Question

An object of density pₐ, volume Vₐ, and weight Wₐ is thrown from a rowboat floating on the surface of a small pond and sinks to the bottom. The weight of the rowboat without the jettisoned object is Wᵦ. Before the object was thrown out, the depth of the pond was hₚ₁ above the pond bottom. After the object sinks, the values of these quantities are hₚ₂ and hᵦ₂. The area of the pond is Aₚ; that of the boat is Aᵦ. Aᵦ may be assumed constant, so that the volume of water displaced by the boat is Aᵦ(hₚ-hᵦ).

Derive an expression for the change in the pond depth (hₚ₂ - hₚ₁). Does the liquid level of the pond rise or fall, or is it indeterminate?

Answer 1

The change in pond depth is determined by the difference in water displacement caused by the rowboat before and after the object is thrown. By applying Archimedes' principle, we can assess the impact of the object's weight on the pond's water level.

Step-by-step explanation:

To understand the change in the pond depth (hp2 - hp1) when an object of density pa and volume Va is thrown from a rowboat, we need to consider the principle of displacement.

Initially, the boat with mass (including the object) displaces a certain volume of water, Aβ(hp1 - hβ1), where Aβ is the area of the boat and hβ1 is the height of the boat's bottom above the pond bottom before the object was thrown.

After the object is thrown, the volume of water displaced changes to Aβ(hp2 - hβ2). The weight of the object in water is equal to the weight of the water it displaces (Archimedes' principle), which is the product of the density of water (pw), gravity (g), and the volume of water displaced by the object (Va).