A solid floats in liquid A with half its volume immersed and in liquid B with 2/3'of its volume immersed. The densities of the liquids A and B are in the ratio of

1. Understanding Matter: Mass, Volume, and Density (basic)

At its most fundamental level, matter is defined as anything that possesses mass and occupies space, which we call volume. Imagine a handful of cotton and a small iron nail; while the cotton might look 'bigger' (occupying more volume), the iron nail feels 'heavier' because it contains more mass in a smaller space. This relationship between mass and the space it occupies is what we call Density. Scientifically, density is the mass present per unit volume of a substance Science, Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p. 140. Mathematically, it is expressed as:

Density = Mass / Volume

A crucial characteristic of density is that it is an intrinsic property—it doesn't change based on the shape or size of the object. Whether you have a giant block of gold or a tiny gold ring, the density remains the same because the atoms are packed in the same way. However, density is sensitive to external factors like temperature and pressure. For instance, as you heat most substances, they expand (increasing volume), which causes their density to decrease Science, Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p. 140.

To make comparisons easier, scientists use Relative Density. This is a ratio that compares the density of a substance to the density of water at a specific temperature. Because it is a ratio of two similar quantities (density divided by density), it is a dimensionless number, meaning it has no units Science, Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p. 141. This concept is vital in geography and planetary science; for example, did you know that among all the planets in our solar system, Earth is the densest? Physical Geography by PMF IAS, The Solar System, p. 26. Understanding density helps us explain why iron sinks in water while massive wooden logs float, and why oil sits on top of water.

Sources: Science, Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.140; Science, Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.141; Science, Class VIII . NCERT(Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.146; Physical Geography by PMF IAS, The Solar System, p.26

2. Pressure in Fluids: Pascal’s Law (intermediate)

When we talk about Pressure in Fluids, we must first recognize that fluids (liquids and gases) behave very differently from solids. While a solid brick only exerts pressure on the surface it rests upon, a fluid exerts pressure in all directions. This is because fluid molecules are in constant motion, colliding with each other and the boundaries of their container. As noted in your studies, liquids exert significant pressure not just on the bottom, but also on the walls of the container Science, Class VIII. NCERT (Revised ed 2025), Chapter 6: Pressure, Winds, Storms, and Cyclones, p.84.

Pascal’s Law is the foundational principle that describes how this pressure behaves in a closed system. It states that if you apply pressure to any point of an incompressible fluid in a confined space, that pressure change is transmitted equally and undiminished to every single portion of the fluid and to the walls of the container. Since pressure (P) is defined as Force (F) divided by Area (A), or P = F/A, this principle allows us to perform incredible feats of mechanical advantage Science, Class VIII. NCERT (Revised ed 2025), Chapter 6: Pressure, Winds, Storms, and Cyclones, p.83.

The most common application of this is the hydraulic lift. Imagine two pistons connected by a fluid-filled pipe—one small (Area A₁) and one large (Area A₂). If you apply a small force (F₁) to the small piston, the pressure created (P = F₁/A₁) travels through the liquid and hits the large piston. Because the pressure must be the same everywhere, the larger area (A₂) results in a much larger output force (F₂). This is how a person can lift a heavy car using a simple hydraulic jack!

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 6: Pressure, Winds, Storms, and Cyclones, p.83; Science, Class VIII. NCERT (Revised ed 2025), Chapter 6: Pressure, Winds, Storms, and Cyclones, p.84

3. Fluid Dynamics: Viscosity and Surface Tension (intermediate)

In our journey through mechanics, we transition from rigid solids to fluids—substances like liquids and gases that lack a fixed shape and flow to fill their containers. While liquids have a definite volume, their particles are free to move, which gives rise to unique mechanical properties like Viscosity and Surface Tension (Science, Class VIII, NCERT, Particulate Nature of Matter, p.104). These properties are governed by the interparticle forces of attraction acting within the fluid.

Viscosity can be thought of as "internal friction" or a fluid's resistance to flow. Imagine pouring water versus pouring honey; honey is much more viscous because its internal layers resist sliding over one another. This resistance occurs because the molecules exert drag on each other. Interestingly, temperature significantly affects this: for most liquids, as you increase the temperature, viscosity decreases because the particles gain enough energy to overcome those attractive forces. In contrast, Surface Tension is a phenomenon occurring at the surface of a liquid where the molecules are pulled inward by cohesive forces, creating a tension that makes the surface behave like a stretched elastic membrane. This is why small insects can walk on water and why raindrops naturally form spheres—the shape that minimizes surface area.

Feature	Viscosity	Surface Tension
Nature	Resistance to flow/internal friction.	Tension at the liquid's surface.
Cause	Internal friction between fluid layers.	Inward cohesive forces on surface molecules.
Example	Oil flowing slower than water.	A needle floating on water's surface.

We can manipulate these properties for practical use. For instance, soaps and detergents are used to reduce the surface tension of water. By breaking down these cohesive forces, the water can better wet the surface of fabric and surround oil particles to lift them away (Science, Class VIII, NCERT, Particulate Nature of Matter, p.111). Understanding these forces is crucial because they work alongside buoyancy—the upward force exerted by a liquid that determines whether an object floats or sinks (Science, Class VIII, NCERT, Exploring Forces, p.76).

Sources: Science, Class VIII, NCERT, Particulate Nature of Matter, p.104; Science, Class VIII, NCERT, Particulate Nature of Matter, p.111; Science, Class VIII, NCERT, Exploring Forces, p.76

4. Capillary Action and Meniscus (intermediate)

To understand Capillary Action, we must first look at the invisible 'tug-of-war' happening at the molecular level between two forces: Adhesion (the attraction between different types of molecules, like water and glass) and Cohesion (the attraction between similar molecules, like water molecules to each other). When you place a thin tube into a liquid, if the adhesive force between the liquid and the tube wall is stronger than the internal cohesive force, the liquid 'climbs' the walls. This spontaneous movement of a liquid through narrow spaces, often against the pull of gravity, is what we call capillary action. As noted in basic physics, the particles of liquids are free to move and can 'stick' to the walls of a container if the conditions are right Science Class VIII, Particulate Nature of Matter, p.104. This interaction creates a curved surface at the top of the liquid column called a Meniscus. The shape of this curve tells us which force is winning the tug-of-war. If Adhesion > Cohesion, you get a Concave Meniscus (like water in glass), where the liquid edges are pulled upward. If Cohesion > Adhesion, you get a Convex Meniscus (like mercury in glass), where the liquid pulls away from the walls and curves downward. The narrower the tube (or the 'capillary'), the higher the liquid will rise or fall because a higher proportion of the liquid is in contact with the surface relative to its weight. In the context of the UPSC syllabus, this isn't just a laboratory curiosity; it is a critical driver of environmental change. In agricultural geography, we see this in the Indo-Gangetic plains. In regions like western Haryana and Punjab, when the water table rises or irrigation is excessive, capillary action pulls underground water (and the salts dissolved in it) up to the surface. As the water evaporates, it leaves behind saline and alkaline crusts—locally known as kallar, thur, or reh—rendering fertile land useless Geography of India, Agriculture, p.67. Similarly, in arid biomes where evaporation exceeds precipitation, a process called Calcification occurs; capillary action brings calcium-rich compounds upward to form distinct soil layers Environment and Ecology, Major Crops and Cropping Patterns in India, p.103.

Force Type	Interaction	Resulting Meniscus	Example
Adhesion > Cohesion	Liquid attracted to container	Concave (U-shaped)	Water in a glass tube
Cohesion > Adhesion	Liquid attracted to itself	Convex (Dome-shaped)	Mercury in a glass tube

Sources: Science Class VIII, Particulate Nature of Matter, p.104; Geography of India, Agriculture, p.67; Environment and Ecology, Major Crops and Cropping Patterns in India, p.103

5. Introduction to Buoyancy and Upthrust (basic)

Have you ever noticed how you feel significantly lighter when you step into a swimming pool? Or why a massive wooden log can effortlessly float on a lake while a tiny pebble sinks immediately? This happens because of a fundamental physical force called buoyancy.

When any object is placed in a liquid, the liquid applies a force on that object in the upward direction. This upward push is known as upthrust or buoyant force Science, Class VIII. NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.77. You can feel this force yourself by taking an empty plastic bottle and trying to push it down into a bucket of water; the resistance you feel pushing back against your hand is the water exerting its buoyant force Science, Class VIII. NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.76.

The fate of an object in a fluid—whether it sinks or floats—is decided by a "tug-of-war" between two opposing forces:

Force	Direction	Nature
Gravitational Force (Weight)	Downward	Earth pulling the object toward its center.
Buoyant Force (Upthrust)	Upward	The fluid pushing the object toward the surface.

If the gravitational force is greater than the buoyant force, the object sinks. However, if the two forces are equal, the object floats Science, Class VIII. NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.76. While we often think of buoyancy in terms of water, it applies to all fluids, including gases. For example, in our atmosphere, air in a low-pressure cell rises because the surrounding denser air exerts a buoyant force on it, pushing it upward Physical Geography by PMF IAS, Pressure Systems and Wind System, p.306.

One of the most important factors determining the strength of this upward push is the density of the liquid Science, Class VIII. NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.76. A denser liquid (like saltwater) provides more upthrust than a less dense liquid (like freshwater), which is why it is much easier for a person to float in the sea than in a river.

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.76; Science, Class VIII. NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.77; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.306

6. Archimedes' Principle and Its Applications (intermediate)

At its heart, Archimedes' Principle explains the interaction between gravity pulling an object down and a fluid pushing it up. When any object is immersed—whether partially or fully—in a fluid, it experiences an upward buoyant force. This force is exactly equal to the weight of the fluid that the object displaces Science, Class VIII, NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.76. This is the reason you feel "lighter" when you are in a swimming pool; the water is physically supporting a portion of your weight.

Whether an object sinks or floats depends on the balance between its weight and this buoyant force. We can break this down into three simple scenarios:

• Sinking: If the weight of the displaced liquid is less than the weight of the object, the downward force wins, and the object sinks.
• Equilibrium (Floating): If the weight of the displaced liquid is exactly equal to the weight of the object, it floats Science, Class VIII, NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.76.
• Rising: If the buoyant force is greater than the weight (like holding a beach ball underwater), the object will accelerate toward the surface.

For a floating object, we use the Law of Flotation. It tells us that for an object to stay afloat, its total weight must equal the weight of the liquid it displaces. Mathematically, this is expressed through the densities: (Density of Solid) × (Total Volume) = (Density of Liquid) × (Immersed Volume). This explains why a heavy iron ship floats while a small iron nail sinks; the ship's shape is designed to displace a massive volume of water, creating enough buoyant force to match its weight.

Condition	Relative Density	Result
Weight > Buoyant Force	Object is denser than liquid	Sinks to the bottom
Weight = Buoyant Force	Object density ≤ liquid density	Floats at or below surface

Sources: Science, Class VIII, NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.76

7. The Law of Flotation: Floating Equilibrium (exam-level)

When an object is placed in a fluid, two opposing forces come into play: the gravitational force (weight) pulling it downward and the buoyant force pushing it upward. For an object to achieve floating equilibrium, these two forces must be perfectly balanced. According to the Law of Flotation, a floating object displaces a weight of fluid exactly equal to its own weight. This is a specific application of Archimedes' Principle, which helps us understand why a massive steel ship floats while a small pebbles sinks Science, Class VIII (NCERT 2025), Exploring Forces, p.76.

To master this mathematically, we look at the relationship between density and volume. The equilibrium condition can be expressed as: (Density of Solid, ρₛ) × (Total Volume, Vₜₒₜ) = (Density of Liquid, ρₗ) × (Immersed Volume, Vᵢₘ). This formula tells us that the fraction of an object that stays underwater depends entirely on the ratio of the densities of the object and the liquid. For instance, if you see an object floating with only a small portion submerged, you know instinctively that it is much less dense than the liquid it is in Science, Class VIII (NCERT 2025), The Amazing World of Solutes, Solvents, and Solutions, p.150.

Consider three objects of the same size but different weights. If Object A sinks deeper than Object B, it indicates that Object A is heavier and thus more dense, requiring a larger volume of displaced water to generate enough buoyant force to balance its weight Science, Class VIII (NCERT 2025), Exploring Forces, p.79. This comparative density is why oil always sits on top of water; the water is denser and effectively "pushes" the lighter oil upward Science, Class VIII (NCERT 2025), The Amazing World of Solutes, Solvents, and Solutions, p.150.

Sources: Science, Class VIII (NCERT 2025), Exploring Forces, p.76, 79; Science, Class VIII (NCERT 2025), The Amazing World of Solutes, Solvents, and Solutions, p.150

8. Solving the Original PYQ (exam-level)

Now that you have mastered the building blocks of fluid mechanics, this question allows you to apply the Principle of Floatation in a comparative context. The core concept here is that for any floating object, the weight of the liquid displaced is exactly equal to the weight of the object itself. As per Science, Class VIII, NCERT (Revised ed 2025), this equilibrium means that the product of the liquid's density and the submerged volume remains constant for the same solid. In simpler terms, the denser the liquid, the less volume the object needs to displace to stay afloat. This inverse relationship is the secret to solving this problem efficiently without getting lost in complex algebra.

To find the ratio, let's look at the equilibrium state for both liquids. For liquid A, half the volume is immersed, meaning the liquid's density must be twice that of the solid (ρA = 2ρs). For liquid B, two-thirds is immersed, which means the liquid is 1.5 times as dense as the solid (ρB = 1.5ρs). When you set up the ratio of ρA to ρB, you are comparing 2 to 1.5. By multiplying both sides by two to eliminate the decimal, you arrive at the clear ratio of 4 : 3. The logic flows naturally: since liquid A required less immersion (only 1/2) compared to liquid B (2/3) to support the same weight, liquid A must be the denser of the two, which immediately points you toward (D) 4 : 3.

UPSC examiners often include options like (C) 3 : 4 to trap students who mistakenly assume a direct relationship between the immersed volume and the liquid density. If you simply took the ratio of the volumes (1/2 : 2/3), you would end up with 3 : 4, which is the exact opposite of the correct physical principle. Option (B) 3 : 2 is another common error resulting from inverted fractions during the algebraic manipulation. Always perform a quick mental check: the liquid with the smaller immersed volume must have the higher density value in the ratio. Since A has the smaller immersed volume (0.5 vs 0.66), its part of the ratio must be the larger number.