³]. In contrast, Specific Gravity is the ratio of the density of a substance to the density of a reference substance (usually water). Because it is a ratio of two densities, it is a dimensionless quantity Physical Geography by PMF IAS, The Solar System, p.23.

To help you distinguish between quantities with dimensions and those without, consider this comparison:

Physical Quantity	Nature	Dimensional Formula
Density	Mass / Volume	[ML⁻³]
Specific Gravity	Density Ratio	Dimensionless [M⁰L⁰T⁰]
Focal Length	Measurement of distance	[L]
Refractive Index	Speed Ratio	Dimensionless [M⁰L⁰T⁰]

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.148; Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.149; Physical Geography by PMF IAS, The Solar System, p.23

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental dimensions of Mass [M], Length [L], and Time [T], this question serves as the perfect bridge to application. It requires you to look past the specific names of physical phenomena to identify their underlying dimensional formulas. In the UPSC Preliminary Exam, a favorite strategy of examiners is to present quantities that sound different but share a common physical nature, or to pair a physical property with its dimensionless ratio. Success here depends on your ability to recognize when units cancel out, leaving a quantity with no dimension at all.

To arrive at the correct answer, systematically evaluate the physical nature of each pair. Potential and kinetic energy are both forms of energy, sharing the dimension [ML²T^-2]. Similarly, focal length and height are both basic measurements of length [L], while gravitational and frictional forces are both forces [MLT^-2]. However, Density is mass per unit volume ([ML^-3]), whereas Specific Gravity is a relative ratio of two densities. Because specific gravity is a ratio of identical units, it is dimensionless. Therefore, (B) Density and specific gravity is the only pair that does not have the same dimension, as one has a physical unit and the other is a pure number.

A common UPSC trap is to use descriptive adjectives—like "gravitational" vs "frictional" or "potential" vs "kinetic"—to distract you into thinking the dimensions change. Remember, if the unit of measurement is the same (e.g., both are measured in Newtons or Joules), the dimensions must be identical. As discussed in Physical Geography by PMF IAS, understanding these distinctions, such as density variations in the solar system, is critical for both the Science and Geography sections of the syllabus. Always be on the lookout for ratios; they are almost always the key to identifying dimensionless quantities.