A vessel filled with water is kept on a weighing pan and the scale adjusted to zero. A block of mass $M$ and density $\rho$ is suspended by a massless spring of spring constant $k$. This block is submerged inside into the water in the vessel. What is the reading of the scale?

> [!TLDR] > The weighing scale measures the additional downward force exerted on it. When the block is submerged, it displaces water, and by Newton's third law, the buoyant force exerted by the water on the block corresponds to an equal and opposite downward force exerted by the block on the water. This downward force is transmitted to the scale. The reading is the weight of the displaced water, which is $\boxed{\frac{M\rho_w g}{\rho}}$. A weighing scale measures the normal force exerted on it by the object (or system) placed upon it. When the scale is adjusted to zero with the vessel and water on it, it means it will display any *additional* downward force applied to it. The core concept here is Archimedes' Principle combined with Newton's Third Law. 1. **Archimedes' Principle:** When an object is submerged in a fluid, it experiences an upward buoyant force equal to the weight of the fluid it displaces. 2. **Newton's Third Law:** If the fluid exerts an upward buoyant force on the object, then the object exerts an equal and opposite (downward) force on the fluid. This downward force on the fluid is what ultimately contributes to the scale reading. The spring's role is simply to suspend the block. Since the spring is not resting on the weighing pan, the force it exerts on the block (and thus the block's weight $Mg$) does not directly contribute to the scale reading. The scale only "feels" what is directly on it or what exerts a force on the water within the vessel. Let's break down the problem: 1. **Identify the system being weighed:** The weighing pan measures the total downward force exerted by the "vessel + water" system. Since the scale was adjusted to zero initially, we are interested in the *change* in this downward force when the block is submerged. 2. **Calculate the volume of the block and the buoyant force:** The block has mass $M$ and density $\rho$. The volume of the block, $V$, is given by: $$V = \frac{\text{Mass}}{\text{Density

A vessel filled with water is kept on a weighing pan and the scale adjusted to zero. A block of mass M and den

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Concept understanding — Archimedes Principle

Archimedes' Principle

Push a beach ball underwater in a swimming pool and feel it fight back — the deeper it goes, the harder it pushes up. Or try lifting a heavy stone while it's submerged: it feels far lighter than it does on land. Something about being surrounded by water makes objects seem to lose weight. That "something" is Archimedes' Principle.

The Intuition

A fluid is made of molecules pressing on everything around them, and pressure increases with depth. So the fluid beneath a submerged object pushes up on it harder than the fluid above pushes down — because there is more fluid weighing down from above onto the fluid below. This net upward push is the buoyant force.

The key insight: this upward force doesn't depend on the object's shape or material. It depends only on how much fluid the object displaces — the volume of fluid it pushes out of the way to make room for itself.

Note

Displaced fluid = the volume of the object that is submerged. Fully dunk a 1-litre bottle, and you displace 1 litre of water.

The Precise Statement

F_b = \rho_{\text{fluid}} \, V_{\text{submerged}} \, g

where $F_b$ is the buoyant force, $\rho_{\text{fluid}}$ is the fluid's density, $V_{\text{submerged}}$ is the submerged volume of the object, and $g$ is the acceleration due to gravity.

Archimedes' Principle: any object, wholly or partly immersed in a fluid, experiences an upward buoyant force equal to the weight of the fluid it displaces.

Why This Matters

Observation	What's happening
A ship floats	Its hull shape displaces a huge volume of water. The weight of that displaced water equals the ship's own weight, so it floats.
A stone feels lighter underwater	The water pushes up with a force equal to the weight of water it displaces, partly cancelling gravity.
A helium balloon rises	The balloon displaces air; the weight of that displaced air exceeds the weight of the helium inside, giving a net upward force.

What Decides Sink or Float: Density

Object density greater than fluid density $\Rightarrow$ it sinks (weight exceeds buoyant force).
Object density less than fluid density $\Rightarrow$ it floats (weight is less than buoyant force).
Object density equal to fluid density $\Rightarrow$ it hovers at any depth. …

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The weighing scale measures the additional downward force exerted on it. When the block is submerged, it displaces water, and by Newton's third law, the buoyant force exerted by the water on the block corresponds to an equal and opposite downward force exerted by the block on the water. This downward force is transmitted to the scale. The reading is the weight of the displaced water, which is $\boxed{\frac{M\rho_w g}{\rho}}$ .

A weighing scale measures the normal force exerted on it by the object (or system) placed upon it. When the scale is adjusted to zero with the vessel and water on it, it means it will display any additional downward force applied to it.

The core concept here is Archimedes' Principle combined with Newton's Third Law.

Archimedes' Principle: When an object is submerged in a fluid, it experiences an upward buoyant force equal to the weight of the fluid it displaces.
Newton's Third Law: If the fluid exerts an upward buoyant force on the object, then the object exerts an equal and opposite (downward) force on the fluid. This downward force on the fluid is what ultimately contributes to the scale reading.

The spring's role is simply to suspend the block. Since the spring is not resting on the weighing pan, the force it exerts on the block (and thus the block's weight $Mg$ ) does not directly contribute to the scale reading. The scale only "feels" what is directly on it or what exerts a force on the water within the vessel.

Let's break down the problem:

Identify the system being weighed:

The weighing pan measures the total downward force exerted by the "vessel + water" system. Since the scale was adjusted to zero initially, we are interested in the change in this downward force when the block is submerged.
Calculate the volume of the block and the buoyant force:

The block has mass $M$ and density $\rho$ .

The volume of the block, $V$ , is given by:

V = \frac{\text{Mass}}{\text{Density}} = \frac{M}{\rho}

When the block is submerged in water (let's denote the density of water as $\rho_w$), it displaces a volume of water equal to its own volume $V$.
According to Archimedes' Principle, the upward buoyant force ($F_B$) exerted by the water on the block is the weight of the displaced water:

F_B = V \rho_w g

Substituting the expression for $V$:

F_B = \left(\frac{M}{\rho}\right) \rho_w g

Determine the force exerted by the block on the water: By Newton's Third Law, if the water exerts an upward buoyant force $F_B$ on the block, then the block exerts an equal and opposite downward force on the water. This downward force is also $F_B$ . …

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Q.A vessel filled with water is kept on a weighing pan and the scale adjusted to zero. A block of mass MMM and density ρ\rhoρ is suspended by a massless spring of spring constant kkk. This block is submerged inside into the water in the vessel. What is the reading of the scale?

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Q.A vessel filled with water is kept on a weighing pan and the scale adjusted to zero. A block of mass $M$ and density $\rho$ is suspended by a massless spring of spring constant $k$ . This block is submerged inside into the water in the vessel. What is the reading of the scale?

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