Two identical solid pieces, one of gold and other of silver, when immersed completely in water exhibit equal weights. When weighed in air (given that density of gold is greater than that of silver)

1. Foundations: Mass, Weight, and Density (basic)

Welcome to your first step in mastering mechanics! To understand how objects move and interact, we must first distinguish between three fundamental concepts: mass, weight, and density. In everyday conversation, we often use 'mass' and 'weight' as if they mean the same thing, but in the world of physics—and for your UPSC preparation—keeping them distinct is crucial.

Mass is the measure of the actual quantity of matter present in an object Science, Class VIII, NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.142. It is an intrinsic property, meaning it does not change regardless of where the object is located in the universe. Whether you are on Earth, the Moon, or floating in deep space, your mass remains constant. Its SI unit is the kilogram (kg).

Weight, however, is a force. Specifically, it is the gravitational force with which a planet (like Earth) pulls an object toward its center Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.75. Because gravity varies depending on your location—for instance, the Moon's gravity is about 1/6th of Earth's—your weight changes depending on where you are, even though your mass stays the same. Weight is measured in Newtons (N).

Feature	Mass	Weight
Definition	Quantity of matter in an object	Gravitational pull on an object
SI Unit	Kilogram (kg)	Newton (N)
Constancy	Constant everywhere	Changes with gravity/location

Finally, we have Density. Think of density as how "tightly packed" the matter is within a specific space. It is defined as the mass present in a 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. This helps us understand why a small piece of lead might be much heavier than a large piece of cork; the lead has a much higher density, meaning more mass is squeezed into a smaller volume.

Sources: Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.75; Science, Class VIII, NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.140-142

2. Understanding Buoyancy and Upthrust (basic)

When you try to push an empty plastic bottle into a bucket of water, you feel a distinct resistance pushing back against your hand. If you release the bottle, it bounces back to the surface. This everyday observation reveals a fundamental principle of physics: all liquids exert an upward force on objects immersed in them. This upward push is known as buoyancy or upthrust Science, Class VIII (NCERT 2025), Chapter 5, p.76. While we often notice this in water, it is a property of all fluids, including the air around us. For instance, the surrounding atmosphere exerts a buoyant force on low-pressure air cells, causing them to rise Physical Geography by PMF IAS, Pressure Systems and Wind System, p.306.

Whether an object sinks or floats depends on the "tug-of-war" between two forces: the gravitational force (weight) pulling the object down and the buoyant force pushing it up. To understand how heavy an object feels when submerged, we use the concept of Apparent Weight. When an object is in a liquid, it appears lighter because the upthrust partially cancels out its real weight. We can express this relationship simply:

Apparent Weight = Real Weight − Buoyant Force

The magnitude of this buoyant force depends on two primary factors: the volume of the object submerged (which determines how much liquid is displaced) and the density of the liquid Science, Class VIII (NCERT 2025), Chapter 5, p.76. If two objects have the exact same volume and are placed in the same liquid, they will experience the exact same buoyant force, regardless of what they are made of or how much they weigh in the air.

Scenario	Result
Gravitational Force > Buoyant Force	The object sinks
Gravitational Force = Buoyant Force	The object floats (neutral buoyancy)

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

3. Archimedes' Principle (intermediate)

Welcome to one of the most famous "Eureka!" moments in science history. Archimedes' Principle explains why heavy steel ships float while a small pebble sinks. At its core, the principle states that when an object is fully or partially immersed in a liquid, it experiences an upward force—called buoyant force or upthrust—that is exactly equal to the weight of the liquid the object displaces Science, Class VIII, Chapter 5: Exploring Forces, p.76. Think of it as a "space tax": if you want to occupy space inside a bucket of water, you must push some water out of the way, and that displaced water pushes back on you with a force equal to its own weight.

Whether an object sinks or floats depends on a vertical tug-of-war between two forces: gravitational force (weight) pulling the object down and buoyant force pushing it up Science, Class VIII, Chapter 5: Exploring Forces, p.76. If the weight of the displaced liquid is less than the object's actual weight, the object sinks. However, if the object can displace a weight of liquid equal to its own weight, it will float. This is why objects feel lighter when submerged; we call this the apparent weight. Mathematically, Apparent Weight = Actual Weight - Buoyant Force.

It is crucial to understand that the buoyant force depends on the volume of the submerged part of the object and the density of the liquid, rather than the mass of the object itself Science, Class VIII, Chapter 5: Exploring Forces, p.76. For example, if you have two spheres of the exact same size—one made of solid gold and one made of hollow plastic—they will both experience the exact same buoyant force when fully submerged because they both displace the same volume of water. However, the gold sphere sinks because its weight is much higher than that buoyant force, while the plastic sphere might fly to the surface.

Sources: Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.76; Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.77

4. Relative Density and Material Properties (intermediate)

To understand why objects feel lighter in water, we must first look at Density—the measure of how much mass is packed into a specific volume. For instance, Gold is significantly denser than Silver. This means that if you have a cube of gold and a cube of silver of the exact same size (volume), the gold cube will be much heavier in air because its atoms are more tightly packed.

When these objects are submerged in a liquid, they encounter Archimedes' Principle. This principle states that any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. Crucially, if two objects have the same volume, they displace the same amount of water and therefore experience the exact same buoyant force, regardless of what material they are made of.

Material Property	Gold (Au)	Silver (Ag)
Relative Density	Very High (~19.3)	High (~10.5)
Durability/Softness	Pure (24k) is very soft; usually alloyed to 22k for jewelry.	Lustrous white metal; second to gold in value.
Chemical Reactivity	Highly resistant to corrosion; found in free state.	Resistant to acetic acid but tarnishes with sulfur.

Science, Class X (NCERT 2025 ed.), Metals and Non-metals, p.50, 54

The concept of Apparent Weight is the final piece of the puzzle. It is calculated as:
Apparent Weight = Real Weight (in air) - Buoyant Force.
If two "identical" looking pieces (same volume) of gold and silver are weighed underwater and found to have the same apparent weight, it leads to a fascinating logical conclusion: since their buoyant forces are identical, their real weights in air must also have been identical. This would only happen if the silver piece was actually larger or if the gold piece was hollow, because solid gold is naturally much heavier than solid silver for the same volume Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.76.

Sources: Science, Class X (NCERT 2025 ed.), Metals and Non-metals, p.50, 54; Science, Class VIII (NCERT 2025 ed.), Exploring Forces, p.76; Environment and Ecology, Majid Hussain, Distribution of World Natural Resources, p.34

5. Laws of Floatation and Equilibrium (intermediate)

When you submerge an object in a fluid—whether it's a bucket of water or the air around us—it feels lighter. This isn't an illusion; it's the result of a physical struggle between two opposing forces: the downward pull of Gravity and the upward push of Buoyancy. As noted in Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.76, if the gravitational force (weight) is greater than the buoyant force, the object sinks. However, if these two forces reach an equilibrium where they are equal, the object floats.

The secret to calculating this upward push lies in Archimedes’ Principle. It states that the buoyant force acting on an object is exactly equal to the weight of the fluid that the object displaces. This leads us to the concept of Apparent Weight. When an object is in water, its weight appears to decrease because the water is "carrying" part of the load. The formula is simple: Apparent Weight = Real Weight (in air) – Buoyant Force. If two objects have the exact same volume, they will displace the exact same amount of water and thus experience the identical buoyant force, regardless of what they are made of.

Whether an object ultimately sinks or floats depends heavily on its Density—defined as the mass per unit volume (Science, Class VIII. NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.140). We can summarize the laws of floatation and equilibrium based on the relationship between the density of the object (ρ_obj) and the density of the liquid (ρ_liq):

Condition	Result	Equilibrium State
Weight > Buoyant Force (ρ_obj > ρ_liq)	Sinks	Unstable (falls to bottom)
Weight = Buoyant Force (ρ_obj = ρ_liq)	Floats (Fully Submerged)	Neutral Equilibrium
Weight < Buoyant Force (ρ_obj < ρ_liq)	Floats (Partially Submerged)	Stable Equilibrium

In practical terms, this is why oil floats on water; as highlighted in Science, Class VIII. NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.150, water is denser than oil, meaning the buoyant force provided by water easily overcomes the weight of the oil.

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, 150

6. Apparent Weight Concept (exam-level)

When you lift a stone underwater, it feels significantly lighter than it does in the air. This isn't because the stone has lost mass, but because of a phenomenon called apparent weight. In physics, weight is the measure of the gravitational pull on an object. However, when an object is immersed in a fluid (like water), it experiences two opposing vertical forces: the gravitational force acting downwards and the buoyant force acting upwards Science, Class VIII, Chapter 5, p.76. The weight we perceive or measure with a scale while the object is submerged is the 'apparent weight'.

The relationship is defined by a simple principle: Apparent Weight = Real Weight (in air) - Buoyant Force. According to Archimedes' Principle, the buoyant force is exactly equal to the weight of the liquid displaced by the object Science, Class VIII, Chapter 5, p.76. Therefore, the more volume an object has, the more water it displaces, and the greater the upward 'push' (buoyant force) it receives. This explains why a heavy ship floats while a small pebbles sinks; the ship is designed to displace a weight of water equal to its own massive weight.

To visualize how these forces interact, consider the following scenarios for a fully submerged object:

Condition	Result	Apparent Weight Value
Weight > Buoyant Force	Object Sinks	Positive (but less than real weight)
Weight = Buoyant Force	Object Floats/Neutral Buoyancy	Zero
Weight < Buoyant Force	Object Rises to Surface	Negative (until it reaches equilibrium at the surface)

It is important to remember that buoyant force depends on the volume of the submerged part of the object and the density of the liquid, not on the mass of the object itself. Two objects of the same size (volume) made of different materials—say, one of lead and one of aluminum—will experience the exact same upward buoyant force when submerged, even though their real weights are very different.

Sources: Science, Class VIII (NCERT 2025), Chapter 5: Exploring Forces, p.74; Science, Class VIII (NCERT 2025), Chapter 5: Exploring Forces, p.76

7. Solving the Original PYQ (exam-level)

To solve this, we must synthesize two core concepts you have just mastered: Archimedes’ Principle and the formula for Apparent Weight. Recall that the weight of an object measured while submerged is its actual weight minus the buoyant force (the upward thrust). The question tells us that the weight in water is equal for both pieces. Mathematically, this means: Weight in Air = Weight in Water + Buoyant Force. Since the 'Weight in Water' is identical for both, the piece that experiences a greater buoyant force must necessarily have a greater weight in air.

Now, let’s apply the reasoning regarding density. We are given that gold is denser than silver. For two objects to exhibit the same weight under water despite this density difference, the less dense object (silver) must occupy a larger volume to displace more water. According to Science, Class VIII. NCERT (Revised ed 2025), the buoyant force is equal to the weight of the liquid displaced. Therefore, the larger volume of the silver piece results in a larger buoyant force. To maintain the equilibrium where submerged weights are equal, the silver piece will weigh more in air to compensate for that extra upward push.

UPSC often includes the word "identical" as a linguistic trap to tempt you toward Option (C), making you think the masses must be the same. Another common pitfall is the density trap (Option A), where students reflexively choose gold because it is "heavier" (denser) by nature. However, the key is focusing on the displaced fluid. Always remember: if the weights in water are equal, the object with the lower density must have a larger volume and, consequently, a higher real weight to offset the increased buoyancy.