Animated Solution for Physics - Properties of Solids and Liquids: A boat floating in a water tank is carrying a number of large stones. If the stones are unloaded into water, what will happen to the water level?

Imagine you are sitting in a boat floating in a quiet swimming pool, and the boat is loaded with heavy stones.

If you pick up those stones and throw them overboard into the water, what happens to the water level of the pool?

Does it rise, fall, or remain exactly the same?

This classic physics puzzle, which has intrigued students for generations, is a beautiful application of Archimedes' Principle and the concept of buoyancy.

Let's dive deep into the physics to understand why our intuition often misleads us, and how a rigorous mathematical approach reveals the elegant truth.

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Stones in the Boat

Let's define our variables clearly: - Let the mass of the boat be $M_b$. - Let the mass of the stones be $M_s$. - Let the density of water be ${\rho }_w$. - Let the density of the stones be ${\rho }_s$.

In the first state, the stones are inside the boat, and the entire system is floating.

According to Archimedes' Principle, any floating object displaces a volume of fluid whose weight is exactly equal to the weight of the floating object.

Where $V_1$ is the volume of water displaced in this initial state.

Solving for $V_1$, we get:

This equation tells us that when the stones are floating inside the boat, they displace water based on their weight.

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Stones in the Water

Now, let's throw the stones into the water.

Since the stones are made of a material denser than water (${\rho }_s>{\rho }_w$), they cannot float on their own and will sink to the bottom of the tank.

The boat, now lighter, continues to float on the surface, supporting only its own mass $M_b$.

In this state, the total volume of water displaced, $V_2$, is the sum of two separate parts: 1. The volume of water displaced by the floating boat: 2. The physical volume of the submerged stones resting at the bottom:

Therefore, the total displaced volume in the second state is:

Notice the crucial difference: while submerged, the stones displace water equal to their physical volume, not their weight!

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To find out whether the water level rises or falls, we simply need to compare $V_1$ and $V_2$ by subtracting them:

Notice how the boat's term $\frac{M_b}{{\rho }_w}$ cancels out perfectly, leaving us with:

Since the stones sink in water, we know that:

This inequality guarantees that the term inside the parentheses is strictly positive:

Because the volume of water displaced in the first state ($V_1$) is greater than the volume of water displaced in the second state ($V_2$), the total volume of displaced water decreases when the stones are thrown into the tank.

Consequently, the water level in the tank must fall.