Question

You are in a boat filled with large rocks in the middle of a small pond. You begin to drop the rocks into the water. What happens to the water level of the pond? a) It rises. d) It rises momentarily and then b) It falls. falls when the rocks hit bottom. c) It doesn't change. e) There is not enough information to say.

Short answer

Answer: b) It falls.

Step-by-step solution

Analyze the initial situation

Initially, there is a boat filled with large rocks in the middle of a pond. The boat is floating, meaning that its weight is balanced by the buoyant force of the water.

Understand the concept of buoyant force

Buoyant force is the upward force exerted by a fluid (in this case, water) on a submerged object. It depends on the volume of water displaced by the object. According to Archimedes' principle, the magnitude of this force is equal to the weight of the water displaced by the object.

Consider the weight of the boat and rocks

When the boat is filled with rocks, its weight and the weight of the rocks are supported by the buoyant force from the water. This means that the volume of water displaced by the boat is equal to the total weight of the boat and rocks divided by the density of water.

Evaluate the effect of dropping rocks into the water

When a rock is dropped into the pond, it sinks to the bottom, displacing the water around it and creating a new buoyant force. The volume of water displaced by the rock is equal to the weight of the rock divided by the density of water.

Compare the displacements

Now, we need to compare the volume of water displaced by the rock when it is in the boat to the volume of water displaced when the rock is at the bottom of the pond. If these volumes are equal, then the water level will not change. If the volume displaced by the rock in the boat is greater than when the rock is at the bottom, the water level will fall. If the volume displaced in the boat is smaller, the water level will rise.

Determine the displacements

When the rock is in the boat, it contributes to the weight of the boat, and the volume of water displaced is equal to the total weight (boat and rocks) divided by the density of water. When the rock is at the bottom, it only displaces an amount of water equal to its own volume, as its weight no longer contributes to the buoyant force on the boat. Because rocks are typically denser than water, they displace a smaller volume of water when they are at the bottom of the pond compared to their contribution to the displacement of the boat when they are on it.

Conclude the effect on water level

Since the volume of water displaced by the rock is greater when it is in the boat than when it is at the bottom of the pond, the water level will fall when the rocks are dropped into the water. The correct answer is: b) It falls.

Key concepts

Buoyant Force

When you place an object in water, it feels lighter, doesn't it? This is because of a force called the buoyant force. Archimedes' principle tells us all about it. It states that the buoyant force on an object submerged in a fluid (like water) is equal to the weight of the fluid that is displaced by the object. So, if a boat is floating on a pond with rocks inside it, the weight of the boat and rocks is balanced by the buoyant force.
The buoyant force is why boats and even massive ships can float without sinking. It depends on two main factors:

• The volume of water displaced by the object.
• The density of the fluid, in this case, water.

Whenever you drop a rock into the water from the boat, the situation changes. The rock sinks, and the way it displaces water changes, affecting the buoyant balance.

Displaced Volume

Displaced volume is a key element in understanding buoyancy. When an object is submerged in a fluid, it pushes the fluid out of the way, making space for itself. This "pushed aside" fluid is what we call the displaced volume. The displaced volume will directly influence the buoyant force experienced by the object.
For our exercise, when rocks are placed on the boat, they don't sink, so they displace water as part of the boat's overall displacement. But once they hit the pond’s bottom, they act on their own, sinking and displacing only their volume, independently of the boat.
The crucial distinction here is that rocks, with their typically high density, displace less volume of water when they are at the bottom of the pond compared to when they are on the floating boat. This is because their weight impacts the water displacement differently according to their position.

Density of Water

The density of water might seem like a minor detail, but it carries immense importance in buoyancy calculations. The density is the mass of water per unit volume, often approximated as 1000 kg/m³ for pure water at standard temperature and pressure. It acts as a key factor for determining how much water an object will displace.
From Archimedes’ principle, we know that the buoyant force is the product of the density of the fluid, the gravitational acceleration, and the volume of fluid displaced. Therefore, the greater the density of water, the greater the buoyant force on an object would be, assuming the displaced volume stays constant.
In the rock and boat scenario, understanding the density helps explain why rocks displacing their volume of incredibly dense material like water results in less displacement compared to when they contribute to the boat's weight while floating. This difference is why the water level in the pond falls when the rocks are removed from the boat—which was displacing more water than they do on their own when they sink.