A vessel contains oil (density pj) over a liquid of density p2; a homogeneous sphere of volume V floats with half of its volume immersed in the liquid and the other half in oil. The weight of the sphere is

1. Understanding Density and Relative Density (basic)

Welcome to the first step of your journey into mechanics! To understand why some things float and others sink, we must first master the concept of Density. Simply put, density is the mass present in a unit volume of a substance Science, Class VIII, Chapter: The Amazing World of Solutes, Solvents, and Solutions, p.140. You can think of it as a measure of how tightly packed the particles of a substance are. Mathematically, it is expressed as:

Density = Mass / Volume

It is important to remember that density is an intrinsic property; it doesn't change based on the shape or size of the object. A small iron nail has the same density as a massive iron pillar. However, density is affected by temperature. When a substance is heated, its particles move apart and spread out, causing the volume to increase while the mass stays the same. Because the volume (the denominator in our formula) increases, the overall density decreases upon heating Science, Class VIII, Chapter: The Amazing World of Solutes, Solvents, and Solutions, p.147. This explains why hot air rises—it becomes less dense than the cooler air around it.

Often, scientists prefer to compare the density of a substance to a standard reference, usually water. This comparison is called Relative Density. It is defined as the ratio of the density of a substance to the density of water at a given temperature Science, Class VIII, Chapter: The Amazing World of Solutes, Solvents, and Solutions, p.141. Since it is a ratio of two similar quantities, Relative Density has no units—it is just a pure number. For instance, if the relative density of a material is 2.7, it tells us that the material is 2.7 times denser than water.

Substance	Density (approx)	Relative Density
Water	1 g/cm³	1.0
Aluminium	2.7 g/cm³	2.7
Gold	19.3 g/cm³	19.3

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.147

2. Archimedes' Principle and Upthrust (basic)

Have you ever noticed how you feel lighter when you step into a swimming pool? Or why a heavy iron ship floats while a small iron nail sinks? To understand this, we need to explore two powerful concepts: Upthrust and Archimedes' Principle.

When any object is placed in a liquid, it faces a "tug-of-war" between two forces. Gravity pulls the object downward, while the liquid exerts an upward force to push it out. This upward force is called upthrust or buoyant force Science, Class VIII NCERT, Chapter 5, p. 77. The strength of this upward push isn't random; it follows a specific rule discovered by the Greek scientist Archimedes. He stated that the upward force acting on an object is exactly equal to the weight of the liquid the object displaces Science, Class VIII NCERT, Chapter 5, p. 76.

Think of it this way: if you submerge a ball in a bucket full to the brim, some water will spill out. If you weigh that spilled water, that weight is the exact amount of upward force pushing on the ball. This leads to a simple rule for whether things sink or float:

Scenario	Force Comparison	Result
Weight > Upthrust	The object is heavier than the water it can push aside.	The object sinks.
Weight = Upthrust	The object weighs exactly the same as the water it displaces.	The object floats (at equilibrium).

The amount of upthrust depends on two main things: the volume of the object submerged and the density of the liquid Science, Class VIII NCERT, Chapter 5, p. 76. This is why it is easier to float in the salty Dead Sea than in a freshwater lake—salty water is denser, so the weight of the displaced "salty" water (the upthrust) is greater!

Sources: Science, Class VIII NCERT, Chapter 5: Exploring Forces, p.76; Science, Class VIII NCERT, Chapter 5: Exploring Forces, p.77

3. The Law of Floatation (intermediate)

To understand why a massive steel ship floats while a tiny pebble sinks, we must look at the Law of Floatation, which is a specific application of Archimedes' Principle. At its heart, floatation is a balance of two opposing forces: the gravitational force pulling the object down and the buoyant force (upthrust) pushing it up. As noted in Science, Class VIII. NCERT (Revised ed 2025), Chapter 5, p. 76, an object floats when these two forces are equal. If the downward pull of gravity is stronger than the maximum upward push the liquid can provide, the object sinks.

The buoyant force isn't random; it is exactly equal to the weight of the fluid displaced by the submerged part of the object. This brings us to the crucial role of density (Mass/Volume). For an object to float, its average density must be less than or equal to the density of the fluid. If you have two immiscible liquids (liquids that don't mix, like oil and water), they will stack based on density, with the less dense liquid on top Science, Class VIII. NCERT (Revised ed 2025), Chapter 10, p. 150. If an object is suspended between these two layers, it experiences a "shared" buoyancy.

In a scenario where a sphere is submerged exactly halfway in oil (density ρ₁) and halfway in a denser liquid (density ρ₂), the total upward buoyant force is the sum of the weights of the two different fluids it has displaced. Mathematically, if the total volume is V, the buoyant force from the oil is (V/2)ρ₁g and from the lower liquid is (V/2)ρ₂g. For the sphere to be in equilibrium (floating steadily), its total weight must equal the sum of these two forces: Weight = V(ρ₁ + ρ₂)g / 2.

Scenario	Force Relation	Result
Weight > Buoyant Force	Downward force wins	Sinks to the bottom
Weight = Buoyant Force	Forces are balanced	Floats (at surface or submerged)

Sources: Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.76; 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.150

4. Surface Tension and Capillarity (intermediate)

At the heart of why liquids behave the way they do is the interparticle force of attraction. In the bulk of a liquid, a molecule is pulled equally in all directions by its neighbors. However, at the surface, there are no liquid molecules above to balance the pull from below. This creates a net inward force, causing the surface to behave like a stretched elastic membrane. This phenomenon is known as Surface Tension. It explains why raindrops are spherical (minimizing surface area) and why certain insects can walk on water. Interestingly, we can manipulate this tension; for instance, soap particles reduce surface tension, allowing water to spread better and lift oil stains off fabric, as seen in the cleaning process where soap attaches to oil at one end and water at the other Science, Class VIII, Particulate Nature of Matter, p.111.

When we introduce a very narrow tube (a capillary) into a liquid, we observe Capillary Action—the spontaneous rise or fall of the liquid. This is a tug-of-war between two forces: Adhesion (attraction between unlike molecules, like water and glass) and Cohesion (attraction between like molecules). If adhesion is stronger than cohesion, the liquid 'climbs' the walls. We see a practical application of this in biology, where the smallest vessels, called capillaries, have walls only one-cell thick to facilitate the exchange of materials Science, Class X, Life Processes, p.93. While biological transport involves complex pressure gradients, the physical principle of capillarity ensures fluids can move effectively through such minute spaces.

Understanding these forces also explains why liquids 'stick' to containers. If a container is not perfectly clean, adhesive forces cause some water to cling to the walls, which can slightly alter volume measurements when pouring from one vessel to another Science, Class VIII, Particulate Nature of Matter, p.104. In nature, this principle is vital for plants to transport water from roots to leaves and for the distribution of tissue fluid or lymph through intercellular spaces Science, Class X, Life Processes, p.94.

Force Type	Description	Effect in Capillarity
Cohesion	Attraction between similar molecules.	Tries to keep the liquid together (pulls inward).
Adhesion	Attraction between different molecules.	Tries to pull the liquid along the surface of the solid.

Sources: Science, Class VIII, Particulate Nature of Matter, p.111; Science, Class VIII, Particulate Nature of Matter, p.104; Science, Class X, Life Processes, p.93; Science, Class X, Life Processes, p.94

5. Fluid Dynamics: Bernoulli’s Principle and Viscosity (exam-level)

Fluid Dynamics is the study of how liquids and gases behave when they are in motion. To understand this, we must first look at the internal friction of a fluid, known as Viscosity. Imagine pouring honey versus pouring water; honey flows slowly because it has high viscosity. This resistance occurs because the particles in a liquid, while free to move and take the shape of their container, still experience interparticle forces of attraction that resist sliding past one another Science, Class VIII (NCERT), Particulate Nature of Matter, p. 104. In a practical sense, viscosity acts like mechanical friction between the layers of the fluid.

When a fluid is in steady, horizontal flow, we look to Bernoulli’s Principle, which is essentially the Law of Conservation of Energy applied to fluids. It states that as the speed of a moving fluid increases, the pressure within that fluid decreases. This is why the roof of a house can be blown off during a high-wind storm: the fast-moving air above the roof creates a low-pressure zone, while the stagnant, higher-pressure air inside the house pushes the roof upward. This principle is fundamental to aviation, explaining how the shape of a wing generates "lift."

Before a fluid moves, it exerts Buoyancy. According to Archimedes' Principle, any object submerged in a fluid experiences an upward force equal to the weight of the fluid it displaces Science, Class VIII (NCERT), Chapter 5, p. 76. If an object is floating in equilibrium between two immiscible liquids (like oil floating on water), the total buoyant force is simply the sum of the weights of the two different fluids displaced. This principle of equilibrium is the reason why heavy ships made of steel can float—they are designed to displace a weight of water equal to their own massive weight.

Concept	Primary Driver	Effect on Fluid Behavior
Viscosity	Internal Friction	Determines the resistance to flow (thickness).
Bernoulli’s	Energy Conservation	Higher velocity leads to lower internal pressure.
Archimedes'	Displacement	Upward force equals weight of displaced fluid.

Sources: Science, Class VIII (NCERT), Particulate Nature of Matter, p.104; Science, Class VIII (NCERT), Chapter 5: Exploring Forces, p.76

6. Buoyancy in Immiscible Layered Liquids (exam-level)

When we study buoyancy, we often imagine an object in a single fluid, like a ship in the ocean. However, in many real-world and exam scenarios, an object might be submerged in immiscible layered liquids—liquids that do not mix, such as oil floating on water. Because oil is less dense than water, it sits on top Science, Class VIII. NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.140. When an object is placed in such a system, it experiences an upward force, known as upthrust or buoyant force, from every liquid layer it displaces.

According to Archimedes' Principle, the buoyant force is equal to the weight of the fluid displaced. In a layered system, the total buoyant force is simply the sum of the buoyant forces exerted by each individual layer. If a sphere of volume V is floating such that half its volume (V/2) is in the top layer (oil with density ρ₁) and the other half (V/2) is in the bottom layer (liquid with density ρ₂), the total upthrust is calculated by adding the weight of the oil displaced to the weight of the liquid displaced: (V/2)ρ₁g + (V/2)ρ₂g.

Scenario	Source of Buoyancy	Total Buoyant Force (Fb)
Single Liquid	One fluid displaced	Vdisplaced × ρ × g
Layered Liquids	Multiple fluids displaced	Σ (Vn × ρn × g)

For an object to float in equilibrium within these layers, the downward gravitational force (its weight) must be exactly balanced by this combined upward buoyant force Science, Class VIII. NCERT, Exploring Forces, p.76. This explains why an object might sink through the oil but stop and float at the boundary of the denser liquid—it finds a position where the sum of the displaced weights equals its own weight.

Sources: Science, Class VIII. NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.140; Science, Class VIII. NCERT, Exploring Forces, p.76; Science, Class VIII. NCERT, Exploring Forces, p.77

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamentals of fluid mechanics, this question serves as a perfect application of Archimedes' Principle and the concept of Static Equilibrium. In your lessons, you learned that for any object to float, the downward force of its weight must be exactly balanced by the upward buoyant force. The key takeaway here is that buoyancy is additive; when an object is submerged in multiple layers of immiscible fluids, the total upward thrust is simply the sum of the weights of the individual fluids displaced by the object's respective volumes in each layer.

To solve this, we analyze the sphere in two parts. First, the bottom half (volume V/2) displaces the denser liquid, contributing a buoyant force of (V/2)ρ₂g. Simultaneously, the top half (volume V/2) displaces the oil, contributing (V/2)ρ₁g. By summing these two forces to find the total buoyancy—which must equal the sphere's weight for it to float—we factor out the common terms to arrive at the correct expression: V(ρ₂ + ρ₁)g/2. This logic mirrors the investigative approach found in NCERT Science, Class VIII (Revised ed 2025), where we learn that displacement is the heart of flotation.

When tackling UPSC science questions, be wary of common "distractor" patterns seen in the other options. Option (A) uses a subtraction sign, a trap for students who confuse buoyant force with apparent weight or pressure differences. Option (C) conveniently ignores the fact that only half of the volume is in each liquid, a mistake made when one rushes and applies the total volume V to both densities. Finally, Option (D) often includes typographical variations to test your attention to detail regarding the specific variables (like ρ₁ vs ρs) defined in the problem. Always remember: Total Buoyancy = Weight of Displaced Fluid 1 + Weight of Displaced Fluid 2.