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 falls when b) It falls. 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

Initial State

The boat with the rocks is floating in the pond, so the weight of the boat and rocks equals the weight of the water displaced. Let's call the volume of water displaced by the boat with the rocks V1.

Final State

When the rocks are dropped into the pond, they sink and displace a certain volume of water. Let's call the volume of water displaced by the rocks V2. The boat now has a lesser weight, so it will also displace a lesser volume of water. Let's call the volume displaced by the boat after the rocks are dropped V3.

Comparing Volumes

Since we are interested in comparing the total initial displacement (V1) to the final displacement (V2 + V3), we can write this equation: V1 = V2 + V3 Now, it's important to consider the rocks' volume and density. If the rocks are denser than water, then the volume of water they displace when sinking (V2) will be less than when they were in the boat. Therefore, the total volume of water displaced by the boat (V3) will also decrease, causing the water level of the pond to drop.

Answer

As a result, the water level of the pond will fall (option b: It falls).

Key concepts

Archimedes' Principle

Understanding Archimedes' principle is essential when analyzing why objects float or sink in water. This principle states that an object submerged in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. This is exactly what occurs when the rocks are still inside the boat in our given exercise. The force that counteracts gravity and keeps the boat afloat is equivalent to the weight of the water volume displaced by the boat and rocks together, known as V1.

In practice, this means that if you submerge an object in water, it will experience an upward force that can lead to floating if this force is greater than the weight of the object. However, factors like the object's density, which we will discuss in the next section, play a crucial role in determining whether it will remain afloat or sink to the bottom. Archimedes' principle is not only a theoretical concept but also a real-world phenomenon that affects everything from ships sailing in the ocean to the ice cubes bobbing in your glass of lemonade.

Density and Buoyancy

Density, often expressed as mass per unit volume, is a critical factor when it comes to buoyancy, which is the ability of an object to float in a fluid. Buoyancy is directly tied to Archimedes' principle, as it describes the upward force exerted on objects when they displace a fluid. If an object's density is less than that of the surrounding fluid, it will float; if it's more, it will sink.

In the context of our exercise, the boat floats because its overall density, including the rocks within it, goes below that of water once the water displacement is taken into account. However, when the rocks are removed from the boat, they sink due to their higher density relative to water. The displacement caused by the rocks alone, called V2, will be less than when they supported the boat's buoyancy because the displaced volume correlates with the object's weight, not its size. Hence, when the rocks are dropped into the pond and sink, the overall density considerations demonstrate why the water level would indeed fall.

Volume Displacement

Volume displacement is a key element in the study of buoyancy and is tightly woven with Archimedes' principle. It refers to the amount of fluid moved out of the way when an object is submerged. The displaced water volume has a direct relationship to the object's weight in the fluid. Therefore, volume displacement can illustrate how the water level will fluctaneously change in different scenarios, like in our exercise with the rocks and the boat.

Initially, the displacement V1 includes the weight of both the boat and rocks, representing a certain water level in the pond. After dropping the rocks into the pond, they displace a smaller volume V2 compared to V1 due to their higher density. Consequently, the boat alone now displaces an even lesser volume V3, because it weighs less without the rocks. When we combine V2 and V3, which represent the final state, this volume is less than the original V1, resulting in a lower water level. Hence, by understanding volume displacement, students can see why, when dense objects are removed from a floating vessel and sink to the bottom of a body of water, the overall water level declines.